- C. D'Apice, P. Kogut, R.Manzo, A.Parisi, Variational Model with Nonstandard Growth
Condition in Image Restoration and Contrast Enhancement,
Communications on Applied Mathematics and Computation, https://doi.org/10.1007/s42967-024-00382-1. [Show Abstract]
We propose a new variational model in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional and discuss its applications to the simultaneous contrast enhancement and denoising of color images. The characteristic feature of the proposed model is that we deal with a constrained non-convex minimization problem that lives in variable Sobolev-Orlicz spaces where the variable exponent is unknown a priori and it depends on a particular function that belongs to the domain of objective functional. In contrast to the standard approach, we do not apply any spatial regularization to the image gradient. We discuss the consistency of the variational model, give the scheme for its regularization, derive the corresponding optimality system, and propose an iterative algorithm for practical implementations.

- P. Kogut, Ya. Kohut, R. Manzo, Existence Result and Approximation of an Optimal Control Problem for the Perona-Malik Equation, Ricerche di Matematica, (73) (2024), 1945-1962, DOI 10.1007/s11587-022-00730-4. [Show Abstract]
We discuss the existence of solutions to an optimal control problem for the Cauchy-Neumann boundary value problem for the evolutionary Perona-Malik equations. The control variable v is taken as a distributed control. The optimal control problem is to minimize the discrepancy between a given distribution ud ∈ L2 (Ω) and the current system state. We deal with such case of non-linearity when we cannot expect to have a solution of the original boundary value problem for each admissible control. Instead of this we make use of a variant of its approximation using the model with fictitious control in coefficients of the principle elliptic operator. We introduce a special family of regularized optimization problems for linear parabolic equations and show that each of these problems is consistent, well-posed, and their solutions allow to attain (in the limit) an optimal solution of the original problem as the parameter of regularization tends to zero.

- C. D'Apice, P. Kogut, R.Manzo, C.Pipino, Variational Approach to Simultaneous Fusion and Denoising of Color Images with Different Spacial Resolution, Communications in Mathematical Sciences, (22) 4 (2024), pp. 1099–1132. [Show Abstract]
We propose a new variational model in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional and discuss its applications to the simultaneous fusion and denoising of color images with different spacial resolution. The characteristic feature of the proposed model is that we deal with a constrained minimization problem that lives in variable Sobolev-Orlicz spaces where the variable exponent, which is associated with non-standard growth, is unknown a priori and it depends on a particular function that belongs to the domain of objective functional. In view of this, we discuss the consistency of the proposed model, give the scheme for its regularization, derive the corresponding optimality system, and propose an iterative algorithm for practical implementations.

- N. Ivanchuk, P. Kogut, O. Kupenko, Generalized Active Contour Model in an Anisotropic Variable Exponent Sobolev Space, Journal of Optimization, Differential Equations and Their Applications (JODEA), 32 (1), 2024, 1–32.
**K_2024_1PDF**[Show Abstract]Mostly motivated by the practical applications especially in the field of satellite remote sensing of agricultural territories, we develop a novel approach to the domain decomposition basing on the anisotropic version of the Chan-Vese active contour model. With that in mind we propose a new statement of this problem in variable Sobolev spaces, provide a rigorous mathematical analysis of the proposed optimization problem, establish sufficient conditions of its consistency and solvability, show that the objective functional is Gateaux differentiable, and derive the corresponding optimality conditions. To illustrate the validity of the obtained results, we give some examples of numerical simulations with the real satellite images.

- V.L. Borsch, P.I. Kogut, On Kinds of Weak Solutions to an Initial Boundary Value Problem for 1D
Linear Degenerate Wave Equation, accepted for publication in Journal of Mathematical
Physics, Analysis, Geometry, 2024. [Show Abstract]
In this article, we discuss the existence and uniqueness of mild, variational, and the so-called non-variational solutions to an initial-boundary value problem (IBVP) for linear wave equation with strong interior degeneracy of the coefficient in the principal part of the elliptic operator. The objective is to provide a well-posedness analysis of the IBVP and find out how the density property of smooth functions in the corresponding weighted Sobolev space affects the uniqueness of its solutions. We show that, in general, the uniqueness of solutions may be violated if the 'degree of degeneracy' corresponds to the strong degeneracy case.

- S. Barbeiro, P. Csomós, M. Ehrhardt, J. Giesselmann,
M. Gugat, P. Kogut, T. Kunkel, I. Nakić, J. Tambača,
Z. Tomljanović, H. Yücel, E. Zuazua, Dynamics of and on networks:
Numerics, control, applications, COST Action 18232 Mathematical Models for Interacting Dynamics on Networks, WG 5: Numerical Methods and Applications, White Paper, 2024. [Show Abstract]
Many physical, biological, chemical, financial, and social phenomena can be described by dynamical systems and frequently the dynamics are coupled in a networked structure. The nature of these connections can be quite different, ranging from situations where connec tions express interactions between certain agents in multi-agent systems to situations where connections are gas pipes in gas transport networks. Very often the differential equations arising in applications have a rather complex structure so that analytical solution formulas are not available; in this case, numerical simulations are a natural tool to gain insights into the solution behavior. Working Group 5: Numerical Methods and Applications of COST Action 18232 brought together experts from different fields of mathematics in order to devise and analyse novel numerical schemes for networked dynamics. This took place in close cooperation with the more analysis-driven investigations in other working groups. Due to the different natures of the studied applications the resultant models displayed a variety of mathematical structures. Thus, different problem-adapted numerical schemes needed to be developed and different techniques needed to be used for their analysis. With an eye towards providing suitable tools for different applications the working group members applied methods from such diverse fields of mathematics as numerical analysis, control theory, semi-group theory, and homogenization. The following white paper showcases progress made by the members of the working group in the last years and relates it to the state of the art in the corresponding areas.

- P. Kogut, A Note on Some Properties of Unbounded Bilinear Forms Associated with
Skew-Symmetric $L^q(\Omega)$- Matrices, Researches in Mathematics, (32) (1) 2024, 83-92. [Show Abstract]
We study the bilinear forms on the space of measurable p-integrable functions which are generated by skew-symmetric matrices with unbounded coefficients. We give an example showing that if a skew- symmetric matrix contains a locally unbounded Lq -elements, then the corresponding quadratic forms can be alternating. These questions are closely related with the existence issues of the Nuemann boundary value problem for p-Laplace elliptic equations with non-symmetric and locally unbounded anisotropic diffusion matrices.

- N. Ivanchuk, P. Kogut,, On Active Contour Model in an Anisotropic
Variable Exponent Sobolev Space and its Application, XIII МІЖНАРОДНА НАУКОВО-ПРАКТИЧНА КОНФЕРЕНЦІЯ "МАТЕМАТИКА. ІНФОРМАЦІЙНІ ТЕХНОЛОГІЇ.ОСВІТА", ЛУЦЬК-СВІТЯЗЬ
31 травня – 2 червня 2024, ТЕЗИ ДОПОВІДЕЙ, 2024, 20-21. [Show Abstract]
We develop a novel approach to the domain decomposition basi- ng on the novel anisotropic version of the Chan-Vese active contour model for a special domain decomposition problem which is moti- vated by huge number of different applications especially the satellite image segmentation under the remote sensing of the Earth surface. In particular, in agriculture, one of fundamental problems is to provi- de a disjunctive decomposition of a fixed field Ω ⊂ R2 into finite number of nonempty subsets Ω = Ω1 ∪ Ω2 ∪ · · · ∪ ΩK such that each of these subsets could be associated with a distinctive value of some agricultural index (NDVI, PVI, LAI, or other ones). The information obtained in this way is an important source of knowledge that can be used for environmental resources management. In particular, it allows to find out how well the crops are doing and whether there are problematic zones inside of given fields or zones where the crops are in need of water or nutrients etc. However, the most restrictive obstacle in the construction of such decomposition is the fact that these subdomains should not overlap the borders between fields or contain any fragments of such borders. meaning that they cannot take in even small parts of different fields with arguably different crops. This makes the above mentioned segmentation problem rather challenging.

- C.D'Apice, P.I. Kogut, R. Manzo, M.V. Uvarov, Variational Model with Nonstandard Growth Conditions for Restoration of Satellite Optical Images via Their Co-Registration with Synthetic Aperture Radar, European Journal of Applied Mathematics, Volume 34, Issue 1, 2023, pp. 77–105, https://doi.org/10.1017/S0956792522000031 [Show Abstract]
In this paper the problem of inpainting of damaged optical images is studied in the case when we have no information about brightness of such images in the damage region. We propose a new variational approach for exact restoration of brightness for optical images using results of their co-registration with Synthetic Aperture Radar (SAR)\footnote[1]{EOS — Spatial Data Analytics, GIS Software, Satellite Imagery — is a cloud-based platform to derive remote sensing data and analyze satellite imagery for business and science purposes.} images of the same regions. We discuss the consistency of the proposed problem, give the scheme for its regularization, derive the corresponding optimality system, and describe in detail the algorithm for the practical implementation of the reconstruction procedure.

- V.L. Borsch, P.I. Kogut, How Can We Manage Repairing a Broken Finite
Vibrating String? Formulations of the Problem, Journal of Optimization, Differential Equations and Their Applications (JODEA), 31 (2), 2023, 88–113. https://model-dnu.dp.ua/index.php/SM/issue/view/JODEA_31_2_23 [Show Abstract]
Three approaches to solve a well-known IBVP posed for vibrating composite string with piece-wise constant properties have been applied. The main issue of the stufy is the number of the matching conditions to be imposed for the solution to the IBVP to be obtained.

- P. Kogut, Ya. Kohut, Optimal sparse control formulation for reconstruction of noise-affected images, Axioms 2023, 12(12), 1073, 22p. (This article belongs to the Special Issue Stability, Approximation, Control and Application); https://doi.org/10.3390/axioms12121073. [Show Abstract]
We discuss the optimal control formulation for enhancement and denoising of satellite multiband images and propose to take it in the form of an $L^1$-control problem for quasi-linear parabolic equation with non-local $p[u]$-Laplacian and with a cost functional of a tracking type. The main characteristic features of the considered parabolic problem is that the variable exponent $p(t,x)$ and the diffusion anisotropic tensor $D(t,x)$ are not well predefined a priori, but instead these characteristics non-locally depend on the form of the solution of this problem, i.e., $p_u=p(t,x,u)$ and $D_u=D(t,x,u)$. We prove the existence of optimal pairs with sparse $L^1$-controls using for that the indirect approach and a special family of approximation problems.

- C.D'Apice, P.I. Kogut, R. Manzo, Qualitative Analysis of an Optimal Sparse
Control Problem for Quasi-Linear Parabolic Equation with Variable Order of
Nonlinearity, Journal of Optimization, Differential Equations and Their Applications (JODEA), 31 (2), 2023, 125–173. https://model-dnu.dp.ua/index.php/SM/issue/view/JODEA_31_2_23
**JODEA_24_DKMPDF**[Show Abstract]In this work, we study a sparse optimal control problem involving a quasi- linear parabolic equation with variable order of nonlinearity as state equation and with a pointwise control constraints. We show that in the case of the cost functional with the terminal term of the tracking type, the proposed optimal control problem is ill-posed, in general. In view of this, we provide a sufficiently mild relaxation of the proposed problem and establish the existence of optimal solutions for the relaxed version. Using the perturbation technique and the concept of variational convergence of minimization problems, we study the attainability of optimal pairs to the relaxed problem by optimal solutions of the special approximating problems. We also discuss the optimality conditions for approximating problems and provide their substantiation.

- C.D'Apice, P.I. Kogut, O.Kupenko, R. Manzo, On Variational Problem with Nonstandard Growth
Functional and Its Applications to Image Processing, Journal of Mathematical Imaging and Vision, 65 (3), 2023, 472–491. https://doi.org/10.1007/s10851-022-01131-w. [Show Abstract]
We propose a new variational model in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional and discuss its applications to image processing. The characteristic feature of the proposed model is that the variable exponent, which is associated with non-standard growth, is unknown a priori and it depends on a particular function that belongs to the domain of objective functional. So, we deal with a constrained minimization problem that lives in variable Sobolev-Orlicz spaces. It makes this minimization problem rather challenging. In view of this, we discuss the consistency of the proposed model, give the scheme for its regularization, derive the corresponding optimality system, and propose the iterative algorithm for practical implementations. In particular, we apply this approach to the image denosing problem and compare the result of restoration with the Total Variation (TV) method.

- C.D'Apice, P.I. Kogut, R. Manzo, On Coupled Two-Level Variational Problem in Sobolev-Orlicz Space, Differential and Integral Equations, (36) (7-8) (2023), 621-660. [Show Abstract]
We study a coupled two-level variational model in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional and discuss its consistence and solvability issues. At the first level, we deal with the so-called temporal interpolation problem that can be cast as a state constrained optimal control problem for anisotropic convection-diffusion equation with two types of control functions — distributed $L^2$-control and $BV$-bounded control in coefficients. At the second level we have a constrained minimization problem with the nonstandard growth energy functional that lives in a variable Sobolev-Orlicz space. The characteristic feature of the proposed model is the fact that the variable exponent, which is associated with non-standard growth in the objective functional, is unknown a priori and it depends on the solution of the first-level optimal control problem.

- C.D'Apice, P.I. Kogut, R. Manzo, A two-level variational algorithm in the Sobolev-Orlicz space to predict daily surface reflectance at LANDSAT high spatial resolution and MODIS temporal frequency, Journal of Computational and Applied Mathematics, Id 115339, 2023, Available online 22 May 2023, https://www.sciencedirect.com/science/article/pii/S0377042723002832. [Show Abstract]
We propose a new two-level variational model in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional and discuss its applications to the spatiotemporal interpolation of multi-spectral satellite images. At the first level, we deal with the temporal interpolation problem that can be cast as a state constrained optimal control problem for anisotropic convection-diffusion equation, whereas at the second level we solve a constrained minimization problem with the nonstandard growth energy functional that lives in variable Sobolev-Orlicz spaces. The characteristic feature of the proposed model is the fact that the variable exponent, which is associated with non-standard growth in spatial interpolation problem, is unknown a priori and it depends on the solution of the first-level optimal control problem. It makes this spatiotemporal interpolation problem rather challenging. In view of this, we discuss the consistency of the proposed model, discuss the existence of optimal solutions, derive the corresponding optimality systems, and propose the scheme for their practical implementations. In particular, we apply this approach to the well-known prediction problem of the Daily MODIS Surface Reflectance at the Landsat-Like Resolution.

- N. Ivanchuk, P. Kogut, P. Martyniuk, Data fusion of satellite imagery for generation of daily cloud free images at high resolution level, arXiv:2302.12495 [math.OC]. [Show Abstract]
In this paper we discuss a new variational approach to the Date Fusion problem of multi-spectral satellite images from Sentinel-2 and MODIS that have been captured at different resolution level and, arguably, on different days. The crucial point of our approach that the MODIS image is cloud-free whereas the images from Sentinel-2 can be corrupted by clouds or noise.

- N. Ivanchuk, P. Kogut, P. Martyniuk, On generation of daily cloud free images at high resolution level, In: Zgurovsky, M., Pankratova, N. (eds) System Analysis and Artificial Intelligence . Studies in Computational Intelligence, vol 1107. Springer, Cham, 2023, 203-232. https://doi.org/10.1007/978-3-031-37450-0_13 2023.
- N. Ivanchuk, P. Kogut, P. Martyniuk,Sentinel-2 and MODIS data fusion for generation of daily cloud-free images at high resolution level, Proceedings of COLINS-2023: 7th International Conference on Computational Linguistics and Intelligent Systems, April 20–21, 2023, Kharkiv, Ukraine, 3403, pp. 87–98. [Show Abstract]
We discuss a new variational approach to the Date Fusion problem of multi-spectral satellite images from Sentinel-2 and MODIS that have been captured at different resolution level and, arguably, on different days. The crucial assumption to our approach is that the MODIS image has to be cloud-free whereas the images from Sentinel-2 can be corrupted by clouds or noise. We formulate the data fusion problem as the two-level optimization problem. We discuss the well thoroughness and consistency of the proposed variational model. We also derive some optimality conditions and supply our approach by results of numerical simulations with the real satellite images.

- V.L. Borsch, P.I. Kogut, Can a Finite Degenerate ‘String’ Hear Itself?
Numerical Solutions to a Simplified IBVP, Journal of Optimization, Differential Equations and Their Applications (JODEA), 31 (1) (2023), 95-110. [Show Abstract]
Some discrete models for a simplified (compared to that published earlier in JODEA, 28 (1) (2020), 1 – 42) initial boundary value problem for a 1D linear degenerate wave equation, posed in a space-time rectangle and solved earlier exactly (JODEA, 30 (1) (2022), 89 – 121), have been considered. It has been demonstrated that the correct evaluation of the degenerate grid flux can be possible.

