Asymptotic analysis of optimal control problems in thick multi-structures

P. I. Kogut

Faculty of Technical Cybernetics, Dniepropetrovsk National Technical University, Acad. Lazarjana str., 2, DIIT, 49010 Dniepropetrovsk, Ukraine

T. A. Mel’nik

Faculty of Mathematics & Mechanics, Taras Shevchenko University of Kyiv, Volodymyrska str., 64, 01033 Kyiv, Ukraine

Abstract

We study the asymptotic behaviour of an optimal control problem in a plane thick multi-structure of type 2 : 1 : 1 for linear elliptic equation subject to control constraints when the number N = a/ε of attached thin domains increases and their thickness ε decreases. We derive conditions under which the limiting (homogenized) optimal control problem can be made explicit.

Key words: optimal control problem, homogenization, thick multi-structure, variational convergence, extension operator

1 Introduction and the statement of the problem

Our aim is to investigate optimal control problems in thick multi-structures. In this paper we propose the abstract scheme to study such problems and consider the most simple problem to show the main ideas and to make the formulas and calculations clearer and shorter.

Email addresses: kogut@a-teleport.com ( P. I. Kogut ),

melnik@imath.kiev.ua ( T. A. Mel’nik).

Preprint submitted to Elsevier Science 16 June 2004

1.1 Thick multi-structures

A thick multi-structure (or thick junctions) Ω_{ε} of type k : m : d is the union of some domain Ω_{0} ∈ ℝ^{n} (junction’s body) and a large number of ε-periodically situated thin domains along some manifold (the joint zone) on the boundary of junction’s body. Here ε is a small parameter and the type k : m : d of the junction refers to the limiting dimensions of the body, the joint zone, and each attached thin domains. This classification was given by T.A. Mel’nyk and S.A. Nazarov.

Such multi-structures are prototypes of widely used engineering constructions such as long bridges on supports, frameworks of houses, industrial installations, spaceship grids as well as many other physical and biological systems with very distinct characteristic scales.

The goal of the study of boundary value problems in thick multi-structures is to describe the asymptotic behaviour of the solutions as ε → 0, i.e., when the number of attached thin domains increases and their thickness decreases.

Thick junctions have special character of the connectedness: there are points in a thick junction, which are at a short distance of order O(ε), but the length of all curves, which connect these points in the junction, is order O(1). As a result, many new difficulties appear in the asymptotic investigation of boundary value problems in thick multi-structures, namely,

• the loss of the coerciveness of differential operators in the limit passage as ε → 0;

• the absence of extension operators that would be bounded uniformly in ε in the Sobolev space W_{2}^{1};

• the power behaviour of junction-layer solutions at infinity;

and new qualitative properties appear in the asymptotic behaviour of the solutions. The type of a thick junction determines the scheme of investigation (see [6]-[8]). The reader can find in these papers extensive reviews on this theme.

The statement of the problem. A model plane thick multi-structure Ω_{ε} of type 2 : 1 : 1 consists of the junction’s body

Ω_{0} = {x ∈ ℝ^{2} : 0 < x_{1} < a, 0 < x_{2} < γ(x_{1}) },

and a large number N of the thin rods G_{ε} = ∪_{j=0}^{N−1}G_{ε}^{j},

G_{ε}^{j} = {x ∈ ℝ^{2} : |x_{1}−ε (j+1/2)| < εh/2, x_{2} ∈ (−l, 0]}, j = 0, 1, …, N−1,

i.e., Ω_{ε} = Ω_{0} ∪ G_{ε}. Here γ ∈ C^{1}([0, a]), 0 < γ_{0} = min_{[0,a]} γ; the fixed number h ∈ (0, 1); N is a large positive integer, therefore ε = a/N is a small discrete parameter, which characterizes the distance between the rods and their thickness.

Denote by Γ_{ε} the union of the bases

Γ_{ε}^{j} = {x ∈ ℝ^{2} : |x_{1} − ε (j + 1/2)| < εh/2, x_{2} = −l }, j = 0, 1, …, N − 1,

of the thin rods and by S_{ε} the union of their vertical sides

S_{ε}^{±}(j) = {x : x_{1} = ε(j + (1 ± h)/2), −l < x_{2} < 0}, j = 0, 1, …, N − 1.

In Ω_{ε} we consider the following optimal control problem: find (u_{ε}^{0}, y_{ε}^{0}) ∈ H^{1}(Γ_{ε}) × H^{1}(Ω_{ε}) such that

I_{ε} (u_{ε}^{0}, y_{ε}^{0}) = inf_{u_{ε},y_{ε}} I_{ε}(u_{ε}, y_{ε}),  (1)

where

I_{ε}(u_{ε}, y_{ε}) = ∫_{Ω_{0}}(y_{ε} − q_{0})^{2}dx + ∫_{Γ_{ε}} u_{ε}^{2} dx_{1},  (2)

the function q_{0} is some given function from L^{2}(Ω_{0}), the control function u_{ε} belongs to some closed subset U_{ε} of the Sobolev space H^{1}(Γ_{ε}) (see (7)) and the function y_{ε} ∈ H^{1}(Ω_{ε}) is a weak solution to the following boundary value problem

\begin{aligned}−Δ_{x} y_{ε}(x) & = & f_{ε}(x), & x ∈ Ω_{ε}, \\ ∂_{ν}y_{ε}(x) & = & −ε k_{0} y_{ε}(x), & x ∈ S_{ε}, \\ y_{ε}(x) & = & u_{ε}(x_{1}), & x ∈ Γ_{ε}, \\ ∂_{ν}y_{ε}(x) & = & 0, & x ∈ ∂Ω_{ε} \ (Γ_{ε} ∪ S_{ε}) . \end{aligned}  (3)

Here ∂_{ν} = ∂/∂ν is the outward normal derivative; the constant k_{0} ≥ 0.

