- 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.

- 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.

- 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.

- 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, 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.

- 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.

- 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.

- 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.

- 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, 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.

- 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, 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.

- 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.

- 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, 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.

- 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.

- 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.

- 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.

- 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.

- 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.

- 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.

- 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.

- 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.

- 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.

- 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.

- 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.

- 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, 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, 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.
- 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, 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, 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, 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. 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, 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, 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 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. I. Когут, Варiацiйна S-збiжнiсть задач мiнiмiзацiї та її геометрична iнтерпретацiя, Доп. НАН України, 1997, N 6. – С. 89–93.
- 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.
- 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).

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