- P. Kogut, Ya. Kohut, Optimal sparse control formulation for reconstruction of noise-affected images, Preprints 2023, 2023081846. https://doi.org/10.20944/preprints202308.1846.v1 [Show Abstract]
In this article, we discuss the optimal control formulation for enhancement and denoising of satellite multiband images and propose to take it in the form of an $L^1$-control problem for quasi-linear parabolic equation with non-local $p[u]$-Laplacian and with a cost functional of a tracking type. The characteristic feature of the considered class of Cauchy-Neumann parabolic problems is the fact that the variable exponent $p(t,x)$ and the anisotropic diffusion tensor $D(t,x)$ are not well predefined a priori, but instead these characteristic non-locally depend on a solution of this problem, i.e., $p=p(t,x,u)$ and $D=D(t,x,u)$. We prove the existence of optimal pairs with sparse controls using for that the indirect approach and a special family of approximation problems.

- V.L. Borsch, P.I. Kogut, On Non-Variational Solutions of the Intial-Boundary Value Problem for Linear Strongly Degenerate 1-D Hyperbolic Equation, Proceedings of INTERNATIONAL WORKSHOP ON CURRENT TRENDS IN ANALYSIS AND APPROXIMATION THEORY, 18th July, 2023, Rome, Italy, 22-25.
- C. D'Apice, P.I. Kogut, R. Manzo, M.V. Uvarov: On Variational Problem with Nonstandard Growth Conditions and Its Applications to Image Processing, Proceedings of the International Conference onNumerical Analysis and Applied Mathematics 2021 (ICNAAM-2021), Rodi, Grecia, 20-26 Settembre 2021, ISBN 978-0-7354-4589-5, DOI 10.1063/5.0162072, Vol. 2849, No. 1, 320004-1-320004-4, 2023
**DAKOMAUV2021SubmittedPDF**[Show Abstract]We propose a new variational model in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional and discuss its applications to image processing. The characteristic feature of the proposed model is that the variable exponent, which is associated with non-standard growth, is unknown a priori and it depends on a particular function that belongs to the domain of objective functional. So, we deal with a constrained minimization problem that lives in variable Sobolev-Orlicz spaces. Because of this the standard approaches are no longer applicable in its study, especially with respect to the existence of minimizers and the study of their basic properties. It makes this minimization problem rather challenging. In view of this, we discuss the consistency of the proposed model, give the scheme for its regularization, derive the corresponding optimality system, and propose the iterative algorithm for practical implementations. In particular, we apply this approach to the image denosing problem and show that in all prototypical examples we obtain better results when compared to the Total Variation (TV) method.

- P.I. Kogut, R. Manzo: Some Results on Optimal Control Problem for Ill-Posed Nonlinear Elliptic Equations, Proceedings of the International Conference on Numerical Analysis and Applied Mathematics 2021 (ICNAAM-2021), Rodi, Grecia, 20-26 Settembre 2021, ISBN 978-0-7354-4589-5, Vol. 2849, No. 1, DOI 10.1063/5.0162073, pp. 410005-1-410005-4, 2023;
**KOMA2021SubmittedPDF**[Show Abstract]We deal with an optimal control problem for or one class of non-linear elliptic equations with an exponential type ofnon-linearity. The boundary control is a density of surface traction acting on a part of boundary of an open domain. The aim is to minimize the discrepancy between a given distribution and the current system state. A special family of regularized optimization problems is introduced using a variant of the classical Tikhonov regularization and it is proven that each of these problem is consistent, well-posed, and their solutions allow to attain (in the limit) an optimal solution of the original problem as the parameter of regularization tends to zero. As a consequence, we prove the existence of optimal solutions to the original problem and propose a way for their approximation.

- P.I. Kogut, O.P. Kupenko, G. Leugering, On Boundary Exact Controllability of One-Dimensional Wave Equations with Weak and Strong Interior Degeneration, Mathematical Methods in the Applied Sciences., 45(2) 2022, pp. 770–792 [Show Abstract]
In this paper we study exact boundary controllability for a linear wave equation with strong and weak interior degeneration of the coefficient in the principle part of the elliptic operator. The objective is to provide a well-posedness analysis of the corresponding system and derive conditions for its controllability through boundary actions. Passing to a relaxed version of the original problem, we discuss existence and uniqueness of solutions, and using the HUM method we derive conditions on the rate of degeneracy for both exact boundary controllability and the lack thereof.

- P.I. Kogut, O.P. Kupenko, G. Leugering, Well-Posedness and Boundary Observability of Strongly Degenerate Hyperbolic Systems on Star-Shaped Planar Network, Pure and Applied Functional Analysis, 7 (5) 2022, 1767-1796. [Show Abstract]
In this article, we discuss well-posedness and boundary observability of the linear wave equation on a star-shaped planar network with degeneration in the leading coefficient related to the stiffness. In particular, we discuss the existence of weak and strong solutions and derive an observability inequality with the observability time depending on the degree of degeneration. In analogy with \cite{Cannarsa}, where one-sided damage of a single wave equation has been considered, we obtain observability for a class of weak damage and the lack of observability for strong damage. We use a representation of the metric graph by singular measures and work in corresponding weighted Sobolev spaces with respect to such measures.

- V.V. Hnatushenko, P.I. Kogut, M.V. Uvarov, Variational Approach for Rigid Co-Registration of Optical/SAR Satellite Images in Agricultural Areas, Journal of Computational and Applied Mathematics, 2022, Volume 400, 15 January 2022, Id 113742, 15p.doi: 10.1016/j.cam.2021.113742 [Show Abstract]
In this paper the problem of Synthetic Aperture Radar (SAR) and optical satel- lite images co-registration is considered. Because of the distinct natures of SAR and optical images, there exist huge radiometric and geometric differences be- tween such images. As a result, the traditional registration approaches are no longer applicable in this case and it makes the registration process challenging. Mostly motivated by the crop field monitoring problem, we propose a new vari- ational approach to the co-registration of SAR and optical images. We discuss the consistency of the proposed statement of this problem, propose the scheme for its regularization, derive the corresponding optimality system, and describe in detail the algorithm for the practical implementation of co-registration pro- cedure. To evaluate the performance of the proposed approach, we illustrate its crucial steps by numerous applications.

- P. Kogut, Ya. Kohut, R. Manzo, Fictitious Controls and Approximation of an Optimal Control Problem for Perona-Malik Equation, Journal of Optimization, Differential Equations and Their Applications (JODEA), 30 (1) (2022), 42-70. [Show Abstract]
We discuss the existence of solutions to an optimal control problem for the Cauchy-Neumann boundary value problem for the evolutionary Perona-Malik equations. The control variable $v$ is taken as a distributed control. The optimal control problem is to minimize the discrepancy between a given distribution $u_d\in L^2(\Omega)$ and the current system state. We deal with such case of non-linearity when we cannot expect to have a solution of the original boundary value problem for each admissible control. Instead of this we make use of a variant of its approximation using the model with fictitious control in coefficients of the principle elliptic operator. We introduce a special family of regularized optimization problems for linear parabolic equations and show that each of these problems is consistent, well-posed, and their solutions allow to attain (in the limit) an optimal solution of the original problem as the parameter of regularization tends to zero.

- P. Kogut, Ya.Kohut, R. Manzo, Some Results on Optimal Control Problem for Perona-Malik
Equation, Proceedings of the 20th International Conference of Numerical Analysis and Applied Mathematics, ICNAAM 2022, 19–25 September 2022,Location: Rhodes, Greece.
**ICNAAM2022.pdfPDF**[Show Abstract]We discuss the existence of solutions to an optimal control problem for the Cauchy-Neumann boundary value problem for the evolutionary Perona-Malik equations. The control variable $v$ is taken as a distributed control. The optimal control problem is to minimize the discrepancy between a given distribution $u_d\in L^2(\Omega)$ and the current system state. We deal with such case of non-linearity when we cannot expect to have a solution of the original boundary value problem for each admissible control. Instead of this we make use of a variant of its approximation using the model with fictitious control in coefficients of the principle elliptic operator. We introduce a special family of regularized optimization problems for linear parabolic equations and show that each of these problems is consistent, well-posed, and their solutions allow to attain (in the limit) an optimal solution of the original problem as the parameter of regularization tends to zero.

- P. Kogut, O. Kupenko, R. Manzo, C. Pipino On Variational Model in Sobolev-Orlicz Spaces for
Spatiotemporal Interpolation of Multi-Spectral
Satellite Images, Proceedings of the IEEE International Conference on System Analysis & Intelligent Computing (SAIC) 04-07 October, 2022,Location: Kyiv, Ukraine, 134-139. 10.1109/SAIC57818.2022.9922923 [Show Abstract]
We propose a new two-level variational model in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional and discuss its applications to the spatiotemporal interpolation of multi-spectral satellite images. At the first level, we deal with the temporal interpolation problem that can be cast as a state constrained optimal control problem for anisotropic convection-diffusion equation, whereas at the second level we solve a constrained minimization problem with a nonstandard growth energy functional that lives in variable Sobolev-Orlicz spaces. The characteristic feature of the proposed model is the fact that the variable exponent, which is associated with non-standard growth in spatial interpolation problem, is unknown a priori and it depends on the solution of the first-level optimal control problem. It makes this spatiotemporal interpolation problem rather challenging. In view of this, we discuss the consistency of the proposed model, study the existence of optimal solutions, and derive the corresponding optimality systems. In particular, we apply this approach to the well-known prediction problem of the Daily MODIS Surface Reflectance at the Landsat-Like Resolution.

- P. Kogut, Ya. Kohut, N. Parfinovych, Solvability Issues for Some Noncoercive and Nonmonotone Parabolic\\ Equations Arising in the Image Denoising Problems, Journal of Optimization, Differential Equations and Their Applications (JODEA), 30 (2) (2022), 19-48. [Show Abstract]
This paper is devoted to the solvability of an initial-boundary value problem for second-order parabolic equations in divergence form with variable order of nonlinearity. The characteristic feature of the considered class of Cauchy-Neumann parabolic problem is the fact that the variable exponent $p(t,x)$ and the anisotropic diffusion tensor $D(t,x)$ are not well predefined a priori, but instead these characteristic depend on a solution of this problem, i.e., $p=p(t,x,u)$ and $D=D(t,x,u)$. Recently, it has been shown that the similar models appear in a natural way as the optimality conditions for some variational problems related to the image restoration technique. However, from practical point of view, in this case some principle difficulties can appear because of the absence of the corresponding rigorous mathematically substantiation. Thus, in this paper, we study the solvability issues of the Cauchy-Neumann parabolic boundary value problem for which the corresponding principle operator is strongly non-linear, non-monotone, has a variable order of nonlinearity, and satisfies a nonstandard coercivity and boundedness conditions that do not fall within the scope of the classical method of monotone operators. To construct a weak solution, we apply the technique of passing to the limit in a special approximation scheme and the Schauder fixed-point theorem.

- V.L. Borsch, P.I. Kogut, Can a finite degenerate string `hear' itself? Exact solutions to a simplified IBVP, Journal of Optimization, Differential Equations and Their Applications (JODEA), 30 (2) (2022), 89-121. [Show Abstract]
The separation of variables based solution to a simplified (compared to that published earlier in JODEA, \textbf{28} (1) (2020), 1\,–\,42) \ibvp{} for a 1D linear degenerate wave equation, posed in a space-time rectangle, has been presented in a fully complete form. Degeneracy of the equation is due to vanishing its coefficient in an interior point of the spatial segment being the side of the rectangle. For the sake of convenience, the solution is interpreted as a vibrating `string'. The solution obtained in the case of weak degeneracy is smooth and bounded, whereas that in the case of strong degeneracy is piece-wise smooth and unbounded in a neighborhood of the point of degeneracy, nevertheless being satisfied some regularity conditions, including square-integrability. In both cases the travelling waves pass through the point of degeneracy, and this phenomenon is referred to as an ability of the `string' to hear itself. The total energy of the `string' is shown to conserve in both cases of degeneracy, provided the ends of the `string' are fixed, though the above vibrating `string' analogy fails in the case of strong degeneracy. The total energy conservation implies the uniqueness of the solution to the problem in both cases of degeneracy.

- C. D'Apice, U. De Maio, P.I. Kogut, Optimal Boundary Control Problem for Ill-Posed Elliptic Equation in Domains with Rugous Boundary. I. Existence Result and Optimality Conditions, Journal of Optimal Control: Applications and Methods, Vol. 42, Issue 1, 2021, 30-53. DOI:10.1002/oca.2660. [Show Abstract]
The main purpose is the study of optimal control problem in a domain with rough boundary for the mixed Dirichlet-Neumann boundary value problem for the strongly non-linear elliptic equation with exponential nonlinearity. A density of surface traction $u$ acting on a part of rough boundary is taken as a control. The optimal control problem is to minimize the discrepancy between a given distribution and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for a given control. After having defined a suitable functional class in which we look for solutions, we prove the consistency of the original optimal control problem and show that it admits a unique optimal solution. Then we derive a first order optimality system assuming the optimal solution is slightly more regular.

- V.V. Hnatushenko, P.I. Kogut, M.V. Uvarov, On Satellite Image Segmentation via Piecewise Constant Approximation of Selective Smoothed Target Mapping, Applied Mathematics and Computation, Vol.389, 2021, Id 125615, 26p, doi.org/10.1016/j.amc.2020.125615 [Show Abstract]
Mostly motivated by the crop field classification problem and the automated computational methodology for the extraction of agricultural crop fields from the satellite data, we propose a new technique for the satellite image segmentation which is based on the concept of a piecewise constant approximation of the slope-based vegetation indices. We discuss in details the consistency of the new statement of segmentation problem and its solvability. We focus our main intension on the rigor mathematical substantiation of the proposed approach, deriving the corresponding optimality conditions, and show that the new optimization problem is rather flexible and powerful model to the study of variational image segmentation problems. We illustrate the accuracy and efficiency of the proposed algorithm by numerical experiences with images that have been delivered by satellite Sentinel-2.

- V.V. Hnatushenko, P.I. Kogut, M.V. Uvarov, On Flexible Co-Registration of Optical and SAR Satellite Images, in "Lecture Notes in Computational Intelligence and Decision Making" (series "Advances in Intelligent Systems and Computing", Editors: Babichev, S., Lytvynenko, V., Wójcik, W., Vyshemyrskaya, S. (Eds.) ), Springer, 2021, 515–534. [Show Abstract]
We consider the problem of Synthetic Aperture Radar (SAR) and optical satellite images co-registration in its special statement. Because of the distinct natures of SAR and optical images, there exist huge radiometric and geometric differences between such images. Their structure, intensity of pixels, and texture are drastically different. As a result, the traditional registration approaches are no longer applicable in this case and it makes the registration process challenging. Mostly motivated by the crop field monitoring problem, we propose a new statement for the co-registration of SAR and optical images. We discuss the consistency of the proposed statement of this problem, propose the scheme for its regularization, and derive the corresponding optimality system.

- V. L. Borsch, P.I. Kogut, Solutions to a Simplified Initial Boundary Value
Problem for 1D Hyperbolic Equation with Interior Degeneracy, Journal of Optimization, Differential Equations and Their Applications (JODEA), 29 (1) (2021), 1-31. [Show Abstract]
A 1-parameter initial boundary value problem (IBVP) for a linear homogeneous degenerate wave equation (JODEA, 28(1), 1\,–\,42) in a space-time rectangle is considered. The origin of degeneracy is the power law coefficient function with respect to the spatial distance to the symmetry line of the rectangle, the exponent being the only parameter of the problem, ranging in (0,1) and (1,2) and producing the weak and strong degeneracy respectively. In the case of weak degeneracy separation of variables is used in the rectangle to obtain the unique bounded continuous solution to the IBVP, having the continuous flux. In the case of strong degeneracy the IBVP splits into the two derived IBVPs posed respectively in left and right half-rectangles and solved separately using separation of variables. Continuous matching of the obtained left and right families of bounded solutions to the IBVPs results in a linear integro-differential equation of convolution type. The Laplace transformation is used to solve the equation and obtain a family of bounded solutions to the IBVP, having the continuous flux and depending on one undetermined function.

- P.I. Kogut, O.P. Kupenko, N. Uvarov, On Increasing of Resolution of Satellite Images via Their Fusion with Imagery at Higher Resolution, Journal of Optimization, Differential Equations and Their Applications (JODEA), 29 (1) (2021), 54-78. [Show Abstract]
In this paper we propose a new statement of the spatial increasing resolution problem of MODIS-like multi-spectral images via their fusion with Lansat-like imagery at higher resolution. We give a precise definition of the solution to the indicated problem, postulate assumptions that we impose at the initial data, establish existence and uniqueness result, and derive the corresponding necessary optimality conditions. For illustration, we supply the proposed approach by results of numerical simulations with real-life satellite images.

- P.I. Kogut, O.P. Kupenko, G. Leugering, On Boundary Null Contrtollability of Strongly Degenerate Hyperbolic Systems on Star-Shaped Planar Networks, Journal of Optimization, Differential Equations and Their Applications (JODEA), 29 (2) (2021), 92-118. [Show Abstract]
In this paper we discuss the problem of boundary exact null controllability for weakly and strongly degenerate linear wave equation defined on star-shaped planar network. The network is represented by a singular measure in a bounded planar domain. The novelty of this article lies in the degeneration of the leading coefficient representative of the material properties at the common node of network. We discuss the existence of weak and strong solutions to the degenerate hyperbolic problem and establish the corresponding controllability properties.