We can regard without loss of generality that the right-hand side f_{ε} belongs to L_{2}(Ω_{1}), where Ω_{1} is the interior of \overline{Ω_{0}} ∪ \overline{D}, D = (0, a) × (−l, 0) is a rectangle that is filled up by the thin rods in the limit passage. We also assume that

f_{ε} → f_{0} in L_{2}(Ω_{1}) as ε → 0.  (4)

The aim of our research is to study the asymptotic behaviour of the problem (1)-(3) as ε → 0, i.e., when the number of attached thin rods increases and their thickness decreases. It should be stressed here that if the small parameter ε is changed then all objects of this control problem (the domain Ω_{ε}, the constraint set U_{ε}, the functional I_{ε} and the set, where we seek its infimum) are also changed.

2 The existence and uniqueness of the solution to problem (1)-(3)

First we recall that for every fixed value ε and for every function u_{ε} ∈ H^{1}(Γ_{ε}), in accordance with the main results of the theory of boundary value problems, there exists a unique function y_{ε} ∈ H^{1}(Ω_{ε}) such that the following integral identity holds

∫_{Ω_{ε}} ∇y_{ε} · ∇φ dx + ε k_{0} ∫_{S_{ε}} y_{ε} φ dx_{2} = ∫_{Ω_{ε}} f_{ε}(x) φ(x) dx ∀ φ ∈ H^{1}(Ω_{ε}; Γ_{ε}),

(5)

and the trace of y_{ε} equals u_{ε} on Γ_{ε}. The function y_{ε} is called a weak solution of the problem (3). Here H^{1}(Ω_{ε}; Γ_{ε}) = {φ ∈ H^{1}(Ω_{ε}) : φ = 0 on Γ_{ε}}. In addition, the solution y_{ε} satisfies the inequality

∥y_{ε}∥_{H^{1}(Ω_{ε})} ≤ c_{1} (∥f_{ε}∥_{L^{2}(Ω_{ε})} + ∥u_{ε}∥_{H^{1}(Γ_{ε})}) .  (6)

Note that the constant c_{1} in (6) is independent of the small parameter ε. This fact immediately follows from the following results.

Lemma 2.1.There exist positive constants C and ε_{0} such that for all values ε ∈ (0, ε_{0}) the following inequality holds

∫_{Ω_{ε}} ψ^{2}(x) dx ≤ C ∫_{Ω_{ε}} |∇ψ(x)|^{2} dx, ∀ ψ ∈ H^{1}(Ω_{ε}; Γ_{ε}).  (7)

Lemma 2.2. For ε small enough, the usual norm ∥ · ∥_{H^{1}(Ω_{ε})} in the Sobolev space H^{1}(Ω_{ε}) and the following norm

∥v∥_{ε,k_{0}} = (∫_{Ω_{ε}} |∇v|^{2} dx + ε k_{0} ∫_{S_{ε}} v^{2} dx_{2})^{1/2} (k_{0} > 0)

are equivalent for ε small enough, i.e., there exist constants C_{1} > 0, C_{2} > 0 and ε_{0} such that for all ε ∈ (0, ε_{0}) the following inequalities

C_{1}∥v∥_{H^{1}(Ω_{ε})} ≤ ∥v∥_{ε,k_{0}} ≤ C_{2}∥v∥_{H^{1}(Ω_{ε})}  (8)

hold for any function v ∈ H^{1}(Ω_{ε}).

Remark 2.1. Hereinafter all constants {c_{i}, C_{i}} in asymptotic inequalities are independent on the parameter ε.

Let U_{0} be the following subset of H^{1}(0, a)

U_{0} = {u ∈ H^{1}(0, a) : ∥u∥_{H^{1}(0,a)} ≤ C_{0}} .

For every ε = a/N we define the set of admissible controls U_{ε} as the restrictions of functions from U_{0} on Γ_{ε}, that is

U_{ε} = {u|_{Γ_{ε}} : ∀ u ∈ U_{0}} .  (9)

It is easy to see that for every u_{ε} ∈ U_{ε} we have u_{ε} ∈ H^{1}(Γ_{ε}) and ∥u_{ε}∥_{H^{1}(Γ_{ε})} ≤ C_{0}. Moreover, there exists an extension operator P_{ε} : H^{1}(Γ_{ε}) → H^{1}(0, a) such that

P_{ε}(u_{ε})|_{Γ_{ε}} = u_{ε}, P_{ε}(u_{ε}) ∈ U_{0}.  (10)

Of course, such an operator is not unique. In what follows, we shall call the sets U_{ε} as the regular constraint sets. Using the standard approach it is east to prove the following property.

Proposition 2.3. For every the set U_{ε} is convex and closed with respect to the weak topology of H^{1}(Γ_{ε}).

Definition 2.4. The pair ”control-state” (u_{ε}, y_{ε}) ∈ H^{1}(Γ_{ε}) × H^{1}(Ω_{ε}) is called admissible for the problem (1)-(3) if u_{ε} ∈ U_{ε} and y_{ε} is the weak solution to problem (3) whose trace on Γ_{ε} is equal to u_{ε}.

We denote by Ξ_{ε} the set of all admissible pairs, i.e.

Ξ_{ε} = \{(u_{ε}, y_{ε}) | \begin{matrix}u_{ε} ∈ U_{ε}, y_{ε} ∈ H^{1}(Ω_{ε}), \\ ∫_{Ω_{ε}} ∇y_{ε} · ∇ϕ dx + εk_{0} ∫_{S_{ε}} y_{ε}ϕ dx_{2} \\ = ∫_{Ω_{ε}} f_{ε}(x)ϕ(x) dx, ∀ ϕ ∈ H^{1}(Ω_{ε}, Γ_{ε}), \\ y_{ε}|_{Γ_{ε}} = u_{ε}. \end{matrix}\}  (11)

As it is immediately follows from (11), the sets Ξ_{ε} are non-empty and convex. Furthermore, for every fixed value ε the set Ξ_{ε} is τ_{ε}-closed, where by τ_{ε} we denote the product of the weak topologies for H^{1}(Γ_{ε}) and H^{1}(Ω_{ε}). In fact, let {(u_{ε,n}, y_{ε,n}) ∈ Ξ_{ε} : n ∈ ℕ} be a sequence of control-states such that u_{ε,n} → u_{ε}^{∗} weakly in H^{1}(Γ_{ε}) and y_{ε,n} → y_{ε}^{∗} weakly in H^{1}(Ω_{ε}) as n → +∞. Then u_{ε}^{∗} ∈ U_{ε} by Proposition 2.1. Now, passing to the limit in the integral identity (5) for y_{ε,n} and taking into account the compactness of the trace operator from H^{1}(Ω_{ε}) into L^{2}(Γ_{ε}), we conclude that y_{ε}^{∗} is a weak solution of problem (3) with the Dirichlet condition y_{ε}^{∗} = u_{ε}^{∗} on Γ_{ε}. Thus (u_{ε}^{∗}, y_{ε}^{∗}) ∈ Ξ_{ε}, i.e. the set Ξ_{ε} is τ_{ε}- closed.