- P. Khanenko, P. Kogut, M. Uvarov, On Variational Model for the Restoration of
Background Surface in Clouds Corrupted Satellite Optical Images, Proceedings of the 11th IEEE International Conference on Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications, IDAACS 2021, September 22-25, 2021, Cracow, Poland, Vol.1, 2021, 536–541. [Show Abstract]
Sensitivity to weather conditions, and specially to clouds, is a severe limiting factor to the use of optical remote sensing for Earth monitoring applications. Typically, the optical satellite images are often corrupted because of poor weather conditions. As a rule, the measure of degradation of optical images is such that one can not rely even on the brightness inside of the damaged regions. As a result, some subdomains of such images become absolutely invisible. So, there is a great deal of missing information in optical remote sensing data. In this work, we propose a new variational approach for exact restoration of multi- spectral satellite optical images. We discuss the consistency of the proposed variational model, give the scheme for its regularization, derive the corresponding optimality system, and discuss the algorithm for the practical implementation of the reconstruction procedure. Experimental results are very promising and they show a significant gain over baseline methods using the reconstruction through linear interpolation between data available at temporally-close time instants.

- P. Khanenko, P. Kogut, M. Uvarov, On Variational Problem with Nonstandard Growth Conditions for the Restoration of Clouds Corrupted Satellite Images, CEUR Workshop Proceedings, the 2nd International Workshop on Computational and Information Technologies for Risk-Informed Systems, CITRisk–2021, September 16-17, 2021, Kherson, Ukraine, Volume 3101, 6–25, 2021. [Show Abstract]
Sensitivity to weather conditions, and specially to clouds, is a severe limiting factor to the use of optical remote sensing for Earth monitoring applications. Typically, the optical satellite images are often corrupted because of poor weather conditions. As a rule, the measure of degradation of optical images is such that one can not rely even on the brightness inside of the damaged regions. As a result, some subdomains of such images become absolutely invisible. So, there is a great deal of missing information in optical remote sensing data. In this work, we propose a new variational approach for exact restoration of multispectral satellite optical images. We discuss the consistency of the proposed variational model, give the scheme for its regularization, derive the corresponding optimality system, and discuss the algorithm for the practical implementation of the reconstruction procedure. Experimental results are very promising and they show a significant gain over baseline methods using the reconstruction through linear interpolation between data available at temporally-close time instants.

- C. D'Apice, U. De Maio, P.I. Kogut, An Indirect Approach to the Existence of Quasi-Optimal Controls in Coefficients for Multi-Dimensional Thermistor Problem, in "Contemporary Approaches and Methods in Fundamental Mathematics and Mechanics", Editors: Sadovnichiy, Victor A., Zgurovsky, Michael (Eds.), Chapter 24, Springer, 2020, 489-522. doi:10.1007/978-3-030-50302-4_24. [Show Abstract]
The paper studies a problem of an optimal control in coefficients for the system of two coupled elliptic equations also known as thermistor problem which provides a simultaneous description of the electric field $u=u(x)$ and temperature $\theta(x)$. The coefficients of operator $\mathrm{div}\,\left(A(x)\, \nabla \, \theta(x)\right)$ are used as the controls in $L^\infty(\Omega)$. The optimal control problem is to minimize the discrepancy between a given distribution $\theta_d\in L^r(\Omega)$ and the temperature of thermistor $\theta\in W^{1,\gamma}_0(\Omega)$ by choosing an appropriate anisotropic heat conductivity matrix $B$. Basing on the perturbation theory of extremal problems and the concept of fictitious controls, we propose an \textquotedblleft approximation approach\textquotedblright\ and discuss the existence of the so-called quasi-optimal and optimal solutions to the given problem.

- U. De Maio, P.I. Kogut, G. Zecca, On Optimal $L^1$-Control in Coefficients for Quasi-Linear Dirichlet Elliptic Boundary Value Problem with $BMO$-Anisotropic $p$-Laplacian, Mathematical Control and Related Fields, 10(4), 2020, 827-854. doi: 10.3934/mcrf.2020021. [Show Abstract]
We study an optimal control problem for a quasi-linear elliptic equation with anisotropic p-Laplace operator in its principle part and $L^1$-controls in coefficient of the low-order term. As characteristic feature of such optimal control problem is a specification of the matrix of anisotropy in BMO-space. Since we cannot expect to have a solution of the state equation in the classical Sobolev space, we specify a suitable functional class in which we look for solutions and prove existence of optimal pairs using the approximation procedure and compactness arguments in variable spaces.

- U. De Maio, P.I. Kogut, R. Manzo, Asymptotic Analysis of an Optimal Boundary Control Problem for Ill-Posed Elliptic Equation in Domains with Rugous Boundary, Asymptotic Analysis, 118(3), pp. 209-234, 2020, doi:10.3233/ASY-191570. [Show Abstract]
We study an optimal control problem for the mixed Dirichlet-Neumann boundary value problem for the strongly non-linear elliptic equation with exponential nonlinearity in a domain with rugous boundary. A density of surface traction $u$ acting on a part of rugous boundary is taken as a control. The optimal control problem is to minimize the discrepancy between a given distribution and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for a given control. After having defined a suitable class of the weak solutions, we provide asymptotic analysis of the above mentioned optimal control problem posed in a family of perturbed domains and give the characterization of the limiting behavior of its optimal solutions.

- P.I. Kogut, O.P. Kupenko, G. Leugering, On Boundary Exact Controllability of Weakly and Strongly Degenerate Wave Equations, Authorea. July 04, 2020.
DOI: 10.22541/au.159386605.59045806. [Show Abstract]
In this paper we study the exact boundary controllability issues for a linear hyperbolic equation with strongly and weakly degenerate coefficients in the principle part of the elliptic operator. The objective is to provide the well-posedness analysis of the corresponding controlled system and derive conditions of its conrollability through boundary actions. Passing to a relaxed version of the original problem, we discuss existence and uniqueness of its solutions, and using the HUM method we derive conditions on the rate of degeneracy when the exact boundary controllability of the corresponding degenerate control problem takes place.

- P.I. Kogut, O.P. Kupenko, R. Manzo, On Regularization of Optimal Control Problem for an Ill-Posed Strongly Nonlinear Elliptic Equation with an Exponential Type of Non-Linearity, Abstract and Applied Analysis, Volume 2020 |Article ID 7418707, 2020, 20p. https://doi.org/10.1155/2020/7418707 [Show Abstract]
We discuss the existence of solutions to an optimal control problem for one class of non-linear elliptic equations with an exponential type of nonlinearity. We deal with such case of nonlinearity when we cannot expect to have a solution of the Dirichlet boundary value problem in the standard functional space for all admissible controls. Instead of this we make use of a variant of the classical Tikhonov regularization in optimal control under PDEs which allows for a certain flexibility in dealing with non-linearities in the state equation. In particular, we eliminate the differential constraints between control and state and allow such pairs run freely in their respective sets of feasibility by introducing some additional variable which plays the role of "defect" that appears in the original state equation. We show that this special residual function can be determined in a unique way. In order to stay not too far from the initial optimal control problem, we introduce a special family of regularized optimization problems. We show that each of these problem is consistent, well-posed, and their solutions allow to attain (in the limit) an optimal solution of the original problem as the parameter of regularization tends to zero. As a consequence, we establish sufficient conditions of the existence of optimal solutions to the given class of nonlinear Dirichlet BVP and propose the way for their approximation.

- P.I. Kogut, O.P. Kupenko On Tikhonov Regularization of Optimal Distributed Control Problem for an Ill-Posed Elliptic Equation with $p$-Laplace Operator and $L^1$-type of Non-Linearity, in "Contemporary Approaches and Methods in Fundamental Mathematics and Mechanics", Editors: Sadovnichiy, Victor A., Zgurovsky, Michael (Eds.), Chapter 6, Springer, 2020, 91-111. doi:10.1007/978-3-030-50302-4_6. [Show Abstract]
We discuss the existence of solutions to an optimal control problem for the Dirichlet boundary value problem for strongly non-linear p-Laplace equations with $L_1$-type of nonlinearity. The control variable u is taken as a distributed control. The optimal control problem is to minimize the discrepancy between a given distribution $y_d\in L^p(\Omega)$ and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the original boundary value problem for each admissible control. Instead of this we make use of a variant of the classical Tikhonov regularization. We eliminate the differential constraints between control and state and allow such pairs run freely in their respective sets of feasibility by introducing some additional variable which plays the role of \textquotedblleft defect\textquotedblright. We show that this special residual function can be determined in a unique way. We introduce a special family of regularized optimization problems and show that each of these problem is consistent, well-posed, and their solutions allow to attain (in the limit) an optimal solution of the original problem as the parameter of regularization tends to zero. As a consequence, we establish sufficient conditions of the existence of optimal solutions to the given class of nonlinear Dirichlet BVP and propose the way for their approximation.

- V.L. Borsch, P.I. Kogut, G. Leugering, On an Initial Boundary-Value Problem for 1D Hyperbolic Equation with Interior Degeneracy: Series Solutions with the Continuously Differentiable Fluxes, J. of Optimization, Differential Equations and Their Applications (JODEA), Vol.28, Issue 1, 2020, 1-42.
**BKL2020PDF** - P.I. Kogut, M.V. Uvarov, Variational Approach for the Reconstruction of Damaged Optical Satellite Images Through Their Co-Registration with Synthetic Aperture Radar, J. of Optimization, Differential Equations and Their Applications (JODEA), Vol.28, Issue 2, 2020, 47-79. doi:10.15421/142004 [Show Abstract]
In this paper the problem of reconstruction of damaged multi-band optical images is studied in the case where we have no information about brightness of such images in the damage region. Mostly motivated by the crop field monitoring problem, we propose a new variational approach for exact reconstruction of damaged multi-band images using results of their co-registration with Synthetic Aperture Radar (SAR) images of the same regions. We discuss the consistency of the proposed problem, give the scheme for its regularization, derive the corresponding optimality system, and describe in detail the algorithm for the practical implementation of the reconstruction procedure.

- V.L. Borsch, P.I. Kogut, The Exact Bounded Solution to an Initial Boundary
Value Problem for 1D Hyperbolic Equation with Interior Degeneracy. I. Separation
of Variables, J. of Optimization, Differential Equations and Their Applications (JODEA), Vol.28, Issue 2, 2020, 2-20. [Show Abstract]
A 1-parameter initial boundary value problem for the linear homogeneous degenerate wave equation in the space-time rectangle with vanished coefficient in an interior point is considered. The IBVP is splitted into three auxiliary IBVPs, involving two undetermined functions h_i(t,\alpha). The auxiliary IBVPs are solved using the method of separation of variables. The matching conditions to gain continuity of the solution to the IBVP and its flux are imposed on the solutions to the auxiliary IBVPs to derive a linear convolution integro-differential system with respect to the unknown functions h_i(t,\alpha), i=1,2.

- P.I. Kogut, O.P. Kupenko, On Optimal Control Problem for an Ill-Posed Strongly Nonlinear Elliptic Equation with $p$-Laplace Operator and $L^1$-Type of Nonlinearity , Discrete and Continuous Dynamical Systems, Series B, Vol.24, Issue 3, 2019, 1273–1295. [Show Abstract]
We study an optimal control problem for one class of non-linear elliptic equations with $p$-Laplace operator and $L^1$-nonlinearity. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for any given control. After defining a suitable functional class in which we look for solutions, we reformulate the original problem and prove the existence of optimal pairs. In order to ensure the validity of such reformulation, we provide its substantiation using a special family of fictitious optimal control problems. The idea to involve the fictitious optimization problems was mainly inspired by the brilliant book of V.S. Mel'nik and V.I. Ivanenko "Variational Methods in Control Problems for the Systems with Distributed Parameters"\ , Kyiv, 1998.

- P.I.Kogut, O.P. Kupenko, On Approximation of an Optimal Control Problem for Ill-Posed Strongly Nonlinear Elliptic Equation with $p$-Laplace Operator, in "Modern Mathematics and Mechanics.
Fundamentals, Problems and Challenges", Editors: Sadovnichiy, Victor A., Zgurovsky, Michael (Eds.), Springer, 2019, Chapter 23, 445-480. [Show Abstract]
We study an optimal control problem for one class of non-linear elliptic equations with $p$-Laplace operator and $L^1$-nonlinearity in their right-hand side. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for any given control. After defining a suitable functional class in which we look for solutions and a special cost functional, we prove the existence of optimal pairs. In order to handle the inherent degeneracy of the p-Laplacian and strong non-linearity in the right-hand side of elliptic equation, we use a two-parametric $(\e,k)$-regularization of $p$-Laplace operator, where we approximate it by a bounded monotone operator, and involve a special fictitious optimization problem. We derive existence of optimal solutions to the parametrized optimization problems at each $(\e,k)$-level of approximation. We also deduce the differentiability of the state for approximating problem with respect to the controls and obtain an optimality system based on the Lagrange principle. Further we discuss the asymptotic behaviour of the optimal solutions to regularized problems as the parameters $\e$ and $k$ tend to zero and infinity, respectively.

- I. Belmas, P. Kogut, D. Kolosov, V. Samusia, S. Onyshchenko, Rigidity of elastic shell of rubber-cable belt during displacement of cables relatively to drum , E3S Web Conf. , Vol.109, Article number 005, 2019, 1–14. [Show Abstract]
Analytical dependencies for determining the rigidity and parameters of a stress-strain state of a rubber layer located between the cables and the drum from their mutual shear are established in a closed form. A method of determining the dispersion of deviation of calculated displacements from established values of elastic shell material is developed. Obtained results allow considering the shape of an elastic layer of rubber, estimate the level of reliability of the results when calculating a stress-strain state of rubber-cable tractive elements from the shear of cables relatively to the drum. Established dependencies can be used for determining a stress-strain state of a layer of elastic material, caused by shear of cables relatively to the drum of a machine, and formulating the condition of strength of a rubber-cable tractive element loaded with shear forces on a drum. Possibility of determination of a lower boundary of the efficiency coefficient in a process of interaction of a rubber-cable rope (belt) with pulleys and drums, both driving and driven, lined with elastic materials and with hard operating surfaces, allows improving the accuracy of calculations of drives of lifting and transporting machines with such executing elements.

- P.I. Kogut, On Optimal and Quasi-Optimal Controls in Coefficients for Multi-Dimensional Thermistor Problem with Mixed Dirichlet-Neumann Boundary Conditions , Control and Cybernetics, Vol. 48, Issue 1, 2019, 31-68. [Show Abstract]
In this paper we deal with an optimal control problem in coefficients for the system of two coupled elliptic equations also known as thermistor problem which provides a simultaneous description of the electric field $u=u(x)$ and temperature $\theta(x)$. The coefficients of operator $\mathrm{div}\,\left(B(x)\, \nabla \, \theta(x)\right)$ are used as the controls in $L^\infty(\Omega)$. The optimal control problem is to minimize the discrepancy between a given distribution $\theta_d\in L^r(\Omega)$ and the temperature of thermistor $\theta\in W^{1,\gamma}_0(\Omega)$ by choosing an appropriate anisotropic heat conductivity matrix $B$. Basing on the perturbation theory of extremal problems and the concept of fictitious controls, we propose an \textquotedblleft approximation approach\textquotedblright\ and discuss the existence of the so-called quasi-optimal and optimal solutions to the given problem.

- A.V. Sladkowski, Y. O. Kyrychenko, P. Kogut, V. Samusia, D. Kolosov, Innovative designs of pumping deep-water hydrolifts based on progressive multiphase non-equilibrium models , Naukovyi Visnyk NHU (Dnipro, Ukraine), Number 2, 2019, 51–57. [Show Abstract]
Development of methodological support for a unique method of a pumping deep-water hydrolift (PDH) for transporting multiphase flows with a solid fraction and innovative pumping systems for its implementation in complex conditions of great depths.

- P.I. Kogut, O.P. Kupenko, G. Leugering, Y. Wang, A Note on Weighted Sobolev Spaces Related to Weakly and Strongly Degenerate Differential Operators, J. of Optimization, Differential Equations and Their Applications (JODEA), Vol.27, Issue 2, 2019, 1-22. [Show Abstract]
In this paper we discuss some issues related to Poincar\'{e}'s inequality for a special class of weighted Sobolev spaces. A common feature of these spaces is that they can be naturally associated with differential operators with variable diffusion coefficients that are not uniformly elliptic. We give a classification of these spaces in the $1$-D case bases on a measure of degeneracy of the corresponding weight coefficient and study their key properties.

- V.V. Hnatushenko, P.I. Kogut, M.V. Uvarov, On Optimal $\mathbf{2}$-D Domain Segmentation Problem via Piecewise Smooth Approximation of Selective Target Mappings, J. of Optimization, Differential Equations and Their Applications (JODEA), Vol.27, Issue 2, 2019, 39-74. [Show Abstract]
In this paper we propose a new technique for the solution of the image segmentation problem which is based on the concept of a piecewise smooth approximation of some target functional. We discuss in details the consistency of the new statement of segmentation problem and its solvability. We focus our main intension on the rigor mathematical substantiation of the proposed approach, deriving the corresponding optimality conditions, and show that the new optimization problem is rather flexible and powerful model to the study of variational image segmentation problems. We illustrate the accuracy and efficiency of the proposed algorithm by numerical experiences.