Theorem 2.5.[Existence and uniqueness] For every value of ε the optimal control problem (2)-(3) has a unique solution, i.e., there exists a unique pair (u_{ε}^{0}, y_{ε}^{0}) ∈ Ξ_{ε} such that

I_{ε}(u_{ε}^{0}, y_{ε}^{0}) = inf_{(u_{ε},y_{ε})∈Ξ_{ε}} I_{ε}(u_{ε}, y_{ε}).

Proof. We attack this problem by means of the so-called direct methods of the Calculus of Variations. Let {(u_{ε,n}, y_{ε,n})} be any minimizing sequence for the problem (1)-(3). Since u_{ε,n} ∈ U_{ε} and due to the estimate (6) and condition (4), we obtain

∥y_{ε,n}∥_{H^{1}(Ω_{ε})} ≤ C_{1},

where the constant C_{1} > 0 doesn’t depend on n.

So, the sequence {(u_{ε,n}, y_{ε,n}) : n ∈ ℕ} ⊂ Ξ_{ε} is bounded in H^{1}(Γ_{ε}) × H^{1}(Ω_{ε}). Without loss of generality, we can suppose that u_{ε,n} → u_{ε}^{0} weakly in H^{1}(Γ_{ε}) and y_{ε,n} → y_{ε}^{0} weakly in H^{1}(Ω_{ε}) as n → +∞. Since the set Ξ_{ε} is closed with respect to the τ_{ε}-topology for H^{1}(Γ_{ε}) × H^{1}(Ω_{ε}), we have that (u_{ε}^{0}, y_{ε}^{0}) ∈ Ξ_{ε}.

The fact that the limit pair (u_{ε}^{0}, y_{ε}^{0}) is an optimal solution for original problem (1)-(3) immediately follows from the sequential τ_{ε}-semicontinuity property of the cost functional I_{ε}. Indeed, by construction of the sequence {(u_{ε,n}, y_{ε,n})}_{n∈ℕ} we have

inf_{(u,y) ∈ Ξ_{ε}} I_{ε}(u, y) = lim_{ε→ 0} I_{ε} (u_{ε,n}, y_{ε,n}) ≥ I_{ε}(u_{ε}^{0}, y_{ε}^{0}),

i.e. I_{ε}(u_{ε}^{0}, y_{ε}^{0}) = inf_{(u,y) ∈ Ξ_{ε}} I_{ε}(u, y). Taking into account that the cost functional I_{ε} is strictly convex, we have: (u_{ε}^{0}, y_{ε}^{0}) is a unique optimal pair. The theorem is proved.

3 Technique of homogenization

One of the main questions of our paper is devoted to the study of the asymptotic behaviour of the optimal control problem (1)-(3) as ε → 0. For this we represent the original problem (1)-(3) for various values of ε in the form of the sequence of corresponding constrained minimization problems

{⟨ inf_{(u,y) ∈Ξ_{ε}} I_{ε}(u, y)⟩ ; ε = a/N → 0} ,  (12)

where the cost functional I_{ε} : Ξ_{ε} → ℝ and the set of admissible pairs are defined in (2) and (11), respectively.

As follows from the representation (12) the question of defining an appropriate limit optimal control problem can be reduced to the analysis of the limit properties of the sequence (12). This will be done through the concept of variational convergence of constrained minimization problems [1],[3],[4],[5]. Before the short representation of the necessary formalism of such convergence we note that whatever the procedure for the limit analysis is, but it has to preserve the main variational property: both optimal pair and minimal value of the cost functional for the the original problem (1)-(3) converge to the corresponding characteristics of a limit optimal control problem as ε tends to zero.

Thus, it has to be chosen the most natural topology for the homogenization that takes into account the above mentioned variational property ([3], [5]), i.e with respect to which the sequence of optimal pairs {(u_{ε}^{0}, y_{ε}^{0}) ∈ Ξ_{ε}; ε → 0} is a sequentially compact. However, it should be stressed that in our case for every fixed ε admissible pairs (u_{ε}, y_{ε}) ∈ Ξ_{ε} belong to the corresponding functional spaces H^{1}(Γ_{ε}) × H^{1}(Ω_{ε}) depending on the small parameter ε.

The choice of topology for homogenization. In view of Khruslov’s classification, the scale of spaces {H^{1}(Γ_{ε}) × H^{1}(Ω_{ε})}_{ε>0} is not strongly connected with H^{1}(0, a)×H^{1}(Ω_{1}) (here Ω_{1} = int (\overline{Ω_{0}} ∪ \overline{D})) because there is not any uniformly bounded sequence of corresponding extension operators Λ_{ε} : H^{1}(Γ_{ε}) × H^{1}(Ω_{ε}) → H^{1}(0, a) × H^{1}(Ω_{1}). However, as it follows from the definition of the sets U_{ε}, for every ε = a/N and for any control function u_{ε} ∈ U_{ε} there exists an operator P_{ε} : H^{1}(Γ_{ε}) → H^{1}(0, a) such that

P_{ε}(u_{ε}) ∈ U_{0} = {u ∈ H^{1}(0, a) : ∥u∥_{H^{1}(0,a)} ≤ C_{0}} .

In view of this we introduce the following concept.

Definition 3.1. We shall say that a sequence of control functions {u_{ε} ∈ U_{ε}}_{ε>0} is weakly convergent to a function u^{∗} with respect to the space H^{1}(0, a) if some corresponding sequence of its images {P_{ε}(u_{ε})}_{ε>0} converges to u^{∗} weakly in H^{1}(0, a).

As for the correctness of this definition, we suppose that there are two different corresponding sequences {P_{ε}^{(1)}(u_{ε})}_{ε>0} and {P_{ε}^{(2)}(u_{ε})}_{ε>0} such that

P_{ε}^{(1)}(u_{ε}) → u_{1}^{∗} and P_{ε}^{(2)}(u_{ε}) → u_{2}^{∗} weakly in H^{1}(0, a) as ε → 0.

Let χ_{h} χ_{h}(·) be the 1-periodic function such that

χ_{h}(t) = \{ \begin{matrix}1, & |t − 1/2| < h/2, \\ 0, & h/2 ≤ |t − 1/2| ≤ 1/2. \end{matrix}  (13)

It is easy to see that

χ_{h}(x_{1}/ε) → ∫_{0}^{1} χ_{h}(t) dt = h weakly in L^{2}(0, a) as ε → 0.

Then, from the following integral identity

∫_{0}^{a} χ_{h}(x_{1}/ε)P_{ε}^{(1)}(u_{ε}) φ dx_{1} = ∫_{0}^{a} χ_{h}(x_{1}/ε)P_{ε}^{(2)}(u_{ε}) φ dx_{1}, ∀ φ ∈ H^{1}(0, a),

it follows in the limit passage as ε → 0 that

h ∫_{0}^{a} u_{1}^{∗} φ dx_{1} = h ∫_{0}^{a} u_{2}^{∗} φ dx_{1}, ∀ φ ∈ H^{1}(0, a).

This means that u_{1}^{∗} = u_{2}^{∗} a.e. in (0, a).

Denote by

ỹ_{ε}(x) := \begin{cases}y_{ε}(x), & x ∈ Ω_{ε}, \\ 0, & x ∈ Ω_{1} \ Ω_{ε}, \end{cases}

where y_{ε} ∈ H^{1}(Ω_{ε}) and let us introduce also the following notations

y_{ε}^{+}(x) := y_{ε}(x), x ∈ Ω_{0};  ỹ_{ε}^{−}(x) := ỹ_{ε}(x), x ∈ D.

Let {u_{e}^{0}, y_{ε}^{0}}_{ε>0} be a sequence of optimal solutions of the original problem. Then, in view of the initial suppositions and estimate (6) we conclude that for every ε > 0 it takes a place the following inequality

∥y_{ε}^{0}∥_{H^{1}(Ω_{0})} + ∥ỹ_{ε}^{0}∥_{L^{2}(D)} ≤ c_{1} (∥f_{ε}∥_{L^{2}(Ω_{1})} + C_{0}) ,

or in the another terms

∥y_{ε}^{0+}∥_{H^{1}(Ω_{0})} + ∥ \binom{}{y_{ε}^{0}}^{−} ∥_{L^{2}(D)} ≤ c_{1} (∥f_{ε}∥_{L^{2}(Ω_{1})} + C_{0}) .

Thus, the sequences \{y_{ε}^{0+}\}_{ε>0} and \{\binom{}{y_{ε}^{0}}^{−}\}_{ε>0} are sequentially compact with respect to the weak topologies of H^{1}(Ω_{0}) and L^{2}(D), respectively. Due to this fact we determine the following notions.

Definition 3.2. We say that the sequence {y_{ε} ∈ H^{1}(Ω_{ε})}_{ε>0} is weakly convergent to a function y_{∗} = (y_{∗}^{+}, y_{∗}^{−}) with respect to the space H^{1}(Ω_{0}) × L^{2}(D) as ε tends to zero (in symbols, y_{ε} ⇝ y_{∗}), if

• y_{ε}^{+} ⟶ y_{∗}^{+} weakly in H^{1}(Ω_{0});

• ỹ_{ε}^{−} ⟶ hy_{∗}^{−} weakly in L^{2}(D);

• y_{∗}^{+}(x_{1}, 0) = y_{∗}^{−}(x_{1}, 0) a.e. on (0, a).

Here the coefficient h is coming as the weak-* limit in L^{∞}(0, a) of the characteristic functions χ_{h}(·/ε) (see (13)).

Definition 3.3. We say that a sequence of admissible pairs {(u_{ε}, y_{ε}) ∈ Ξ_{ε}}_{ε>0} is w-convergent to a triplet (u, y^{+}, y^{−}) (in symbols, (u_{ε}, y_{ε}) ⟹^{w} (u, y^{+}, y^{−})) with respect to the space H^{1}(0, a) × H^{1}(Ω_{0}) × L^{2}(D) as ε tends to zero, if

(1) u_{ε} → u in the sense of Definition 3.1;

(2) y_{ε} ⇝ y = (y^{+}, y^{−}) in the sense of Definition 3.2.

The following assertion is concerned to the w-compactness property of such sequences.

Proposition 3.4. Let {(u_{ε}, y_{ε}) ∈ Ξ_{ε}}_{ε>0} be any sequence of admissible pairs for the original problem (1)–(3), (9). Then there exist a subsequence {(u_{ε′}, y_{ε′})}_{ε′>0} and a pair (u, y) such that (u_{ε′}, y_{ε′}) ⟹^{w} (u, y).

The definition of homogenized problem and its properties. First of all we introduce the convergence concept of the sets of admissible pairs Ξ_{ε} as ε → 0. As follows from the previous results, each of these sets Ξ_{ε} belongs to the corresponding Sobolev space H^{1}(Γ_{ε}) × H^{1}(Ω_{ε}). So, we put as the basis of such convergence the sequential version of the set w-convergence in Kuratowski’s sense.