- C. D'Apice, U. De Maio, P.I. Kogut, Thermistor Problem: Multi-Dimensional Modelling, Optimization, and Approximation, 32nd EUROPEAN CONFERENCE ON MODELLING AND SIMULATION, May 22nd - May 25th, 2018, Wilhelmshaven, Germany, DOI:https://doi.org/10.7148/2018-0348.
**CUP_2018bPDF**[Show Abstract]We consider a problem of an optimal control in coefficients for the system of two coupled elliptic equations also known as thermistor problem which provides a simultaneous description of the electric field $u=u(x)$ and temperature $\theta(x)$. The coefficients of operator $\mathrm{div}\,\left(A(x)\, \nabla \, \theta(x)\right)$ are used as the controls in $L^\infty(\Omega)$. The optimal control problem is to minimize the discrepancy between a given distribution $\theta_d\in L^r(\Omega)$ and the temperature of thermistor $\theta\in W^{1,\gamma}_0(\Omega)$ by choosing an appropriate anisotropic heat conductivity matrix $B$. Basing on the perturbation theory of extremal problems and the concept of fictitious controls, we propose an \textquotedblleft approximation approach\textquotedblright\ and discuss the existence of the so-called quasi-optimal and optimal solutions to the given problem.

- C. D'Apice, U. De Maio, P.I. Kogut, On Optimal Control of Quasi-Linear Elliptic Equation with Variable $p(x)$-Laplacian, The 2018 International Conference of Applied and Engineering Mathematics, London, U.K., 4-6 July, 2018.
**CUP_2018aPDF**[Show Abstract]We consider an optimal control problem for quasi-linear elliptic equation containing the $p$-Laplacian with variable exponent $p=p(x)$. The exponent $p(x)$ are used as the controls in $L^1(\Omega)$. The optimal control problem is to minimize the discrepancy between a given distribution $y_d\in L^\alpha(\Omega)$ and the current system state $y\in W^{1,p(\cdot)}_0(\Omega)$ by choosing an appropriate exponent $p(x)$. Basing on the perturbation theory of extremal problems, we study the existence of optimal pairs and propose the ways for relaxation of the original optimization problem.

- P.I.Kogut, O.P. Kupenko, On Approximation of State-Constrained Optimal Control Problem in Coefficients for $p$-Biharmonic Equation, J. of Optimization, Differential Equations and Their Applications (JODEA), Vol.26, Issue 1, 2018, 45-71. [Show Abstract]
We study a Dirichlet-Navier optimal design problem for a quasi-linear monotone $p$-biharmonic equation with control and state constraints. The coefficient of the $p$-biharmonic operator we take as a design variable in $BV(\Omega)\cap L^\infty(\Omega)$. In order to handle the inherent degeneracy of the p-Laplacian and the pointwise state constraints, we use regularization and relaxation approaches. We derive existence and uniqueness of solutions to the underlying boundary value problem and the optimal control problem. In fact, we introduce a two-parameter model for the weighted $p$-biharmonic operator and Henig approximation of the ordering cone. Further we discuss the asymptotic behaviour of the solutions to regularized problem on each $(\e,k)$-level as the parameters tend to zero and infinity, respectively.

- C. D'Apice, M.-P. D'Arienzo, P.I. Kogut, R. Manzo, On Boundary Optimal Control Problem for an Arterial System: Existence of Feasible Solutions, Journal of Evolution Equations, 2018, Vol.18, Issue 4, 1745–1786. [Show Abstract]
We discuss a control constrained boundary optimal control problem for the Boussinesq-type system arising in the study of the dynamics of an arterial network. We suppose that a control object is described by an initial-boundary value problem for $1D$ system of pseudo-parabolic nonlinear equations with an unbounded coefficient in the principal part and Robin boundary conditions. The main question we discuss in this part of paper is about topological and algebraical properties of the set of feasible solutions. Following Faedo-Galerkin method, we establish the existence of weak solutions to the corresponding initial-boundary value problem and show that these solutions possess some special extra regularity properties which play a crucial role in the proof of solvability of the original optimal control problem.

- P.I.Kogut, R. Manzo, M.V. Poliakov, On Existence of Bounded Feasible
Solutions to Neumann Boundary Control Problem for p-Laplace Equation with
Exponential Type of Nonlinearity, J. of Optimization, Differential Equations and Their Applications (JODEA), Vol.26, Issue 1, 2018, 8-28. [Show Abstract]
We study an optimal control problem for mixed Dirichlet-Neumann boundary value problem for the strongly non-linear elliptic equation with $p$-Laplace operator and $L^1$-nonlinearity in its right-hand side. A distribution $u$ acting on a part of boundary of open domain is taken as a boundary control. The optimal control problem is to minimize the discrepancy between a given distribution $y_d\in L^2(\Omega)$ and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for any admissible control. After defining a suitable functional class in which we look for solutions and assuming that this problem admits at least one feasible solution, we prove the existence of optimal pairs. We derive also conditions when the set of feasible solutions has a nonempty intersection with the space of bounded distributions $L^\infty(\Omega)$.

- P.I. Kogut, O.P. Kupenko, On Holder dependence of the coeffcients
on the resolvent of Dirichlet boundary value problem with anisotropic p-Laplacian , Visnyk DNU. Series: Mathematical Modelling, Dnipropetrovsk: DNU, Issue 9, No. 8, 2017, p. 40–57. [Show Abstract]
We discuss the Holder continuity property for the inverse mapping that identies the diffusivity matrix A(x) in the main part of anisotropic p-Laplace equation as a function of resolvent operator. In particular, we prove that, within a chosen class of non-smooth admissible matrices the resolvent determines the anisotropic diffusivity in a unique manner and the corresponding inverse mapping is Holder continuous in suitable topologies.

- P.I. Kogut, Yu.A. Maksimenkova, On regularity of weak solutions to one class
of initial-boundary value problem with pseudo-differential operators , Visnyk DNU. Series: Mathematical Modelling, Dnipropetrovsk: DNU, Issue 9, No. 8, 2017, p. 70-108. [Show Abstract]
We discuss the solvability and some extra regularity properties for the weak solutions to one class of the initial-boundary value problem arising in the study of the dynamics of an arterial system.

- P.I. Kogut, Yu. Kovalenko, Asymptotic Analysis of an Optimal Boundary Control Problem for Linear Elliptic Equation in Domain with a Rough Boundary , Differential Equations and their Applications, Dnipro: DNU, MMF, Issue 3, No. 1, 2017, p. 53–83.
**KK1_2017PDF**[Show Abstract]We study the asymptotic behaviour of an optimal boundary control problem for az linear elliptic equation in two-dimensional domain $\Omega_\e$ with mixed types of boundary conditions. We assume that the boundary of domain $\Omega_\e$ contains a highly oscillating part with respect to $\e$, and we suppose that the control influence is realized via the Neumann boundary condition posed on the highly oscillating part of boundary. We present some ideas and results concerning the asymptotic analysis of such problems as $\e\to 0$ and derive conditions under which the homogenized problem can be recovered in an explicit form. We show that the mathematical description of the homogenized optimal boundary control problem is different from the original one. These differences appear not only in the limit cost functional, geometry of a limit domain, and Neumann boundary conditions, but also in the control constraints.

- P.I. Kogut, E.G. Shcherbytska, On Homogenization of an Optimal Control Problem for Quasi-Linear Elliptic Equation in One-Dimensional Setting , Differential Equations and their Applications, Dnipro: DNU, MMF, Issue 3, No. 1, 2017, p. 21–52.
**KS1_2017PDF**[Show Abstract]We are concerned with the asymptotic analysis of an optimal control problem for $1$-D partial differential equation with $\e$-periodic coefficients both in the principle part of generalized $p$-Laplace elliptic operator and in the cost functional, as the period $\e$ tends to zero. We focus on the optimal control problem for quasi-linear elliptic equation with mixed (Neumann and Dirichlet) boundary conditions, $L^1$-bounded distributed control, and a Radon measure in the right hand side of the original equation. Using approaches of the homogenization theory and utilizing the Sobolev embedding theorems, we show that the original problem tends to the optimal control problem with clearly defined structure for a one-dimensional homogenized elliptic equation containing the standard $p$-Laplace operator, and its solution can be approximated by the optimal solution to the original problem in an appropriate topology as small parameter of periodicity $\e$ tends to zero.

- T.S. Byra, P.I. Kogut, H\"older continuity of the resolvent of one class of boundary value problems for elliptic equations on the coefficients of generalized $p$-Laplace operator , Differential Equations and their Applications, Dnipro: DNU, MMF, Issue 3, No. 1, 2017, p. 3–20.
**KB1_2017PDF**[Show Abstract]We discuss the H\"lder continuity property for the inverse mapping that identifies the diffusivity coefficient $\sigma(x)$ in the main part of generalized $p$-Laplace equation as a function of resolvent operator. In particular, we prove that, within a chosen class of non-smooth admissible functions the resolvent determines the diffusivity in a unique manner and the corresponding inverse mapping is H\"lder continuous in the suitable topologies.

- T. Horsin, P. Kogut, On unbounded optimal controls in
coefficients for ill-posed elliptic Dirichlet
boundary value problems , Asymptotic Analysis, Volume 98, 2016, 155-188. [Show Abstract]
We consider an optimal control problem associated to Dirichlet boundary value problem for linear elliptic equations on a bounded domain $\Omega$. We take the matrix-valued coefficients $A=A^{sym}+A^{skew}$ of such system as a control in $L^1(\Omega;\mathbb{S}^N_{sym})\oplus L^{2p}(\Omega;\mathbb{S}^N_{skew})$. One of the important features of the class of admissible controls is the fact that the matrices $A(x)$ are unbounded on $\Omega$ and eigenvalues of the symmetric parts $A^{sym}$ may vanish in $\Omega$. In spite of the fact that the equations of this type can exhibit non-uniqueness of weak solutions, the corresponding OCP is well-possed and admits at least one solution. At the same time, optimal solutions to such problem can inherit a singular character of the matrices $A^{skew}$. We indicate two types of optimal solutions to the above problem and show that one of them can not be attained by optimal solutions of regularized problems for coercive elliptic equations with bounded coefficients, using the Steklov smoothing of matrix-valued controls $A$.

- E. Casas, P.I. Kogut, G. Leugering, Approximation of Optimal Control Problems in the Coefficient for the $p$-Laplace Equation. I. Convergence Result, SIAM J. on Control and Optimization, 54(3), 2016, 1406-1422. [Show Abstract]
We study a Dirichlet optimal control problem for a quasilinear monotone elliptic equation, the so-called weighted p-Laplace problem. The coefficient of the $p$-Laplacian, the weight $u$, we take as a control in $BV(\Omega)\cap L^\infty(\Omega)$. In this article, we use box-type constraints for the control, such that there is strictly positive lower and some upper bound. In order to handle the inherent degeneracy of the p-Laplacian, we use a regularization, sometimes referred to as the $\e$-p-Laplacian. We derive existence and uniqueness of solutions to the underlying boundary value problem and the optimal control problem. In fact, we introduce a two-parameter model for the weighted $\e$-p-Laplacian, where we approximate the non-linearity by a bounded monotone function, parametrized by $k$. Further we discuss the asymptotic behaviour of the solutions to regularized problem on each $(\e,k)$-level as the parameters tend to zero and infinity, respectively.

- T. Horsin, P. Kogut, O. Wilk, : Optime L2-Control Problem In Coefficients For A Linea Elliptic Equation. II. Approximation Of Solutions And Optimality Conditions , Mathematical Control and Related Fields, Vol. 6, No.4, 2016, 595-628. [Show Abstract]
In this paper we study we study a Dirichlet optimal control prob- lem associated with a linear elliptic equation the coefficients of which we take as controls in the class of integrable functions. The characteristic feature of this control object is the fact that the skew-symmetric part of matrix-valued control A(x) belongs to L2-space (rather than Linfinty). In spite of the fact that the equations of this type can exhibit non-uniqueness of weak solutions, the corresponding OCP, under rather general assumptions on the class of admissi- ble controls, is well-posed and admits a nonempty set of solutions [9]. However, the optimal solutions to such problem may have a singular character. We show that some of optimal solutions can be attainable by solutions of special optimal control problems in perforated domains with fictitious boundary controls on the holes.

- C. D’Apice; P.I. Kogut; R. Manzo, On Optimization of a Highly Re-Entrant Production System. Networks and Heterogeneous Media, Vol 11, Issue 3, September 2016. p.415-445. [Show Abstract]
We discuss the optimal control problem stated as the minimization in the $L^2$-sense of the mismatch between the actual out-flux and a demand forecast for a hyperbolic conservation law that models a highly re-entrant production system. The output of the factory is described as a function of the work in progress and the position of the switch dispatch point (SDP) where we separate the beginning of the factory employing a push policy from the end of the factory, which uses a quasi-pull policy. The main question we discuss in this paper is about the optimal choice of the input in-flux, push and quasi-pull constituents, and the position of SDP.

- P.I.Kogut, O.P. Kupenko, Optimality Conditions for $L^1$-Control
in Coefficients of a Degenerate Nonlinear
Elliptic Equation, in "Advances in Dynamical Systems and Control", Chapter 24, Springer, 2016, 429-469. [Show Abstract]
We study an optimal control problem for a nonlinear elliptic equation of p-Laplace type with a coefficient in the leading order of differentiation taken as control in $L^1(\Omega)$. We allow such controls to vanish on zero Lebesgue sets. As a result, we deal with degenerate elliptic Dirichlet problems that can exhibit the Lavrentiev phenomenon and non-uniqueness of weak solutions. Moreover, the non-differentiability of the term $|\nabla y|^{p-2}\nabla y$ at 0 implying the non-differentiability of the state $y(u)$ with respect to the control necessitates refined concepts in order to derive optimality conditions.

- P.I. Kogut, A.O. Putchenko , On approximate solutions to one class of non-linear
Dirichlet elliptic boundary value problems , Visnyk DNU. Series: Mathematical Modelling, Dnipropetrovsk: DNU, Issue 8, No. 8, 2016, p. 27–55. [Show Abstract]
We discuss the existence of weak solutions to one class of Dirichlet boundary value problems (BVP) for non-linear elliptic equation. Because of the specific of nonlinearity, we cannot a priori expect to have a solution in the standard functional space. Instead of this we show that the original BVP admits the so-called approximate weak solution. To do so, we introduce a special family of perturbed optimal control problems (OCPs) where the class of fictitious controls are closely related with the properties of distribution in right-hand side of the elliptic equation, and we show that optimal solutions of such problems allow to attain (in the limit) some approximate solutions as the parameter of perturbation $\e>0$ tends to zero. The main questions we discuss in this paper touch on solvability of perturbed OCPs, uniqueness of their solutions, asymptotic properties of optimal pairs as the perturbation parameter $\e>0$ tends to zero, and deriving of optimality conditions for the perturbed OCPs. As a consequence, we obtain the sufficient conditions of the existence of weak solutions to the given class of nonlinear Dirichlet BVP and propose the way for their approximation.

- P.I. Kogut, P.I. Tkachenko , On optimal control problem for nonlinear elliptic
equation with variable p(x)-Laplacian , Visnyk DNU. Series: Mathematical Modelling, Dnipropetrovsk: DNU, Issue 8, No. 8, 2016, p. 71–92. [Show Abstract]
An optimal control problem for Dirichlet boundary value problem with nonlinear elliptic equation containing $p(x)$-Laplacian is considered. It is shown that this problem has at least one solution. The corresponding theoretic framework regarding Sobolev-Orlicz spaces is given.

- P.I. Kogut, G. Leugering, R. Schiel On Henig Regularization of Material
Design Problems for Quasi-Linear p-Biharmonic Equation, Applied Mathematics, 7, 2016, 1547-1570 (Published Online August 2016 in SciRes. http://www.scirp.org/journal/am). [Show Abstract]
We study a Dirichlet optimal design problem for a quasi-linear monotone p-biharmonic equation with control and state constraints. We take the coefficient of the p-biharmonic operator as a design variable in BV(Ω). In this article, we discuss the relaxation of such problem.

- P.I. Kogut; R. Manzo; A.O. Putchenko, On approximate solutions to the Neumann elliptic boundary value problem with non-linearity of exponential type. Boundary Value Problems, Vol 2016, Issue 1, December 2016. p.1-32 (doi:10.1186/s13661-016-0717-1).
**KMP20161186**[Show Abstract]We discuss the existence of weak solutions to one class of Neumann boundary value problems (BVP) for non-linear elliptic equation. We introduce a special family of perturbed optimal control problems (OCPs) where the class of fictitious controls are closely related with the properties of distribution in right-hand side of the elliptic equation, and we show that optimal solutions of such problems allow to attain (in the limit) some approximate solutions as the parameter of perturbation $\e>0$ tends to zero. The main questions we discuss in this paper touch on solvability of perturbed OCPs, uniqueness of their solutions, asymptotic properties of optimal pairs as the perturbation parameter $\e>0$ tends to zero, and deriving of optimality conditions for the perturbed OCPs. As a consequence, we obtain the sufficient conditions of the existence of weak solutions to the given class of nonlinear Neumann BVP and propose the way for their approximation.