Definition 3.5. We say that a set Ξ_{0} is the sequential w-limit in the Kuratowski’s sense (or K(w)-limit) of the sequence

{Ξ_{ε} ⊂ H^{1}(Γ_{ε}) × H^{1}(Ω_{ε})}_{ε>0}  (14)

with respect to the space H^{1}(0, a)×H^{1}(Ω_{0})×L^{2}(D) if the following conditions are satisfied:

(1) for every triplet (u, y^{+}, y^{−}) ∈ Ξ_{0} there exist a constant ε^{0} > 0 and a sequence {(u_{ε}, y_{ε})} w-converging to (u, y^{+}, y^{−}) such that (u_{ε}, y_{ε}) ∈ Ξ_{ε} for every ε ≤ ε;

(2) if {Ξ_{ε_{k}}}_{k∈ ℕ} is a subsequence of {Ξ_{ε}}_{ε>0} and {(u_{ε_{k}}, y_{ε_{k}})} is a sequence w-converging to (u, y^{+}, y^{−}) such that (u_{ε_{k}}, y_{ε_{k}}) ∈ Ξ_{ε_{k}} for every k ∈ ℕ, then (u, y^{+}, y^{−}) ∈ Ξ_{0}.

Let Ξ_{0} be a K(w)-limit of some sequence of sets (14), and let {(u_{ε}, y_{ε})}_{ε>0} be any sequence of pairs such that

(u_{ε}, y_{ε}) ∈ Ξ_{ε} for every ε = 1/N.

Then it is obvious that every w-cluster ”triplet” of this sequence belongs to the K(w)-limit set Ξ_{0}. This observation allows us to give the simplest example of K(w)-convergent sequence for which K(w)-limit can be computed by using just the Definition 3.5. Indeed, let {u_{ε}, y_{ε}}_{ε>0} be any sequence of admissible pairs for the original optimal control problem (1)–(3). We set

Ξ̂_{ε} = {(u_{γ}, y_{γ}) : γ = 1/k ≤ 1/n = ε} for every ε = 1/n.

Then \left\{{ \atop Ξ_{ε}}\right\} K(w)-converges to the set of all K(w)-cluster triplets of \{u_{ε}, y_{ε}\}_{ε>0}.

Let us now return to the main object of this subsection, namely, to the sequence of constrained minimization problems (12). To characterize a ”limit” minimization problem of such sequence as ε tends to zero we introduce the following concept:

Definition 3.6. We say that a minimization problem on H^{1}(0, a) × H^{1}(Ω_{0}) × L^{2}(D)

⟨ inf_{(u,y^{+},y^{−})∈ Ξ_{0}} I_{0}(u, y^{+}, y^{−})⟩  (15)

is the variational limit of the sequence (12) with respect to the w-convergence if the following conditions are satisfied:

(i) Ξ_{0} is a K(w)-limit of the sets {Ξ_{ε}}; (ii) for every triplet (u, y^{+}, y^{−}) ∈ Ξ_{0} and for every sequence of admissible

pairs {(u_{k}, y_{k})}_{k∈ ℕ}, which is w-converging to (u, y^{+}, y^{−}) and such that (u_{k}, y_{k}) ∈ Ξ_{ε_{k}} for some ε_{k} → 0 as k → ∞, it is

I_{0}(u, y^{+}, y^{−}) ≤ lim inf_{k→∞} I_{ε_{k}}(u_{k}, y_{k});  (16)

(iii) for every triplet (u, y^{+}, y^{−}) ∈ Ξ_{0} there exists a sequence {(u_{ε}, y_{ε})}_{ε>0} such that

(u_{ε}, y_{ε}) ∈ Ξ_{ε} for every ε > 0;

(u_{ε}, y_{ε}) ⟹^{w} (u, y^{+}, y^{−});

I_{0}(u, y^{+}, y^{−}) ≥ lim sup_{ε→0} I_{ε}(u_{ε}, y_{ε}).  (17)

Remark 3.1. Note that this definition can be interpreted as the natural extension of the well-known notion of Γ-convergence. Indeed, in the case when Ω_{ε} = Ω_{0} (i.e. D = ∅), the limit functional I_{0} : Ξ_{0} → R in (15) coincides with the sequential Γ-limit of the sequence {P_{Ξ_{ε}}I_{ε} : H^{1}(0, a) × H^{1}(Ω_{0}) → R)} with respect to the product of corresponding weak topologies, where it is denoted

P_{Ξ_{ε}}I_{ε}(u, y) = \{ \begin{matrix}I_{ε}(u, y), & (u, y) ∈ Ξ_{ε}; \\ +∞, & otherwise. \end{matrix}

We do not want here to enter into details of Γ-convergence theory, but only to emphasize that the variational properties of w-limit cost functional I_{0} : Ξ_{0} → R in the sense of Definition 3.6 are similar to the ones of Γ-limit. In fact, in the following theorem we will prove that the variational w-convergence of the sequence (12) to the problem (15) implies the convergence of the minimum values of I_{ε} on Ξ_{ε} to the minimum one of I_{0} on Ξ_{0}, and moreover, we’ll prove that in this case every w-cluster ”point” of the sequence of the minimizers of I_{ε} is a minimizer of I_{0}.

Theorem 3.7. Assume that the constrained minimization problem (15) is the variational w-limit of the sequence (12) and this problem has a non-empty set of solutions. Furthermore, for every ε > 0, let (u_{ε}^{0}, y_{ε}^{0}) ∈ Ξ_{ε} be a solution of the corresponding problem (12) such that this one is relatively w-compact. Then every its w-cluster triplet (u^{0}, (y^{0})^{+}, (y^{0})^{−}) is a minimizer of I_{0} on Ξ_{0}, and furthermore

I_{0} (u^{0}, (y^{0})^{+}, (y^{0})^{−}) = lim_{k→∞} I_{ε_{k}}(u_{ε_{k}}^{0}, y_{ε_{k}}^{0}),  (18)
inf_{(u, y^{+}, y^{−})∈ Ξ_{0}} I_{0}(u, y^{+}, y^{−}) = lim_{ε→0} inf_{(u,y)∈ Ξ_{ε}} I_{ε}(u, y).  (19)

Proof. Let {(u_{ε_{k}}^{0}, y_{ε_{k}}^{0})}_{k∈ ℕ} be a w-convergent subsequence of the sequence of minimizers, and let (u^{∗}, (y^{∗})^{+}, (y^{∗})^{−}) be its w-limit. Then, by Definition 3.5, we have (u^{∗}, (y^{∗})^{+}, (y^{∗})^{−}) ∈ Ξ_{0}. Moreover, due to the property (ii) of Definition 3.6, one gets

lim inf_{k→∞} min_{(u,y)∈ Ξ_{ε_{k}}} I_{ε_{k}}(u, y) = lim inf_{k→∞} I_{ε_{k}}(u_{ε_{k}}^{0}, y_{ε_{k}}^{0})

≥ I_{0}(u^{∗}, (y^{∗})^{+}, (y^{∗})^{−}) ≥ min_{(u, y^{+}, y^{−})∈ Ξ_{0}} I_{0}(u, y^{+}, y^{−}).  (20)

Let (u^{0}, (y^{0})^{+}, (y^{0})^{−}) ∈ Ξ_{0} be any solution of the limit problem (15). It follows that there exist a constant ε^{0} > 0 and a sequence {(u_{ε}, y_{ε})}, w-converging to (u^{0}, (y^{0})^{+}, (y^{0})^{−}), such that (u_{ε}, y_{ε}) ∈ Ξ_{ε} for all ε ≤ ε^{0}, and

I_{0}(u^{0}, (y^{0})^{+}, (y^{0})^{−}) ≥ lim sup_{ε→0} I_{ε}(u_{ε}, y_{ε}).

Using this fact we have

min_{(u, y^{+}, y^{−})∈Ξ_{0}} I_{0}(u, y^{+}, y^{−}) = I_{0}(u^{0}, (y^{0})^{+}, (y^{0})^{−})

≥ lim sup_{ε→0} I_{ε}(u_{ε}, y_{ε}) ≥ lim sup_{ε→0} min_{(u,y) ∈Ξ_{ε}} I_{ε}(u, y)

≥ lim sup_{k→∞} min_{(u,y) ∈ Ξ_{ε_{k}}} I_{ε_{k}}(u, y) = lim sup_{k→∞} I_{ε_{k}}(u_{ε_{k}}^{0}, y_{ε_{k}}^{0}).  (21)

Therefore, in view of (20) we have obtained the following inequality

lim inf_{k→∞} I_{ε_{k}}(u_{ε_{k}}^{0}, y_{ε_{k}}^{0}) ≥ lim sup_{k→∞} I_{ε_{k}}(u_{ε_{k}}^{0}, y_{ε_{k}}^{0}).

In a result, combining (20) and (21), we may conclude

I_{0}(u^{∗}, (y^{∗})^{+}, (y^{∗})^{−}) = min_{(u, y^{+}, y^{−})∈ Ξ_{0}} I_{0}(u, y^{+}, y^{−}),

I_{0}(u^{∗}, (y^{∗})^{+}, (y^{∗})^{−}) = lim_{ε→0} min_{(u,y) ∈ Ξ_{ε}} I_{ε}(u, y).

Thus we have obtained the required properties. This proof is complete.

4 Convergence theorems

Now as ε → 0 it is natural to ask what happens to the family (12) in the limit. Does there exist a limit of such sequence, and, if so, can it be determined?

What form form can it take and what kind of optimal control problem does this limit correspond?

At first we note that in accordance with Lemmas 2.1—2.2, and Proposition 3.4 the sequence of admissible pairs {Ξ_{ε} ⊂ H^{1}(Γ_{ε}) × H^{1}(Ω_{ε})}_{ε>0} is such that

sup_{ε>0} sup_{(u_{ε},y_{ε})∈ Ξ_{ε}} [∥u_{ε}∥_{H^{1}(Γ_{ε})} + ∥y_{ε}∥_{H^{1}(Ω_{ε})}] ≤ C_{0} + C_{1}.

From this, using the notation from previous section, we immediately have

sup_{ε>0} sup_{(u_{ε},y_{ε})∈ Ξ_{ε}} [∥P_{ε}(u_{ε})∥_{H^{1}(0,a)} + ∥y_{ε}^{+}∥_{H^{1}(Ω_{0})} + ∥ỹ_{ε}^{−}∥_{L^{2}(D)}] ≤ C_{0} + C_{1}.

Thus the images of the sets Ξ_{ε} in the space H^{1}(0, a) × H^{1}(Ω_{0}) × L^{2}(D) are uniformly bounded. Since the space H^{1}(0, a) × H^{1}(Ω_{0}) × L^{2}(D) is separable and reflexive, by Kuratowski compactness theorem, there exists a subsequence of {Ξ_{ε}}, still denoted by {Ξ_{ε}}, which K(w)-converges to a set Ξ_{0} ⊂ H^{1}(0, a)× H^{1}(Ω_{0}) × L^{2}(D). So, in order to recovery the variational limit (15) we have to identify the limit set Ξ_{0} and cost functional I_{0}. As follows from the definition of K(w)-limit, we have to be able to pass to the limit in the integral identity (5) for problem (3) as ε → 0. But, till now it was not known how to pass to the limit with the Robin conditions on S_{ε}. Nevertheless, since the lateral boundaries S_{ε} of the thin rods are plane, it is possible to use for this the approach of the paper [2].

We now in a position to state the main result of this paper.

Theorem 4.1. For the sequence of constrained minimization problems (12) there exists the unique variational limit with respect to the w-convergence

⟨ inf_{(u,y^{+},y^{−})∈ Ξ_{0}} I_{0}(u, y^{+}, y^{−})⟩ ,  (22)

where the set Ξ_{0} ⊂ H^{1}(0, a) × H^{1}(Ω_{0}) × L^{2}(D) and cost functional I_{0} have the following representation

Ξ_{0} = \{(u_{0}, v_{0}^{+}, v_{0}^{−}) | \begin{matrix}−Δ_{x}v_{0}^{+}(x) = f_{0}(x), x ∈ Ω_{0}, \\ −h ∂_{x_{2}x_{2}}^{2}v_{0}^{−}(x) + 2k_{0}v_{0}^{−}(x) = h f_{0}(x), x ∈ D, \\ ∂_{ν}v_{0}^{+}(x) = 0, x ∈ ∂Ω_{0} \ [0, a], \\ v_{0}^{+}(x_{1}, 0) = v_{0}^{−}(x_{1}, 0), x_{1} ∈ [0, a], \\ ∂_{x_{2}}v_{0}^{+}(x_{1}, 0) = h ∂_{x_{2}}v_{0}^{−}(x_{1}, 0), x_{1} ∈ [0, a], \\ v_{0}^{−}(x_{1}, −l) = u^{∗}(x_{1}), x_{1} ∈ [0, a]; \end{matrix}\} (23)