- C. D’Apice; P.I. Kogut, On Optimal Control Problem for Conservation Law Modelling One Class of Highly Re-Entrant Production Systems. Proceeding of the 11th International Symposium on Numerical Analysis of Fluid Flow and Heat Transfer, 19-25 Sep 2016, Ialysos, Greece.
**Proceeding_2016PDF**[Show Abstract]We discuss the optimal control problem stated as the minimization in the $L^2$-sense of the mismatch between the actual out-flux and a demand forecast for a hyperbolic conservation law that models a highly re-entrant production system. The output of the factory is described as a function of the work in progress and the position of the so-called push-pull point (PPP) where we separate the beginning of the factory employing a push policy from the end of the factory, which uses a pull policy.

- P.I. Kogut, G. Leugering, R. Schiel , On the relaxation of state-constrained linear control problems via Henig dilating cones , Control and Cybernetics, Vol. 45, No. 2, 2016, p. 131–162. [Show Abstract]
We discuss a regularization of state-constrained optimal control problem via a Henig relaxation of ordering cones. Considering a state-constrained optimal control problem, the pointwise state constraint is replaced by an inequality condition involving a so-called Henig dilating cone. It is shown that this class of cones provides a reasonable solid approximation of the typically nonsolid ordering cones which correspond to pointwise state constraints. Thereby, constraint qualifications which are based on the existence of interior points can be applied to given problems. Moreover, we characterize admissibility and solvability of the original problem by analyzing the associated relaxed problem. We also show that the optimality system for the original problem can be obtained through the limit passage in the corresponding optimality system for the relaxed problem. As an example of our approach, we derive the optimality conditions for a state constrained Neumann boundary optimal control problem and show that in this case the corresponding Lagrange multipliers are more regular than Borel measures.

- P.I. Kogut, A.O. Putchenko, On Approximate Solutions to One Class of Non-
Linear Dirichlet Elliptic Boundary Value Problems. The Case of L2-Fictitious
Controls , Journal of Differential Equations and their Applications (Research in Partial and Ordinary Differential Equations, Optimization
Theory, and Optimal Control Problems), Dnipro: DNU, Issue 2, No. 1, 2016, p. 3–31. [Show Abstract]
We discuss the existence of weak solutions to one class of Dirichlet boundary value problems (BVP) for non-linear elliptic equation. Because of the specific of nonlinearity, we cannot a priori expect to have a solution in the standard functional space. Instead of this we show that the original BVP admits the so-called approximate weak solutions. To do so, we introduce a special family of perturbed optimal control problems (OCPs). The main question we discuss in this paper is about solvability of perturbed OCPs, uniqueness of their solutions, and asymptotic properties of optimal pairs as the perturbation parameter tends to zero. As a result, we derive the suffcient conditions of the existence of weak solutions to the given class of nonlinear Dirichlet BVP and give a practical way for the approximation of such solutions.

- P. I. Kogut, G. Leugering, Optimal and Approximate Boundary Control of an Elastic Body with Quasistatic Evolution of Damage, Mathematical Methods in the Applied Sciences, Volume 38, Issue 13, pages 2739–2760, 15 September 2015. [Show Abstract]
In this paper we study an optimal control problem for the mixed boundary value problem for an elastic body with quasistatic evolution of an internal damage variable. We suppose that the evolution of microscopic cracks and cavities responsible for the damage is described by a nonlinear parabolic equation. A density of surface traction p acting on a part of boundary of an elastic body S it taken as a boundary control. Since the initial-boundary value problem of this type can exhibit the Lavrentieff phenomenon and non-uniqueness of weak solutions, we deal with the solvability of this problem in the class of weak variational solutions. Using the convergence concept in variable spaces and following the direct method in Calculus of Variations, we prove the existence of solutions to the optimal control problem under rather general assumptions on the quasistatic evolution of damage.

- P. I. Kogut, O. P. Kupenko, G. Leugering, Optimal Control Problems in Coefficients for Nonlinear Dirichlet Problems of Monotone Type:
Optimality Conditions. Part I , Journal of Analysis and
its Applications (ZAA), Volume 34, Issue 1, 2015, 85-108.
**KKL1_2015PDF**[Show Abstract]In this article we study an optimal control problem for a nonlinear monotone Dirichlet problem where the controls are taken as matrix-valued coefficients in $L^\infty(\Omega;\mathbb{R}^{N\times N} )$. For the exemplary case of a tracking cost functional, we derive first order optimality conditions. This first part is concerned with the general case of matrix-valued coefficients under some hypothesis, while the second part focuses on the special class of diagonal matrices.

- P. I. Kogut, O. P. Kupenko, G. Leugering, Optimal Control Problems in Coefficients for Nonlinear Dirichlet Problems of Monotone Type:
Optimality Conditions. Part II , Journal of Analysis and
its Applications (ZAA), Volume 34, Issue 2, 2015, 199-219.
**KKL2_2015PDF**[Show Abstract]We discuss a short-time existence theorem of solutions to the initial value problem for a fourth-order dispersive flow for curves parametrized by the real line into a compact Kähler manifold. Our equations geometrically generalize a physical model describing the motion of a vortex filament or the continuum limit of the Heisenberg spin chain system. Our results are proved by using so-called the energy method. We introduce a bounded gauge transform on the pullback bundle, and make use of local smoothing effect of the dispersive flow a little.

- T. Horsin, P. Kogut, Optimal L^2-Control Problem in Coefficients for a Linear Elliptic
Equation. I. Existence Result , Mathematical Control and Related Fields, Volume 5, Number
1, March 2015, 73-96.
**KT_MCRF_1PDF**[Show Abstract]In this paper we study an optimal control problem (OCP) associated to a linear elliptic equation { on a bounded domain $\Omega$}. The matrix-valued coefficients $A$ of such systems is our control { in $\Omega$ and will be taken in} $L^2(\Omega;\mathbb{R}^{N\times N})$ { which in particular may comprises the case of unboundedness}.{ Concerning the boundary value problems associated to }the equations of this type, one may exhibit non-uniqueness of weak solutions— namely, approximable solutions as well as another type of weak solutions that can not be obtained through the $L^\infty$-approximation of matrix $A$. Following the direct method in the calculus of variations, we show that the given OCP is well-possed and admits at least one solution. At the same time, optimal solutions to such problem may have a singular character in the above sense. In view of this we indicate two types of optimal solutions to the above problem: the so-called variational and non-variational solutions, and show that some of that optimal solutions can be attainable by solutions of special optimal boundary control problems.

- P.I. Kogut, O.P. Kupenko , On Henig regularization of state-constrained optimal control problem for the $p$-Laplace equation , Visnyk DNU. Series: Mathematical Modelling, Dnipropetrovsk: DNU, Issue 7, No. 8, 2015, p. 56–74.
**KK_2015PDF**[Show Abstract]We study a Dirichlet optimal control problem for a quasilinear monotone $p$-Laplace equation with control and state constraints. The coefficient of the $p$-Lapla\-cian, the weight $u$, we take as a control in $L^1(\Omega)$. We discuss a relaxation of such problem following the so-called Henig regularization scheme.

- P.I. Kogut, L.V. Voloshko , On relaxation of state-constrained optimal control problem in coefficients for biharmonic equation, Visnyk DNU. Series: Mathematical Modelling, Dnipropetrovsk: DNU, Issue 7, No. 8, 2015, p. 104–122.
**KV_2015PDF**[Show Abstract]We study a Dirichlet optimal control problem for biharmonic equation with control and state constraints. The coefficient of the biharmonic operator, the weight $u$, we take as a control in $L^1(\Omega)$. We discuss the relaxation approach and show that some optimal solutions to the original problem can be attained in the limit by optimal solutions of some extremal problem for variational inequality with a special penalized cost functional.

- P.I. Kogut, R. Manzo, On existence of weak solutions to a Cauchy problem for one class of conservation laws , Visnyk DNU. Series: Mathematical Modelling, Dnipropetrovsk: DNU, Issue 7, No. 8, 2015, p. 28–55.
**KM_2015PDF**[Show Abstract]We discuss the existence of weak solutions to the Cauchy problem for one class of hyperbolic conservation laws that models a highly re-entrant production system. The output of the factory is described as a function of the work in progress and the position of the so-called push-pull point (PPP) where we separate the beginning of the factory employing a push policy from the end of the factory, which uses a pull policy. The main question we discuss in this paper is about the optimal choice of the input in-flux, push and pull constituents, and the position of PPP.

- T. Horsin, P. I. Kogut, O. Wilk, Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions, ArXiv 1510.0867, 29 Oct 2015. [Show Abstract]
In this paper we study a Dirichlet optimal control problem associated with a linear elliptic equation the coefficients of which we take as controls in the class of integrable functions. The characteristic feature of this control object is the fact that the skew-symmetric part of matrix-valued control $A(x)$ belongs to $L^2$-space (rather than $L^\infty$). In spite of the fact that the equations of this type can exhibit non-uniqueness of weak solutions, the corresponding OCP, under rather general assumptions on the class of admissible controls, is well-posed and admits a nonempty set of solutions. However, the optimal solutions to such problem may have a singular character. We show that some of optimal solutions can be attainable by solutions of special optimal control problems in perforated domains with fictitious boundary controls on the holes. In order to illustrate the difficulties on the approximations of the OCPs due to the possible existence of variational and non-variational solutions, we present some numerical simulations.

- A. V. Dovzhenko, P. I. Kogut, R. Manzo, Epi and Coepi-Analysis of Vector-Valued Mappings, Optimization. A Journal of Mathematical Programming and Operations Research, 2014, Vol 63, Issue 4, P. 535-557. [Show Abstract]
This paper deals with a new characterization of lower semicontinuity of vector-value mappings in normed spaces. We study the link between the lower semicontinuity property of vector-value mappings and the topological properties of their epigraphs and coepigraphs, respectively. We show that if the objective space is partially ordered by a pointed cone with a nonempty interior, then coepigraphs are stable with respect to the procedure of their closure and, moreover, the locally semicompact vector-valued mappings with closed coepigraphs are lower semicontinuous. Using these results we propose some regularization schemes for vector-valued functions. In the case when there are no assumptions on the topological interior of the ordering cone, we introduce a new concept of lower semicontinuity for vector-value mappings, the so-called epi-lower semicontinuity, which is closely related with the closedness of epigraphs of such mappings, and study their main properties. All principal notions and assertions are illustrated by numerous examples.

- P.I. Kogut, R.Manzo, On Quadratic Scalarization of Vector Optimization Problems in Banach Spaces , Applicable Analysis, Vol. 93, No. 5, 2014, 994-1009.
**F2.2013PDF**[Show Abstract]We study vector optimization problems in partially ordered Banach spaces. We suppose that the objective mapping possesses a weakened property of lower semicontinuity and make no assumptions on the interior of the ordering cone. We discuss the "classical" scalarization of vector optimization problems in the form of weighted sum and also we propose another type of scalarization for such problems, the so-called adaptive scalarization, which inherits some ideas of Pascoletti–Serafini approach. As a result, we show that the scalar nonlinear optimization problems can be by-turn approximated by the quadratic minimization problems. Such regularization is especially attractive from a numerical point of view because it gives the possibility to apply rather simple computational methods for the approximation of the entire set of efficient solutions.

- C. D'Apice, U. De Maio, P. I. Kogut, R. Manzo, On the Solvability of an Optimal Control Problem in Coefficients for Ill-Posed Elliptic Boundary Value Problems, Electronic Journal of Differential Equations, Vol. 2014, No. 166, 2014, 1-23. [Show Abstract]
We study an optimal control problem (OCP) associated to a linear elliptic equation $-\mathrm{div}\,\left(A(x)\nabla y+C(x)\nabla y\right)=f$. The characteristic feature of this control object is the fact that the matrix $C(x)$ is skew-symmetric and belongs to $L^2$-space (rather than $L^\infty)$. We adopt a symmetric positive defined matrix $A(x)$ as control in $L^\infty(\Omega;\mathbb{R}^{N\times N})$. In spite of the fact that the equations of this type can exhibit non-uniqueness of weak solutions, we prove that the corresponding OCP, under rather general assumptions on the class of admissible controls, is well-posed and admits a nonempty set of solutions provided a special choice of skew-symmetric $L^2$-matrix $C$. The main trick we apply to the proof of the existence result is the approximation of the original OCP by regularized OCPs in perforated domains with fictitious boundary controls on the holes. As a consequence, we show that some of optimal solutions to the original problem can be attainable by solutions of regularized OCPs.

- S. O. Gorbonos, P. I. Kogut, On Pathological Solution to an Optimal Boundary Control Problem for Linear Parabolic Equation with Continuous Coefficients, Cybernetics and Computer Engineering (Кібернетика та обчислювальна техніка, cб. наук. праць ІК ім. Глушкова НАН України), Vol.176, 2014, p.5-18. [Show Abstract]
We study an optimal boundary control problem (OCP) associated to the linear parabolic equation $y_t-\mathrm{div}\,\left(\nabla y+A(x)\nabla y\right)=f$. The characteristic feature of this equation is the fact that the matrix $A(x)=[a_{ij}(x)]_{i,j=1,\dots,N}$ is skew-symmetric, $a_{ij}(x)–a_{ji}(x)$, locally continous, and belongs to $L^2$-space (rather than $L^\infty)$. We show that under special choice of matrix $A$ and distribution $f$ a unique solution to the original OCP inherits a singular character of the original matrix $A$ and it can not be attainable by the solutions of the similar OCPs with $L^\infty$-approximations of matrix $A$.

- P.I. Kogut, On Approximation of an Optimal Boundary Control Problem for Linear Elliptic Equation with Unbounded Coefficients, to appear in Discrete and Continuous Dynamical Systems - Series A (DCDS-A), Vol.34, No.5, 2014, 2105-2133. [Show Abstract]
We study an optimal boundary control problem (OCP) associated to a linear elliptic equation $-\mathrm{div}\,\left(\nabla y+A(x)\nabla y\right)=f$. The characteristic feature of this equation is the fact that the matrix $A(x)=[a_{ij}(x)]_{i,j=1,\dots,N}$ is skew-symmetric, $a_{ij}(x)–a_{ji}(x)$, measurable, and belongs to $L^2$-space (rather than $L^\infty)$. In spite of the fact that the equations of this type can exhibit non-uniqueness of weak solutions — namely, they have approximable solutions as well as another type of weak solutions that can not be obtained through an approximation of matrix $A$, the corresponding OCP is well-possed and admits a unique solution. At the same time, an optimal solution to such problem can inherit a singular character of the original matrix $A$. We indicate two types of optimal solutions to the above problem: the so-called variational and non-variational solutions, and show that each of that optimal solutions can be attainable by solutions of special optimal boundary control problems.

- P. I. Kogut, G. Leugering, On Existence of Optimal Solutions to Boundary Control Problem for
an Elastic Body with Quasistatic Evolution of Damage, In the book "Continuous and Distributed Systems: Theory and Applications", Series: Solid Mechanics and Its Applications, Springer-Verlag, Berlin, 2014, Vol.211, Chapter 19, p.265-286.
**KL_2013_1PDF**[Show Abstract]We study an optimal control problem for the mixed boundary value problem for an elastic body with quasistatic evolution of an internal damage variable. We use the damage field $\zeta=\zeta(t,x)$ as an internal variable which measures the fractional decrease in the stress-strain response. When $\zeta=1$ the material is damage-free, when $\zeta=0$ the material is completely damaged, and for $0<\zeta<1$ it is partially damaged. We suppose that the evolution of microscopic cracks and cavities responsible for the damage is described by a nonlinear parabolic equation, whereas the model for the stress in elastic body is given as $\vec\sigma=\zeta(t,x) A\ee(\mathbf{u})$. The optimal control problem we consider in this paper is to minimize the appearance of micro-cracks and micro-cavities as a result of the tensile or compressive stresses in the elastic body.

- C. D'Apice, P. I. Kogut, R. Manzo, On Relaxation of State Constrained Optimal Control Problem for a PDE-ODE Model of Supply Chains, Networks and Heterogeneous Media, Volume 9, Issue 3, 2014, Pages 501-518.
**pdfsPDF**[Show Abstract]We discuss the optimal control problem (OCP) stated as the minimization of the queues and the difference between the effective outflow and a desired one for the continuous model of supply chains, consisting of a PDE for the density of processed parts and an ODE for the queue buffer occupancy. The main goal is to consider this problem with pointwise control and state constraints. Using the so-called Henig delation, we propose the relaxation approach to characterize the solvability and regularity of the original problem by analyzing the corresponding relaxed OCP.