I_{0}(u, v_{0}^{+}, v_{0}^{−}) = ∫_{Ω_{0}} (v_{0}^{+} − q_{0})^{2} dx + h ∫_{0}^{a} u^{2} dx_{1}.  (24)

Moreover, the following statements hold:

(1) the sequence of optimal pairs {(u_{ε}^{0}, y_{ε}^{0} ∈ Ξ_{ε})}for the original problem (1)– (3) w-converges to (u_{0}, v_{0}^{+}, v_{0}^{−}) as ε → 0, where (u_{0}, v_{0}^{+}, v_{0}^{−}) is an optimal solution for limit problem (22);

(2) inf_{(u_{ε},y_{ε})∈ Ξ_{ε}} I_{ε}(u_{ε}, y_{ε}) ⟶ min_{(u,v^{+},v^{−})∈ Ξ_{0}} I_{0}(u, v^{+}, v^{−}) = I_{0}(u_{0}, v_{0}^{+}, v_{0}^{−}).

The proof of this theorem is based on verification of all conditions in Definition 3.6 with help of the following result:

Lemma 4.2.Let {u_{ε}}_{ε>0} be any sequence of admissible controls which is convergent to a function u^{∗} with respect to the space H^{1}(0, a). Let {y_{e}} be the corresponding solutions of the problem (3). Then (u_{ε}, y_{ε}) ⇒^{w} (u^{∗}, v_{0}^{+}, v_{0}^{−}) as ε → 0, where

v_{0}(x) = \begin{cases}v_{0}^{+}(x), & x ∈ Ω_{0}, \\ v_{0}^{−}(x), & x ∈ D, \end{cases}  (25)

is the unique weak solution to the following limit problem

\begin{aligned}−Δ_{x}v_{0}^{+}(x) & = & f_{0}(x), & x ∈ Ω_{0}, \\ −h ∂_{x_{2}x_{2}}^{2}v_{0}^{−}(x) + 2k_{0}v_{0}^{−}(x) & = & h f_{0}(x), & x ∈ D, \\ ∂_{ν}v_{0}^{+}(x) & = & 0, & x ∈ ∂Ω_{0} \ [0, a], \\ v_{0}^{+}(x_{1}, 0) & = & v_{0}^{−}(x_{1}, 0), & x_{1} ∈ [0, a], \\ ∂_{x_{2}}v_{0}^{+}(x_{1}, 0) & = & h ∂_{x_{2}}v_{0}^{−}(x_{1}, 0), & x_{1} ∈ [0, a], \\ v_{0}^{−}(x_{1}, −l) & = & u^{∗}(x_{1}), & x_{1} ∈ [0, a]. \end{aligned}  (26)

Proof. For this we shall use the fact that the boundaries S_{ε} are plane. Then the following relation ∂_{x_{2}}(ỹ_{ε}^{−}) = { \atop ∂_{x_{2}}(y_{ε}^{−})} holds in D. In the case, when the boundaries S_{ε} are not plane, we should construct special extension operator P_{Ω_{ε}} : H^{1}(Ω_{ε}) ↦ H^{1}(Ω_{1}).

Because of u_{ε} ∈ U_{ε} and due to (4) and (6), there exists such a subsequence {ε′} of {ε} (still denote by {ε}) such that

y_{ε}^{+} → v_{0}^{+} weakly in H^{1}(Ω_{0}),
ỹ_{ε}^{−} → v^{∗} = h(h^{−1}v^{∗}) =: h v_{0}^{−} weakly in L^{2}(D),

{ \atop ∂_{x_{i}}(y_{ε}^{−})} → γ_{i} weakly in L^{2}(D), i = 1, 2, as ε → 0.

With the help of the notations introduced above, we can rewrite integral identity (5) in the following view

∫_{Ω_{0}} ∇y_{ε}^{+} · ∇φ dx + ∫_{D}({ \atop ∂_{x_{1}}(y_{ε}^{−})} ∂_{x_{1}}φ + { \atop ∂_{x_{2}}(y_{ε}^{−})} ∂_{x_{2}}φ) dx +

+ \frac{2k_{0}}{h} ∫_{D} ỹ_{ε}^{−}φ dx − ε \frac{2k_{0}}{h} ∫_{G_{ε}} Y (x_{1}/ε)(y_{ε} ∂_{x_{1}}φ + φ ∂_{x_{1}}y_{ε}) dx =

= ∫_{Ω_{0}} f_{ε}φ dx + ∫_{D} χ_{h}(x_{1}/ε)f_{ε}φ dx ∀ φ ∈ H^{1}(Ω_{1}; Γ_{0}). (27)

Here Y (t) = −t + \frac{1}{2} + [t] ([t] is the integral part of t), ∂_{x_{1}}v = \frac{∂v}{∂x_{1}} , Γ_{0} = \{x : x_{1} ∈ (0, a), x_{2} = −l}, H^{1}(Ω_{1}; Γ_{0}) = {φ ∈ H^{1}(Ω_{1}) : φ = 0 on Γ_{0}}. Passing in the limit in (27) as ε → 0 and taking into account that the second integral in the second line in (27) is order O(ε), we obtain

∫_{Ω_{0}} ∇v_{0}^{+} · ∇φ dx + ∫_{D}(γ_{1} ∂_{x_{1}}φ + γ_{2} ∂_{x_{2}}φ) dx + 2k_{0} ∫_{D} v_{0}^{−}φ dx =

= ∫_{Ω_{0}} f_{0} φ dx + h ∫_{D} f_{0} φ dx ∀ φ ∈ H^{1}(Ω_{1}; Γ_{0}). (28)

To find γ_{2} we pass to the limit in the following identity

∫_{D} { \atop ∂_{x_{2}}(y_{ε}^{−})} ψ dx = − ∫_{D} ỹ_{ε}^{−} ∂_{x_{2}}ψ dx ∀ ψ ∈ C_{0}^{∞}(D)

as ε → 0. As a result, we get

∫_{D} γ_{2} ψ dx = −h ∫_{D} v_{0}^{−} ∂_{x_{2}}ψ dx ∀ ψ ∈ C_{0}^{∞}(D).