- P.I. Kogut, O.P. Kupenko , Shape stability of optimal control problems in coefficients for coupled system of Hammerstein type , Visnyk DNU. Series: Mathematical Modelling, Dnipropetrovsk: DNU, Issue 6, No. 8, 2014, p. 55-87. [Show Abstract]
In this paper we consider an optimal control problem (OCP) for the coup\-led system of a nonlinear monotone Dirichlet problem with matrix-valued $L^\infty(\Omega;\mathbb{R}^{N\times N} )$-controls in coefficients and a nonlinear equation of Hammerstein type, where solution nonlinearly depends on $L^\infty$-control. Since problems of this type have no solutions in general, we make a special assumption on the coefficients of the state equations and introduce the class of so-called solenoidal admissible controls. Using the direct method in calculus of variations, we prove the existence of an optimal control. We also study the stability of the optimal control problem with respect to the domain perturbation. In particular, we derive the sufficient conditions of the Mosco-stability for the given class of OCPs.

- T. Horsin, P. Kogut, On unbounded optimal controls in coefficients for ill-posed elliptic Dirichlet boundary value problems , Visnyk DNU. Series: Mathematical Modelling, Dnipropetrovsk: DNU, Issue 6, No. 8, 2014, p. 3-38. [Show Abstract]
We consider an optimal control problem associated to Dirichlet boundary value problem for linear elliptic equations on a bounded domain $\Omega$. We take the matrix-valued coefficients $A(x)$ of such system as a control in $L^1(\Omega;\mathbb{R}^{N}\times \mathbb{R}^{N})$. One of the important features of the admissible controls is the fact that the coefficient matrices $A(x)$ are non-symmetric, unbounded on $\Omega$, and eigenvalues of the symmetric part $A^{sym}=(A+A^t)/2$ may vanish in $\Omega$.

- P.I. Kogut, I.G. Balanenko , On a priori feedback control problem for degenerate parabolic equation , Visnyk DNU. Series: Mathematical Modelling, Dnipropetrovsk: DNU, Issue 6, No. 8, 2014, p. 121-136.
**Balanenko_Kogut_2014PDF**[Show Abstract]One class of optimal control problems for degenerate parabolic equation with Dirichlet boundary conditions and a priori feedback law are considered. It is shown that such problems have an optimal solution in the corresponding weighted Sobolev space provided the operators of feedback laws possess special compactness properties.

- S.O. Gorbonos, P.I. Kogut, Necessary optimality conditions to an control problem for parabolic system with unbounded coefficients , Visnyk DNU. Series: Mathematical Modelling, Dnipropetrovsk: DNU, Issue 6, No. 8, 2014, p. 159-171. [Show Abstract]
The necessary optimality conditions to an optimal control problem for parabolic system with unbounded coefficients were developed using the concepts of the generalized right hand directional derivative and quasi-adjoint state.

- S.O. Gorbonos, P.I. Kogut, On non-variational solutions to optimal control problems for linear degenerate parabolic equations , Visnyk DNU. Series: Mathematics, Dnipropetrovsk: DNU, Issue 19, No. 6, 2014, p. 36-51.

- P. I. Kogut, G. Leugering, Matrix-valued L1-optimal control in the coefficients of linear elliptic problems, ZAA (Zeitschrift für Analysis und ihre Anwendungen), Volume 32, Issue 4, 2013, p. 433–456.
**F3.2013PDF**[Show Abstract]In this paper we study an optimal control problem associated to a linear degenerate elliptic equation with mixed boundary conditions. We adopt the square symmetric matrix of coefficients A(x)=[aij(x)]i,j=1,...,N as a control in L1(S;RN(N-1)/2). The characteristic feature of the class of admissible controls is the fact that eigenvalues of the coefficient matrices may vanish in S. The equations of this type can exhibit the Lavrentieff phenomenon and non-uniqueness of weak solutions. Using the convergence concept in variable spaces and following the direct method in the Calculus of variations, we discuss the solvability of this optimal control problem in the class of weak admissible solutions.

- P.I. Kogut, R.Manzo, On Vector-Valued Approximation of State Constrained Optimal Control Problems for Nonlinear Hyperbolic Conservation Laws , Journal of Dynamical and Control Systems, No.2, Vol.19, 2013, 381-404.
**F1.2013PDF**[Show Abstract]We study one class of nonlinear fluid dynamic models with controls in the initial condition and the source term. The model is described by a nonlinear inhomogeneous hyperbolic conservation law with state and control constraints. We consider the case when the greatest lower bound of the cost functional can be unattainable on the set $\Xi$ of admissible pairs or the set $\Xi$ is possibly empty. Using the methods of vector-valued optimization theory, we show that this optimal control problem admits the existence of the so-called weakened approximate solution which can be interpreted as generalized solution to some vector optimization problem of special form.

- І. Г. Баланенко, П. І. Когут, Задача оптимального стартового керування виродженим параболічним рівнянням, Вісник ДНУ. Сер. Моделювання, № 8, Вип. 5, 2013. — С. 47–61. [Show Abstract]
Досліджується задача оптимального керування для виродженого параболічного рівняння зі змішаними крайовими умовами на межі області. Із залученням нерівності типу Харді-Пуанкаре показано, що така задача має єдиний оптимальний розв'язок у вагових просторах Соболєва. Отримано та обґрунтовано необхідні умови оптимальності.

- C. О. Горбонос, П. І. Когут, Варіаційні розв'язки однієї задачі оптимального керування з необмеженими коефіцієнтами, Вісник ДНУ. Сер. Моделювання, № 8, Вип. 5, 2013. — С. 69–83.
**Gorbonos.pdfPDF**[Show Abstract]Досліджено задачу оптимального керування параболічною системою з необмеженими коефіцієнтами. Введено поняття варіаційного розв'язку поставленої задачі керування та встановлено умови його існування.

- P.I.Kogut, On Some Properties of Unbounded Bilinear Forms Associated with Skew-Symmetric L^2(Omega)-Matrices, Вісник ДНУ. Сер. Моделювання, № 8, Вип. 5, 2013. — С. 84–97.
**Kogut_DNU_2013.pdfPDF**[Show Abstract]We study the bilinear forms on the space of measurable square-integrable functions which are generated by skew-symmetric matrices with unbounded coefficients. We show that in the case when a skew-symmetric matrix contains L^2-elements, the corresponding quadratic forms can be alternative. Since these questions are closely related with the existence of a unique solution for linear elliptic equations with unbounded coefficients, we show that the energy identities for weak solutions can be studied in the framework of the corresponding alternative quadratic forms. To this end, we discuss the problems of integration by parts for measurable functions and give a generalization of some formulae for the non-Lipschitz case.

- T. Horsin, P. Kogut, Optimal L^2-Control Problem in Coefficients for a Linear Elliptic Equation, {submitted to Mathematical Control and Related Fields}, 2013, 1-60, arXiv:1306.2513. [Show Abstract]
In this paper we study an optimal control problem (OCP) associated to a linear elliptic equation { on a bounded domain \Omega}. The matrix-valued coefficients A of such systems is our control { in \Omega and will be taken in} L^2(Omega;{R}^{N\times N}) { which in particular may comprise some cases of unboundedness}.{ Concerning the boundary value problems associated to }the equations of this type, one may face non-uniqueness of weak solutions— namely, approximable solutions as well as another type of weak solutions that can not be obtained through the L^\infty-approximation of matrix A. Following the direct method in the calculus of variations, we show that the given OCP is well-posed in the sense that it admits at least one solution. At the same time, optimal solutions to such problem may have a singular character in the above sense. In view of this, we indicate two types of optimal solutions to the above problem: the so-called variational and non-variational solutions, and show that some of that optimal solutions can be attainable by solutions of special optimal boundary control problems.

- P. I. Kogut, G. Leugering, Optimal L1-Control in Coefficients for Dirichlet Elliptic Problems: H-Optimal Solutions, Zeitschrift für Analysis und ihre Anwendungen, (2012), Vol.31, Issue 1, 31-53. [Show Abstract]
In this paper we study a Dirichlet optimal control problem associated with a linear elliptic equation the coefficients of which we take as controls in L1(S). In particular, when the coefficient matrix is taken to satisfy the decomposition B(x)=r(x)A(x) with a scalar function r, we allow the r to degenerate. Such problems are related to various applications in mechanics, conductivity and to an approach in topology optimization, the SIMP-method. Since equations of this type can exhibit the Lavrentieff phenomenon and non-uniqueness of weak solutions, we show that the optimal control problem in the coefficients can be stated in different forms depending on the choice of the class of admissible solutions. Using the direct method in the Calculus of variations, we discuss the solvability of the above optimal control problems in the so-called class of H-admissible solutions.

- P.I.Kogut, T.M. Rudyanova, On $L^1$-Matrices with Degenerate Spectrum and Weak Convergence in Associated Weighted Sobolev Spaces, Вісник ДНУ. Сер. Математика, 2012, Том. 20, № 6/1, 134-144. [Show Abstract]
We study the compactness property of the weak convergence in variable Sobolev spaces of the following sequences $\left\{(A_n,u_n)\in L^1(\Omega;\mathbb{R}^{N\times N})\times W_{A_n}(\Omega;\Gamma_D)\right\}$, where the squared symmetric matrices $A:\Omega\rightarrow \mathbb{R}^{N\times N}$ belong to the Lebesgue space $L^1(\Omega,\mathbb{R}^{N\times N})$ and their eigenvalues may vanish on subdomains of $\Omega$ with zero Lebesgue measure.

- І. Г. Баланенко, П. І. Когут, Про одну задачу оптимального керування для виродженого параболічного рівняння, Вісник ДНУ. Сер. Моделювання, № 8, Вип. 4, 2012. — С. 3–18. [Show Abstract]
Досліджується задача оптимального керування для виродженого параболічного рівняння зі змішаними крайовими умовами на межі області. Із залученням нерівності типу Харді – Пуанкаре показано, що така задача має єдиний оптимальний розв'язок у вагових просторах Соболєва. Отримано та обґрунтовано необхідні умови оптимальності.

- P.I.Kogut, O.P.Kupenko, On Attainability of Optimal Solutions for Linear Elliptic Equations with Unbounded Coefficients, Вісник ДНУ. Сер. Моделювання, № 8, Вип. 4, 2012. — С. 63–82. [Show Abstract]
An optimal boundary control problem associated to a linear elliptic equation describing diffusion in a turbulent flow is studied. The characteristic feature of this equation is the fact that, in applications, the stream matrix $A(x)=[a_{ij}(x)]_{i,j=1,\dots,N}$ is skew-symmetric, $a_{ij}(x)–a_{ji}(x)$, measurable, and belongs to $L^2$-space (rather than $L^\infty)$. An optimal solution to such problem can inherit a singular character of the original stream matrix $A$. We show that optimal solutions can be attainable by solutions of special optimal boundary control problems.

- В.Н. Богомаз, П.I. Когут, Про квадратичну скаляризацію одного класу задач векторної оптимізації в банахових просторах, Вісник ДНУ. Сер. Моделювання, № 8, Вип. 4, 2012. — С. 40–52. [Show Abstract]
Розглядається проблема скаляризації для одного класу задач векторної оптимізації в частково впорядкованих банахових просторах. Припускається, що цільове відображення має ослаблену властивість напівнеперервності знизу та упорядковувальний конус не є тілесним. Запропоновано метод скаляризації, який успадковує ідеї підходу Пасколетті – Серафіні. Показано, що в цьому випадку скалярні задачі оптимізації можна апроксимувати квадратичними задачами мінімізації.

- В.Н. Богомаз, П.I. Когут, О разрешимости одной задачи векторной оптимизации с фазовыми ограничениями, Проблемы управления и информатики, № 3, 2012. — С. 31–45.

- A. В. Довженко, П. I. Когут, Epi-напівнеперервні знизу відображення та їх властивості, Математичні студії, 36(1) 2011, 86-96.
**KD_2011PDF**[Show Abstract]We present a new concept of lower semicontinuity for vector-valued mappings, so-called lower epi-semicontinuity, which is closely related with the closedness of epigraphs of these mappings. We consider the case when the mappings take values in a real Banach space Y partially ordered by a closed convex pointed cone L, and we make no additional assumptions on topological interior of the ordering cone. We also study the comparison of the lower epi-semicontinuity with the classical notion of lower semicontinuity and its generalizetions.

- G. Buttazzo, P. I. Kogut, Weak optimal controls in coefficients for linear elliptic problems, Revista Matematica Complutense, Vol.24, 2011, pp. 83–94.
**file_2PDF**[Show Abstract]In this paper we study an optimal control problem associated to a linear degenerate elliptic equation with mixed boundary conditions. The equations of this type can exhibit the Lavrentieff phenomenon and non-uniqueness of weak solutions. We adopt the weight function as a control in L1(S). Using the direct method in the Calculus of variations, we discuss the solvability of this optimal control problem in the class of weak admissible solutions.

- A.V. Dovzhenko, P.I. Kogut, R. Manzo, On the concept of Г-convergence for locally compact vector-valued mappings, Far East Journal of Applied Mathematics, Vol.60, Issue 1, 2011, p.1-39. [Show Abstract]
The paper deals with a new concept of limit for sequences of locally compact vector-valued mappings in normed spaces. We generalize the well-know concept of $\Gamma$-convergence to the so-called $\Gamma^{\Lambda,\mu}$-convergence in vector-valued case. To this aim we study the link between the lower semicontinuity property of vector-valued mappings and the topological properties of their coepigraphs. We show that if the objective space is partially ordered by a pointed cone with nonempty interior, then coepigraphs are stable with respect to their closure and, moreover, the locally semicompact vector-valued mappings with closed coepigraphs are lower continuous. Using these results we establish the relationship between $\Gamma^{\Lambda,\mu}$-convergence of the sequences of mappings and $K$-convergence of their coepigraphs in the sense of Kuratowski and study the main topological properties of $\Gamma^{\Lambda,\mu}$-limits.

- P. I. Kogut, G. Leugering, Optimal L1-Control in Coefficients for Dirichlet Elliptic Problems: W-Optimal Solutions, Journal of Optimization Theory and Applications, 150(2) (2011), 205-232. [Show Abstract]
In this paper we study a Dirichlet optimal control problem associated with a linear elliptic equation the coefficients of which we take as controls in L1(S). The coefficients may degenerate and, therefore, the problems may exhibit the so-called Lavrentieff phenomenon and non-uniqueness of weak solutions. We consider the solvability of this problem in the class of W-variational solutions. Using the concept of variational convergence of constrained minimization problems in variable spaces, we prove the existence of W-solutions to the optimal control problem and provide way for their approximation. We emphasize that control problems of this type are important in material- and topology optimization as well as in damage or life-cycle optimization.

- П. И. Когут, К вопросу об устойчивости Lp-решений интегральных уравнений Вольтерра, Вісник ДНУ. Сер. Моделювання, Т.19, № 8, Вип. 3, 2011. — С. 3–19.
**Kogut_1_11PDF**[Show Abstract]The stability of linear Volterra integral equations are discussed from the Lyapunov Direct Method point of view.

The characteristic feature of these equations is the fact that their solutions are not continuous functions. We consider the case when the solution class is the Bochner space Lploc(0,∞;X) of locally p-integrable functions. - І. Г. Баланенко, П. І. Когут, Про класифiкацiю розв'язкiв початково-крайових задач для вироджених параболiчних рiвнянь, Вісник ДНУ. Сер. Моделювання, Т.19, № 8, Вип. 3, 2011. — С. 54–72.
**Kogut_1_12PDF**[Show Abstract]The classification of the weak solutions to the Dirichlet initial boundary value problem associated for a linear degenerate parabolic equation has been studied. Some applications to associated optimal control problems in coefficients are discussed.

- P. I. Kogut, R. Manzo, I. V. Nechay, Generalized efficient solutions to one class of vector optimization problems in Banach spaces, Australian Journal of Mathematical Analysis and Applications, 2010, Vol.7, Issue 1., pp. 1–27. [Show Abstract]
In this paper, we study vector optimization problems in Banach spaces for essentially nonlinear operator equations with additional control and state constraints. We assume that an objective mapping possesses a weakened property of lower semicontinuity and make no assumptions on the interior of the ordering cone. Using the penalization approach we derive both sufficient and necessary conditions for the existence of efficient solutions of the above problems. We also prove the existence of the so-called generalized efficient solutions via the scalarization of some penalized vector optimization problem.

- P. I. Kogut, Manzo R., I. V. Nechay, Topological Aspects of Scalarization in Vector Optimization Problems, Australian Journal of Mathematical Analysis and Applications, 2010, Vol.7, Issue 2., pp. 25–49. [Show Abstract]
In this paper, we study vector optimization problems in partially ordered Banach spaces. We suppose that the objective mapping possesses a weakened property of lower semicontinuity and make no assumptions on the interior of the ordering cone. We derive sufficient conditions for existence of efficient solutions of the above problems and discuss the role of topological properties of the objective space. We discuss the scalarization of vector optimization problems when the objective functions are vector-valued mappings with a weakened property of lower semicontinuity. We also prove the existence of the so-called generalized efficient solutions via the scalarization process. All principal notions and assertions are illustrated by numerous examples.