This means that γ_{2} = h ∂_{x_{2}}v_{0}^{−} a.e. in D. To find γ_{1} we consider the integral identity (3) with such a test-function

ϕ(x) = \begin{cases}0, & x ∈ Ω_{0}, \\ ε Y (x_{1}/ε) ψ(x), & x ∈ G_{ε}, \end{cases}

where ψ is an arbitrary function from C_{0}^{∞}(D). It is easy to verify that ϕ ∈ H^{1}(Ω_{ε}) and ∇ϕ(x) = (−ψ(x) + εY (x_{1}/ε)∂_{x_{1}}ψ(x) , εY (x_{1}/ε)∂_{x_{2}}ψ(x)). We have

∫_{G_{ε}} ∂_{x_{1}}y_{ε} ψ dx = ε ∫_{G_{ε}} Y (x_{1}/ε)(∇y_{ε} · ∇ψ − f_{ε} ψ) dx,

from were |∫_{G_{ε}} ∂_{x_{1}}y_{ε} ψ dx| ≤ ε C_{1}(∥y_{ε}∥_{H^{1}(Ω_{ε})} + ∥f_{ε}∥_{L^{2}(Ω_{1})})∥ψ∥_{H^{1}(D)} ≤ ε C_{2}∥ψ∥_{H^{1}(D)}. Thus, on the one hand

lim_{ε→0} ∫_{G_{ε}} ∂_{x_{1}}y_{ε} ψ dx = 0 ∀ ψ ∈ C_{0}^{∞},

from the other hand

lim_{ε→0} ∫_{G_{ε}} ∂_{x_{1}}y_{ε} ψ dx = lim_{ε→0} ∫_{D} { \atop ∂_{x_{1}}(y_{ε}^{−})} ψ dx = ∫_{D} γ_{1} ψ dx ∀ ψ ∈ C_{0}^{∞}.

This means that γ_{1} = 0 a.e. in D.

So, the function v_{0} satisfies the following identity

∫_{Ω_{0}} ∇v_{0}^{+} · ∇φ dx + h ∫_{D} ∂_{x_{2}}v_{0}^{−} ∂_{x_{2}}φ dx + 2k_{0} ∫_{D} v_{0}^{−}φ dx =

= ∫_{Ω_{0}} f_{0}φ dx + h ∫_{D} f_{0}φ dx ∀ φ ∈ H^{1}(Ω_{1}; Γ_{0}). (29)

Now let us show that

v_{0}^{−} = u^{∗} a.e. on Γ_{0}.

Since the trace of y_{ε} is equal to u_{ε} on Γ_{ε}, we have

ỹ_{ε}^{−}(x_{1}, −l) = χ_{h}(x_{1}/ε) P_{ε}(u_{ε})(x_{1}), x_{1} ∈ (0, a),  (30)

where P_{ε}(u_{ε}) ∈ U_{0} and due to the condition of the theorem P_{ε}(u_{ε}) → u^{∗} weakly in H^{1}(0, a) as ε → 0. Therefore, the right-hand side of (30) weakly in L^{2}(0, a) converges to h u^{∗} as ε → 0. To find the limit of the left-hand side, we use the following identity

∫_{0}^{a} ỹ_{ε}^{−}(x_{1}, −l) ψ(x_{1}) dx_{1} = l^{−1} ∫_{0}^{a} ∫_{−l}^{0} ỹ_{ε}^{−}(x_{1}, x_{2}) ψ(x_{1}) dx_{2}dx_{1} −

− l^{−1} ∫_{0}^{a} ∫_{−l}^{0} { \atop ∂_{x_{2}}(y_{ε}^{−})} x_{2} ψ(x_{1})dx_{2}dx_{1}, ∀ ψ ∈ C_{0}^{∞}(0, a), (31)

from where it follows that the weak limit of ỹ_{ε}^{−}(·, −l) is equal to v_{0}^{−}. Similarly, we can show that v_{0}^{+}(x_{1}, 0) = v_{0}^{−}(x_{1}, 0), x_{1} ∈ (0, a).

The identity (29) is the corresponding integral identity for problem (26) in the anizotropic Sobolev space

V (Ω_{1}; Γ_{0}) = {v ∈ L^{2}(Ω_{1}) : ∂_{x_{2}}v ∈ L^{2}(Ω_{1}), ∂_{x_{1}}v ∈ L^{2}(Ω_{0}), v(x_{1}, −l) = 0}

with the scalar product

(u, v)_{V} = ∫_{Ω_{0}} ∇u · ∇v dx + h ∫_{D} ∂_{x_{2}}u ∂_{x_{2}}v dx + 2k_{0} ∫_{D} u v dx.

It should be stressed here that functions from this space have trace on any horizontal interval in \overline{D} and the space H^{1}(Ω_{1}; Γ_{0}) is dense in V (Ω_{1}; Γ_{0}).

So, a function v_{0} ∈ V (Ω_{1}) defined in (25) is a weak solution to problem (26) if it satisfies the following identity

(v_{0}, φ)_{V} = ∫_{Ω_{0}} f_{0}φ dx + h ∫_{D} f_{0}φ dx ∀ φ ∈ V (Ω_{1}; Γ_{0})

and the trace of v_{0} on Γ_{0} equals u^{∗}. Using standard Hilbert space methods, we can state that there exists a unique weak solution v_{0} ∈ V (Ω_{1}) to problem (26). Due to the uniqueness of the solution to problem (26), the above reasoning holds for any subsequence of {ε} chosen at the beginning of the proof. Therefore, the lemma is proved.

It is easy to see that the limit minimization problem (22) can be rewritten in the form of optimal control problem (OCP), which is called homogenized OCP.

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