- I. G. Balanenko, P. I. Kogut H-Optimal control in coefficients for Dirichlet parabolic problems, Вісник ДНУ. Сер. Моделювання, Т.18, № 8, Вип. 2, 2010. — С. 45–63. [Show Abstract]
In this paper we study the Dirichlet optimal control problem associated with a linear parabolic equation the coefficients of which we take as controls in L1(S). Since equations of this type can exhibit the Lavrentieff phenomenon and non-uniqueness of weak solutions, we show that the optimal control problem in the coefficients can be stated in different forms depending on the choice of the class of admissible solutions. Using the direct method in the Calculus of variations, we discuss the solvability of the above optimal control problems in the so-called class of H-admissible solutions.

- C. D'Apice, P. I. Kogut, R. Manzo, Efficient controls for one class of fluid dynamic models, Far East Journal of Applied Mathematics, 2010, Vol.46, No.2, pp. 85–119. [Show Abstract]
We study one class of fluid dynamic models in vector-valued optimization statement. The model consists of a system of two hyperbolic conservation laws with a source term: a non-linear conservation law for the goods density and an evolution equation for the processing rate. We consider the case when an objective space is the Banach space Lploc(RN) partially ordered by the natural ordering cone of positive elements. We derive sufficient conditions for the existence of efficient controls. In the case when the original control problem is not regular and may fail to have the entropy solutions, we discuss the regularization approach and prove the existence of the so-called efficient regularizators to the original vector-valued optimization problem.

- А. В. Довженко, П. І. Когут, Coepi-аналіз властивості напівнеперервності знизу векторнозначних відображень в банахових просторах, Вісник ДНУ. Сер. Математика, Т.18, № 6/1, Вип. 15, 2010. — С. 110–122. [Show Abstract]
В роботі розглядається зв'язок властивості напівнеперервності знизу векторнозначних відображень та їх конадграфіків. За допомогою конадграфіків запропоновано механізм знаходження напівнеперервної знизу регуляризації відображень. Отримано варіаційне подання такої регуляризації.

- P. I. Kogut, Rudyanova T.N. On some smooth approximations on thick periodic multi-structures, Вісник ДНУ. Сер. Математика, Т.18, № 6/1, Вип. 15, 2010. — С. 158–170.
**Kogut_1_15PDF**[Show Abstract]In this paper we study the approximation properties of measurable and square-integrable functions. In particular we show that any L2-bounded functions can be approximated in L2-norm by smooth functions defined on a highly oscillating boundary of thick multi-structures in Rn. We derive the norm estimates for the approximating functions and study their asymptotic behavior.

- Т. А. Божанова, П. І. Когут, Про узагальнені розв'язки однієї оптимізаційної задачі на транспортних мережах, Динамические системы, 2010, Вып.28. — С.46–60. [Show Abstract]
У даній роботі розглядається гідродинамічна модель для транспортного потоку на мережі. У припущенні, що такий потік є керованим процесом, ставиться задача його оптимізації у векторній формі. Розглянуто випадок, коли цільове відображення діє в лебегів простір та є напівнеперервним зверху на області визначення. Залучаючи ідеологію регуляризації за Тихоновим, введено поняття узагальненого розв'язку задачі векторної оптимізації на мережі. Використовуючи той факт, що множина ефективних розв'язків такої задачі є непорожньою та залучаючи процедуру скаляризації, доведено існування узагальнених розв'язків розглянутої задачі векторної оптимізації.

- C. D'Apice, P. I. Kogut, R. Manzo, Efficient Controls for Traffic Flow on Networks, Dynamical and Control Systems, 2010, Vol.16, No.3, pp. 407–437.
**Kogut_1_16PDF**[Show Abstract]We study traffic flow models for road networks in vector-valued optimization statement where the flow is controlled at the nodes of the network. We consider the case when an objective mapping possesses a weakened property of upper semicontinuity and make no assumptions on the interior of the ordering cone. We derive sufficient conditions for the existence of efficient controls of the traffic problem and discuss the scalarization approach to its solution. We also prove the existence of the so-called generalized efficient controls.

- P. I. Kogut, Manzo R., I. V. Nechay, On existence of efficient solutions to vector optimization problems in Banach spaces, Note di Matematica, Vol.30(1), 2010, pp. 25–39.
**Kogut_1_17PDF**[Show Abstract]In this paper, we present a new characterization of lower semicontinuity of vector-valued mappings and apply it to the solvability of vector optimization problems in Banach spaces. With this aim we introduce a class of vector-valued mappings that is more wider than the class of vector-valued mappings with the "typical" properties of lower semi-continuity including quasi and order lower semi-continuity. We show that in this case the corresponding vector optimization problems have non-empty sets of efficient solutions.

- C. D'Apice, P. I. Kogut, R. Manzo, On Approximation of Entropy Solutions for One System of Nonlinear Hyperbolic Conservation Laws with Impulse Source Terms, Journal of Control Science and Engineering, 2010, Vol. 46, No. 2. p.85-119.
**Kogut_1_18PDF**[Show Abstract]We study one class of nonlinear fluid dynamic models with impulse source terms. The model consists of a system of two hyperbolic conservation laws: a nonlinear conservation law for the goods density and a linear evolution equation for the processing rate. We consider the case when influx-rates in the second equation take the form of impulse functions. Using the vanishing viscosity method and the so-called principle of fictitious controls, we show that entropy solutions to the original Cauchy problem can be approximated by optimal solutions of special optimization problems.

- P. I. Kogut, Manzo R., I. V. Nechay, Topological aspects in vector optimization problems, Вісник ДНУ. Сер. Моделювання, Т.17, № 8, Вип. 1, 2009. — С. 61–88. [Show Abstract]
In this paper we study vector optimization problems in partially ordered Banach spaces. We suppose that the objective mapping possesses a weakened property of lower semicontinuity and make no assumptions on the interior of the ordering cone. We derive sufficient conditions for existence of efficient solutions of the above problems and discuss the role of topological properties of the objective space. We discuss the scalarization of vector optimization problems when the objective functions are vector-valued mappings with a weakened property of lower semicontinuity. We also prove the existence of the so-called generalized efficient solutions via the scalarization process. All principal notions and assertions are illustrated by numerous examples.

- V. V. Gotsulenko, P. I. Kogut, Suboptimal boundary control of the Benard system in a cylindrically perforated domain, Ukrainian Mathematical Bulletin, Vol. 6, No. 4, 2009, pp. 431–467.
**Kogut_1_20PDF**[Show Abstract]We study an optimal boundary control problem for the B´enard system in a cylindrically perforated domain. The control is the boundary velocity ﬁeld supported on the “vertical” sides of thin cylinders. We show that an optimal solution to some limit problem in a nonperfo- rated domain can be used as a basis for the construction of the so-called asymptotically suboptimal controls for the original control problem.

- I. G. Balanenko, P. I. Kogut, On existence of the weak optimal BV-controls in coefficients for linear elliptic problems, Вісник ДНУ. Сер. Моделювання, Т.17, № 8, Вип. 1, 2009. — С. 93–103. [Show Abstract]
In this paper we study the optimal control problem associated to a linear degenerate elliptic equation with mixed boundary conditions. We adopt a weight coefficient in the main part of elliptic operator as control in BV(S). Since the equations of this type can exhibit the Lavrentieff phenomenon and non-uniqueness of weak solutions, we show that this optimal control problem is regular. Using the direct method in the Calculus of variations, we discuss the solvability of the above optimal control problems in the class of weak admissible solutions.

- P. I. Kogut, G. Leugering, Homogenization of optimal control problems for one-dimensional elliptic equations on periodic graph, ESAIM: Control, Optimization and Calculus of Variations (COCV), Vol. 15, 2009, pp. 471–498.
**Kogut_1_23PDF**[Show Abstract]We are concerned with the asymptotic analysis of optimal control problems for 1-D partial diﬀerential equations deﬁned on a periodic planar graph, as the period of the graph tends to zero. We focus on optimal control problems for elliptic equations with distributed and boundary controls. Using approaches of the theory of homogenization we show that the original problem on the periodic graph tends to a standard linear quadratic optimal control problem for a two-dimensional homogenized system, and its solution can be used as suboptimal controls for the original problem.

- C. D'Apice, U. De Maio, P. I. Kogut, Boundary velocity suboptimal control of incompressible flow in cylindrically perforated domain, Discrete and Continuous Dynamical Systems, Serie B, Vol.11, No.2, 2009, pp. 283–314.
**Kogut_1_22PDF**[Show Abstract]In this paper we study an optimal boundary control problem for the 3D steady-state Navier-Stokes equation in a cylindrically perforated domain. The control is the boundary velocity field supported on the 'vertical' sides of thin cylinders. We minimize the vorticity of viscous flow through thick perforated domain. We show that an optimal solution to some limit problem in a non-perforated domain can be used as basis for the construction of suboptimal controls for the original control problem. It is worth noticing that the limit problem may take the form of either a variational calculation problem or an optimal control problem for Brinkman's law with another cost functional, depending on the cross-size of thin cylinders.

- В. В. Гоцуленко, П. И. Когут, Субоптимальное граничное управление системой Бенара в цилиндрически перфорированной области, Український математичний вісник, № 4, Том. 6, 2009, С. 436–474.
**Kogut_1_21PDF**[Show Abstract]Для cистемы Бенара рассматривается задача оптимального граничного управления течением вязкой несжимаемой жидкости в обобщенной ячейке Куэтта. Предложена концепция асимптотически субоптимальных управлений в такой задаче и установлена иx структура, когда перфорация цилиндрической области соответствует критическому случаю.

- А. В. Довженко, П. И. Когут, Квазі-неперервна знизу регуляризація відображень в бананових просторах, Вісник ДНУ, Сер. Математика, Т.17, № 6/1, Вип. 14, 2009. — С. 60–72. [Show Abstract]
Запропоновано спосіб квазі-напівнеперервної знизу регуляризації відображень у банахових просторах.

- O. P. Kogut, P. I. Kogut, T. N. Rudyanova, A note on H-convergence, Вісник ДНУ. Сер. Математика, Т.17, № 6/1, Вип. 14, 2009. — С. 150–162.
**visn-09PDF**[Show Abstract]In this paper we study the H-convergence property for the uniformly bounded sequences of square matrices Ae in Linfty(D;Rn x n). We derive the sufficient conditions which guarantee the coincidence of H-limit with the weak-* limit of such sequences in Linfty(D;Rn x n).

- C. D'Apice, U. De Maio, P. I. Kogut, Gap phenomenon in homogenization of parabolic optimal control problems, IMA Journal of Mathematical Control and Information (Cambridge University), Vol.25, 2008, pp. 461–480.
**file_2_01PDF**[Show Abstract]In this paper we study the asymptotic behaviour of a parabolic optimal control problem in a subdomain D e of Rn, whose boundary contains a highly oscillating part. We consider this problem with two different classes of Dirichlet boundary controls, and, as a result, we provide its asymptotic analysis with respect to the different topologies of homogenization. It is shown that the mathematical descriptions of the homogenized optimal control problems have different forms and these differences appear not only in the state equation and boundary conditions but also in the control constrains and the limit cost functional.

- C. D'Apice, U. De Maio, P. I. Kogut, Suboptimal boundary controls for elliptic equations in critically perforated domains, Annales de I'Institut Henri Poincare (C), Nonlinear Analysis, Vol.25, Issue 6, 2008, pp. 1073–1101.
**file_2_02PDF**[Show Abstract]In this paper we study the asymptotic behaviour, as a small parameter e tends to zero, of a class of boundary optimal control problems Pe, set in e-periodically perforated domain. The holes have a critical size with respect to e-sized mesh of periodicity. The support of controls is contained in the set of boundaries of the holes. This set is divided into two parts, on one part the controls are of Dirichlet type; on the other one the controls are of Neumann type. We show that the optimal controls of the homogenized problem can be used as suboptimal ones for the problems Pe.

- P. I. Kogut, G. Leugering, Homogenization of Dirichlet optimal control problems with exact partial controllability constraints, Asymptotic Analysis, Vol.57, No.3-4, 2008, pp. 229–249.
**file_2_03PDF**[Show Abstract]We consider an elliptic distributed quadratic optimal control problem with exact controllability constraints on a part of the domain which, in turn, is parametrized by a small parameter epsilon. The quadratic tracking type functional is defined on the remaining part of the domain. We thus consider a family of optimal control problems with state equality constraints. The purpose of this paper is to study the asymptotic limit of the optimal control problems as the parameter epsilon tends to zero. The analysis presented is in the spirit of the direct approach of the calculus of variations. This is achieved in the framework of relaxed problems. We finally apply the procedure to an optimal control problem on a perforated domain with holes of critical size. It is shown that a strange term in the terminology of Cioranescu and Murat appears in the limiting homogenized problem.

- P. I. Kogut, I. V. Nechay, On scalarization of vector optimization problems in Banach spaces, J. of Automation and Information Sciences, Vol.40, Issue 12, 2008, pp. 1–13.
**file_2_04PDF**[Show Abstract]The main focus of this work is the one class of vector optimization problems with mapping that is not necessarily lower semicontinuous. The sufficient conditions of existence efficient solutions of such problems are given. We show that certain part of set of efficient solutions can be achieved by using the scalarization technique. All main theoretical notions are supported by illustrative examples.

- P. I. Kogut, T. N. Rudyanova, On the weakly-∗ dense subsets in L∞(Ω), Вісник ДНУ, Сер. Математика, Т.16, №6/1, 2008, pp. 149–154.
**file_2_05PDF**[Show Abstract]We study the density property of the compactly supported smooth functions in the space L∞(Ω). We show that this set is dense with respect to the weak-ast convergence in variable spaces.

- П. І. Когут, І. В. Нечай, До питання регуляризації задач векторної оптимізації, Вісник ДНУ, Сер. Математика, Т.16, №6/1, 2008, pp. 99–110. To the question of regularization of vector optimization problems, Researches in Mathematics, Vol 16 (2008) pp. 99–110. https://doi.org/10.15421/240812
**file_2_06DOC**[Show Abstract]Запропоновано спосіб регулярізації одного класу задач нескалярної оптимізації в банахових просторах для випадку, коли векторнозначне відображення не є напівнеперервним знизу у певному сенсі, в наслідок чого, порушуються достатні умови їх розв’язності.

- P. I. Kogut, G. Leugering, Asymptotic analysis of state constrained semilinear optimal control problems, Journal of Opt. Theory and Appl. (JOTA), Vol.135, No.2, 2007, pp. 301–321.
**file_2_07PDF**[Show Abstract]The asymptotic behaviour of state constrained semilinear optimal control problems for distributed parameter systems with variable compact control zones is investigated. We derive conditions under which the limiting problems can be made explicit.

- P. I. Kogut, G. Leugering, On the homogenization of optimal control problems on periodic graphs, Lecture Notes in Pure and Applied Mathematics, Vol.252, 2007, pp. 55–74.
**file_2_08PDF**[Show Abstract]We are concerned with the asymptotic analysis of optimal control problems for 1-D partial differential equations defined on a periodic planar graph, as the period of the graph tends to zero. We focus on optimal control problems for elliptic equations with distributed and boundary controls. Using approaches of the theory of homogenization we show that the original problem on the periodic graph tends to a standard linear quadratic optimal control problem for a two-dimensional homogenized system, and its solution can be used as suboptimal controls for the original problem.

- V. V. Gotsulenko, P. I. Kogut, Boundary control of a laminar flow of a viscous incompressible fluid in the generalized Couette cell. The asymptotic approach, Automation and Remote Control, Vol. 68 , Issue 2, 2007, pp. 267–283.
- П. І. Когут, М. Є. Сердюк, Задачі каркасної інтерполяції статичних зображень, Вісник Херсонського техн. унів., № 2(28), 2007, 153–157.
- P. I. Kogut, T. N. Rudyanova, On the property of homothetic mean value on periodically perforated domains, Вісник ДНУ, Сер. Математика, №8, 2007, 158–166.
**file_2_09PDF**[Show Abstract]In this paper we study the limit properties of one class of periodic functions, as a small parameter tends to zero, defined on $\e$-periodically perforated domain. The holes have a critical size with respect to the sized mesh of periodicity. For this type of functions we derive the so-called homothetic mean-value property.

- П. І. Когут, М. Є. Сердюк, Про задачу реконструкції зображень за їх каркасною інтерполяцією, Системні технології, №3(50), 2007, 35–45.
- П. І. Когут, М. Є. Сердюк, Асимптотичний підхід до задач каркасної інтерполяції статичних зображень, Математичне моделювання, №2(17), 2007, 3–7.
- В. В. Гоцуленко, П. И. Когут, Граничное управление ламинарным течением вязкой несжимаемой жидкости в обобщенной ячейке Куэтта. Асимтотический подход, Автоматика и телемеханика, № 2, 2007, 63–80.

- П. И. Когут, В. Н. Мизерный, Т. Н. Рудянова, К проблеме усреднения задач оптимального управления. 1. Анализ существующих схем, Проблемы управления и информатики, 2006, N.3. — C.10–26.
- П. И. Когут, В. Н. Мизерный, Т. Н. Рудянова, К проблеме усреднения задач оптимального управления. 2. Вариационная сходимость задач условной минимизации, Проблемы управления и информатики, 2006, N.4. — C.9–21.
- C. D'Apice, U. De Maio, P. I. Kogut, On the Lavrentieff phenomenon in the homogenization of an optimal control problem, Доп. НАН України, 2006, N.6. — C. 77-82.
- П. І. Когут, І. В. Нечай, Про розвязність одного классу задач векторної оптимізації, Наукові Вісті ННТУ-КПІ, № 5, 2006, C. 148–158.
- P. I. Kogut, V. M. Mizerny, T. M. Rudjanova, On homogenization of optimal control problems. I. Analysis of existing approaches, J. Automation and Information Sciences, 38(2006), i. 5, 6-22; translated from Problemy Upravlenija I Informatiki No.3, 2006, pp. 10–26.
- P. I. Kogut, V. M. Mizerny, T. M. Rudjanova, On homogenization of optimal control problems. II. Variational convergence of the constrained minimization problems, J. Automation and Information Sciences, 38, i. 7, 2006, pp. 58–72; translated from Problemy Upravlenija I Informatiki No. 4, 58-71(2006).

- П. И. Когут, G*-сходимость нелинейных операторов в теории усреднения управляемых объектов, Системный анализ и кибернетика, 2005, N. 5, C. 42–65; English transl. in Cybernetics and System Analysis, 2005, V. 41, No 5, 670–687.
- П. И. Когут, Т. А. Мельник, Предельный анализ одного класса задач оптимального управления в густых сингулярных соединениях, Проблемы управления и информатики, 2005, N. 2. — C.13–31.
- П. И. Когут, Т. А. Мельник, К усреднению одной задачи оптимального управления в густых сингулярных соединениях, Доп. НАН України, 2005, N.7. — C.52–58;
- P. I. Kogut, T. A. Mel'nik, Limit analysis of one class of optimal control problems in thick singular junctions, J. Automation and Information Sciencis, Vol. 37, i. 1, 2005, 8–23; translated from Problemy Upravlenija I Informatiki No.2, 13-31(2005).
- P. I. Kogut, G∗-convergence of nonlinear operators in the theory of homogenization of control objects, Cybernetics and System Analysis, Vol. 41, No 5, 2005, 670–687; translated from Kibern. Sist. Anal., No.5, 42-65 (2005).

- Н. А. Баламут, П. И. Когут, Об усреднении одной нелинейной задачи оптимального управления, Доп. НАН України, 2004, N. 2. — C. 60–65.
- P. I. Kogut, T. A. Mel'nik, Asymptotic analysis of optimal control problems in thick multi-structures, In the book "Generalized Solutions in Control Problems", Proceeding of the IFAC Workshop GSCP-2004, Pereslavl-Zalessky, Russia, September 21-29,2004. General Theory: Distributed Parameter Systems, p. 265–275.
**file_2_10DVI**

- Н. А. Баламут, П. И. Когут, Предельный анализ одного класса квазилинейных задач оптимального управления с фазовыми ограничениями, Проблемы управления и информатики, 2003, N.4. — C. 29–43; English trans. in J. Automatation and Information Sciences, 2003, V. 35, N 7, 25-38.
- П. И. Когут, В. Н. Мизерный, Т. Н. Рудянова, К усреднению объектов управления, описываемых операторными уравнениями Гаммерштейна. Нерегулярный случай, Проблемы управления и информатики, 2003, N.6. — C.24–34; English transl. in J. Automatation and Information Sciences, 2003, V.35, N. 11, 19-28.
- P. I. Kogut, V. N. Mizernuy, T. N. Rudyanova, On homogenization of cotrolled objects described by operators equation of Gammerstain type, J. Automation and Information Sciences, 35(2003), i. 11, pp. 19–28; translated from Problemy Upravlenija I Informatiki No.6, 24–34(2003).
- N. A. Balamut, P. I. Kogut, Limit analysis of one class of quasilinear problems of optimal control with constraints on the state, J. Automation and Information Sciences, 35(2003), i. 7, pp. 25–38; translated from Problemy Upravlenija I Informatiki No.4, 29–43(2003).

- П. И. Когут, В. Н. Мизерный, Об усреднении экстремальных задач для линейных операторных уравнений Гаммерштейна, Доп. НАН України, 2002, N. 4. — C. 66–71.
- P. I. Kogut, On the homogenization of control systems with non-regular constraints, Нелинейные граничные задачи, 2002, N. 12. — C. 108–118.
- P. I. Kogut, G. Leugering, S-homogenization of optimal control problems in Banach spaces, Mathematische Nachrichten, 2002, V.233-234, pp. 141–169.

- P. I. Kogut, G. Leugering, On S-homogenization of an optimal control problem with control and state constraints, ZAA (J. Mathematical Methods in the Appl. Sci.), 2001, V. 20, No.2, pp. 395–429.
- P. I. Kogut, G. Leugering, Homogenization of optimal control problems in variable domains. Principle of fictitious homogenization, J. Asymptotic Analysis, 2001, V.26, pp. 37–72.
- P. I. Kogut, O. V. Museyko, On the S-Homogenization of Variational Inequalities in Banach Spaces, Доповiдi НАН України, 2001, N 5. — С. 73–76.
- P. I. Kogut, On the topological convergence of sets in partially ordered spaces, Нелинейные граничные задачи, 2001, No.11. — C. 94–97.
- П. И. Когут, О. В. Мусейко, S-усреднение линейных вариационных неравенств в банаховых пространствах. Регулярный случай., Проблемы управления и информатики, 2001,N.5. — C. 27–40; English transl. in J. Automatation and Information Sciences, 2001, V. 35, N. 9, pp. 31–42.
- П. И. Когут, В. Н. Мизерный, К усреднению оптимизационных задач для операторных уравнений Гаммерштейна, Проблемы управления и информатики, 2001, N.6. — C. 37–53; English transl. in J. Automatation and Information Sciences, 2001, V. 33, N. 11, pp. 36–50.
- P. I. Kogut, On the variational S-compactness of conditional minimization problems, J. Math. Sci., New York, 107(2001), , No. 2, 3736–3745 ; translated from Zh. Obchysl. Prykl. Mat., 83, 41-51 (1998).
- P. I. Kogut, V. S. Mel'nyk, On the weak compactness of bounded sets in Banach and loccally convex spaces, Ukrainian Mathematical Journal, 52(2001), No. 6, pp. 837–846; translated from Ukr Mat. Zh. Vol.52, No.6, 731-739(2000).
- P. I. Kogut, O. V. Museyko, On S-homogenization of linear variational inequalities in Banach spaces. Regular case, J. Automation and Information Sciences, 35(2001), I. 9, pp. 31-42; translated from Problemy Upravlenija I Informatiki No.5, 27–40(2001).
- P. I. Kogut, V. N. Mizernuy, On homogenization of optimization problems for operators equations of the Gammersain type, J. Automation and Information Sciences, 33(2001), i. 11, pp. 36–50; translated from Problemy Upravlenija I Informatiki No.6, 37–53(2001).

- П. И. Когут, В. С. Мельник, О слабой компактности ограниченных множеств в банаховых и локально выпуклых пространствах, Укр. математ. журн., 2000, Т. 52, N 6. — С. 731–739; English transl. in Ukrainian Mathematical Journal, 2001, V. 52, No. 6, pp. 837–846.
- П. І. Когут, Т. М. Рудянова, Про варiацiйну V-збiжнiсть загальних задач векторної оптимiзацiї, Доповiдi НАН України, 2000, N 5. — С. 92–97.
- П. И. Когут, Т. Н. Рудянова, Вариационный принцип V-выбора в общих задачах векторной оптимизации, Проблемы управления и информатики, 2000, N 3. — С. 78–87.
- П. І. Когут, Т. М. Рудянова, V-граничний аналiз векторнозначних вiдображень, Укр. математ. журн., 2000, Т. 52, N 12. — С. 1661–1675; English transl. in Ukrainian Mathematical Journal, 2000, V. 52, No. 12, pp. 1896–1912.
- П. И. Когут, О. В. Мусейко, Об одном подходе к усреднению нелинейных вариационных неравенств в банаховых пространствах, Сложные системы управления, 2000, T. 129.— С. 18–28.
- P. I. Kogut, Moro-Yosida approximation of conditional minimization problems and its limit properties, J. Math. Sci., New York 102(2000), No. 6, pp. 3775–3781; translated from Obchisl. Prikl. Mat. 81, 62–69 (1997).
- P. I. Kogut, T. N. Rudyanova, V-limit analysis of vector valued mappings, Ukrainian Mathematical Journal, 52(2000), No.12, pp. 1896–1912; translated from Ukr Mat. Zh., Vol.52, No.12, 1661–1675(2000).

- P. I. Kogut, On the structural representation of S-homogenized optimal control problems, Nonlinear Boundary Value Problems, 1999, V. 9. – pp. 84–91.
- П. И. Когут, Про топологiчну збiжнiсть надграфiкiв в частково впорядкованих векторних просторах, Математичнi студiї, 1999, N 12. — С. 161–170.

- P. I. Kogut, Γ-representation of variational S-limits of constrained minimization problems in normal Hausdorff spaces, Cybern. Syst. Anal. Vol. 34, 1998, No. 1, pp. 86–96 ; translated from Kibern. Sist. Anal. No.1, 104-118 (1998).
- P. I. Kogut, V. S. Mel'nik, On one class of extremum problems for nonlinear operator systems, Cybern. Syst. Anal., Vol. 34, 1998, No.6, pp.894–904; translated from Kibern. Sist. Anal. No.6, 121-132 (1998).
- П. И. Когут, О конструировании метрических пространств на множестве задач условной минимизации, Проблемы управления и информатики, 1998, N 1. – С. 37–47.
- П. И. Когут, О Γ-представлении вариационных S-пределов задач условной минимизации в нормальных хаусдорфовых пространствах, Кибернетика и системный анализ, 1998, N 1. – С. 104–118; English transl. in Cybern. Syst. Anal., 1998, V. 34, No. 1, pp. 86–96.
- P. I. Kogut, On the S-homogenization of optimal control problems, Допов. НАН України, 1998, N 1. – С. 132–136.
- A.I, Egorov, P.I. Kogut, Homogenization of optimal control problem in coefficients for linear elliptic equations, Zhongxing Jixie/Heavy Machinery, Issue 6, 1998, 89-104 [Show Abstract]
The properties of optimal control problems have been considered for elliptic systems, where a factor in rapidly varying coefficients appears as a function of control. The theorem of existence of optimal control has been proved for such problems. The necessary conditions for optimality and structure the homogenized of optimal control problem have been obtained.

- П. И. Когут, Об усреднении одной сингулярно возмущенной задачи оптимального управления, Кибернетика и вычислительная техника, 1998, Вып. 113. – С. 80–89.
- П. И. Когут, В. С. Мельник, Об одном классе экстремальных задач для нелинейных операторных систем, Кибернетика и системный анализ, 1998, N 6. – С. 121–132; English transl. in Cybern. Syst. Anal., 1998, V. 34, No. 6, pp.894-904.
- П. И. Когут, О вариационной S-компактности задач условной минимизации, Журнал обчислювальної та прикладної математики, 1998, N 1, Т. 83. — С. 41–51; English transl. in J. Math. Sci., New York 2001, V. 107, No. 2, pp. 3736-3745.
- P. I. Kogut, On averaging of a singularly perturbed optimal control problem, Cybernetics and Computing Technology, Discrete Control Systems, Vol. 113, 1998, pp. 59–72 ; translated from Kibernetika i Vitchislitelnaja Technika, N.113, 67–81(1998).

- П. И. Когут, Вариационная S-сходимость задач минимизации. 2. Топологические свойства S-пределов, Проблемы управления и информатики, 1997, N 3. – С. 78–90.
- П. I. Когут, Варiацiйна S-збiжнiсть задач мiнiмiзацiї та її геометрична iнтерпретацiя, Доп. НАН України, 1997, N 6. – С. 89–93.
- П. И. Когут, S-сходимость в теории усреднения задач оптимального управления, Укр. математ. журн., 1997, N 11. – С. 1488–1498; English transl. in Ukrainian Mathematical Journal, 1997, V. 49, No. 11, 1671–1682.
- П. И. Когут, S-сходимость задач условной минимизации и ее вариационные свойства, Проблемы управления и информатики, 1997, N 4. – С. 64–79.
- П. И. Когут, Aппроксимация Моро–Иосида задач условной минимизации и ее предельные свойства, Журнал обчислювальної та прикладної математики, 1997, N 1, Т. 81. – С. 62–69; English transl. in J. Math. Sci., New York, 2000, V. 102, No. 6, pp. 3775–3781.
- P. I. Kogut, S-convergence in the theory of homogenization of problems of optimal control, Ukrainian Mathematical Journal, Vol. 49, 1997, No. 11, pp. 1671-1682; translated from Ukr Mat. Zh., Vol.49, No.11, 1488-1498(1997).
- A. I. Egorov, P. I. Kogut, Homogenization of Optimal Control Problems in Coeffitients for Linear Elliptic Equations, J. of . Autom. Information Sci., Vol. 29, 1997, No. 4-5, pp. 37–46; ; translated from Problemy Upravlenija I Informatiki, No.6, 89–104(1995).

- П. И. Когут, Асимптотичний аналiз керованих параболiчних систем з еволюцiєю на границi, Доповiдi НАН України, 1996, N 10. – С. 99–103.
- П. И. Когут, Асимптотики вищих порядкiв розв'язку однiєї задачi оптимального керування розподiленою системою з швидкоосцилюючими коефiцiєнтами, Укр. математ. журнал, 1996, N 7, Т. 48. – С. 940–948; English transl. in Ukr. Math. J., 1996, V.48, N. 7, 1063–1073.
- П. И. Когут, Вариационная S-сходимость задач минимизации. 1. Определение и основные свойства, Проблемы управления и информатики, 1996, N 5. – С. 29–42.
- P. I. Kogut, High-order asymptotics og the solution of a problem of optimal control of a distributed system with rapidly oscillating coefficients, Ukr. Math. J., Vol. 48, 1996, N. 7, pp. 1063–1073; translated from Ukr Mat. Zh., Vol.48, No.7, 940–948(1996).
- P. I. Kogut, Variational convergence of optimal control problems for elliptic systems, J. Automation and Information Sciences, Vol. 28, 1996, N. 3-4, pp. 1063–1073; translated from Problemy Upravlenija I Informatiki, No.3, 72–84(1995).

- П. И. Когут, Вариационная сходимость задач оптимального управления для систем эллиптического типа, Проблеми управления и информатики, 1995, N 3. – С. 72–84.
- А. И. Егоров, П. И. Когут, Усреднение задач оптимального управления коэффициентами в линейных эллиптических уравнениях, Проблемы управления и информатики, 1995, N 6. – С. 89–104.
- P. I. Kogut, On averaging of a singularly perturbed optimal control problem, Cybernetics and Computing Technology discrete control systems, 1995, N 113. — pp. 59–72.
- P. I. Kogut, State exponential stability of a denumerable system of the Volterra integral equations, J. Automation and Information Sciencis, Vol. 27, 1995, No.5-6, pp. 60–75; translated from Avtomatika No.5, 38-47 (1991).

- A. I. Egorov, P. I. Kogut, On the state stability of a system of integro-differential equations of nonstationary aeroelasticity, J.of Mathematical Sciences (Springer, New York), Vol. 70, 1994, N. 1, pp. 1578–1585 ; translated from Vychislitel'naja i Prikladnaja Matematika, N.70, 112–121(1990).

- P. I. Kogut, State exponential stability of a denumerable system of the Volterra integral equations, Sov. J. Autom. Inf. Sci., Vol. 24, 1991, No. 5, pp. 37–46 ; translated from Avtomatika, No.5, 38–47 (1991).
- P.I.Kogut, Exponential stability by the state of the count system of the integral Volterra equation, Avtomatika, No.2, 1991, 38-47 [Show Abstract]
The stability problem of the infinite system of integral equations of Volterra type is considered. Necessary and sufficient conditions of exponential stability with respect to state for such systems are obtained. An extended version of the Lyapunov stability theory for such equations is presented. The proposed approach is based on the state space conception and notions of the Lyapunov second method. The conclusion is made that the obtained results may be applied to processes dynamics of which is described by the Volterra integral equations.

- П. И. Когут, Об экспоненциальной устойчивости по состоянию счетной системы интегральных уравнений Вольтерра, Автоматика, 1990, N 5. – С. 38–47; English transl. in J. of Automation and Information Sciences, 1995, Vol. 27, N 5–6. — P. 60–75 and in Sov. J. Autom. Inf. Sci., 1991, V. 24, No.5, 37–46.
- А. И. Егоров, П. И. Когут, Об устойчивости по состоянию системы интегро-дифференциальных уравнений нестационарной аэроупругости, Вычислительная и прикладная математика, 1990, N.70, pp. 112–121; English transl. in J.of Mathematical Sciences (Springer, New York), 70(1994), N. 1, 1578–1585.

- А. И. Егоров, П. И. Когут, Оптимальный синтез для уравнений нейтрального типа, Докл. АН УССР, Сер.А, 1989, N 5. – С. 64–67.