Approximate Homogenized Synthesis for Distributed Optimal Control Problem with Superposition Type Cost Functional

In this paper, we consider the optimal control problem in the feedback form (synthesis) for a parabolic equation with rapidly oscillating coefficients and not-decomposable quadratic cost functional with superposition type operator. In general, it is not possible to find the exact formula of optimal synthesis for such a problem because the Fourier method can’t be directly applied. But transition to the homogenized parameters greatly simplifies the structure of the problem. Assuming that the problem with the homogenized coefficients already admits optimal synthesis form, we ground approximate optimal control in the feedback form for the initial problem. We give an example of superposition operator for specific conditions in this paper.


Introduction
In this work, we focus on the finding effective methods of control for complicated infinite-dimensional systems, initiated in the works [1], [2], [3].Finding control in the feedback form or synthesis plays important role here.In [4] it was proposed and substantiated a procedure for constructing approximate optimal synthesis for a wide class of distributed processes in micro-inhomogeneous medium, investigated earlier in [5].We use some known facts on G-convergence theory from [6], [7].In this paper, we consider the optimal control problem in the feedback form for a parabolic equation with rapidly oscillating coefficients and not-decomposable quadratic cost functional with superposition or Nemyckii type operator.In general, to find the exact formula of optimal synthesis is not possible for such a problem because we can not directly apply the Fourier method.But the transition to the homogenized parameters greatly simplifies the structure of the problem.Assuming that the problem with the homogenized coefficients already admits optimal synthesis form, we ground approximate optimal control in the feedback form for the initial problem.We give an example of Nemyckii operator for specific conditions in this paper.

Setting of the problem
Let Ω ⊂ R n be a bounded domain and let ε ∈ (0, 1) be a small parameter.In the cylinder Q = (0, T ) × Ω controlled process {y, u} is described by the problem 234 with a cost functional where ) , a is measurable, symmetric, periodic matrix, satisfying conditions of uniform ellipticity and boundedness: ) is a Caratheodory function, and there exist functions , and constant C > 0, independent of ε ∈ (0, 1), such that for all ξ ∈ R and almost all x ∈ Ω the following inequalities hold Under these conditions [9] Nemyckii's operator q ε (x, •) : L 2 (Ω) → L 2 (Ω) is continuous.Hence, by conditions (1.3), (1.4) and properties of solutions of the problem (1.1) (see Lemma 2.1) we obtain that the problem (1.1), (1.2) has solution {ȳ ε , ūε } (optimal process) in class W (0, T ) × L 2 (Q), where W (0, T ) is a class of functions y ∈ L 2 ( 0, T ; H 1 0 (Ω) ) , which have generalized derivatives with respect to t from class L 2 ( 0, T ; H −1 (Ω) ) [1].In general case, we are not able to find the exact optimal feedback law for the problem (1.1), (1.2).However, in many cases [5] a transition to homogenized parameters simplifies the structure of the problem.We will assume that the problem with homogenized coefficients already admits optimal feedback control of the form u[t, x, y (t, x)].
The main goal of this paper is to prove the fact that the form u[t, x, y (t, x)] realizes an approximate feedback control in initial problem (1.1), (1.2), i. e. for any η > 0 for ε > 0 small enough, where y ε is a solution of the problem (1.1), (1.2) with control u[t, x, y ε ].

Main results
We shall use ∥ • ∥ to denote the norm and ( • , • ) to denote the inner product in L 2 (Ω).Let us assume that there exists a Caratheodory function uniformly for |ξ| ≤ r. (2.1) We refer to the following problem as an homogenized one for the problem (1.1), (1.2).Here a constant matrix a 0 is homogenized for a ε [6], In further arguments we will use the following result about convergence of parabolic operators which is the consequence of G-convergence of there exists a measurable map u : there exist constants such that for all t ∈ [0, T ] and y, z ∈ L 2 (Ω) the inequalities hold (2.7) Before we formulate the main result, we give a typical example of the function q ε : Ω × R → R, for which the conditions (1.4), (2.1), (2.5) -(2.7) hold. Example.
The last formula yields Further, using feedback law (2.6), we consider the problem (2.9) Under conditions (2.7) the problem (2.8) has a unique solution y ε in the class W (0, T ) [8].
The main result of this article is the following theorem.

Proof
At the beginning we show that as ε → 0 both the solution y ε of the problem (2.9) and the solution ȳε of the problem (1.1), (1.2) tend to y in some sense, where {y, u} is the optimal process in the problem (2.2), (2.3).We consider first the problem (2.9).For almost all (a.a.) t ∈ (0, T ), the following estimate holds for the solution y ε (2.11) Using Gronwall's Lemma, from (2.11) we obtain that the sequence {y ε } is bounded in W (0, T ).Then, by Compactness Lemma [8] there exists a function z ∈ W (0, T ), such that along subsequence ) . (2.12) From this and from (2.7) we derive that (2.13) From Lemma 2.1 we obtain that z is a solution of the problem (2.9) with operator A 0 and initial data y 0 , and as (2.14) Since the optimal control problem (2.2), (2.3) has a unique solution {y, u} and formula u(t, x) = u[t, x, y(t, x)] is valid for control u, then y is a solution of the problem (2.9) with operator A 0 and initial data y 0 .However, this problem also has a unique solution, so y ≡ z, and moreover, the convergences (2.12) -(2.14) hold as ε → 0 (not only along subsequence).

Conclusion
In this paper, we investigated the optimal control problem for a parabolic equation with rapidly oscillating coefficients and a special type cost functional with superposition or, another words, Nemyckii type operator.For good understanding we give an example of Nemyckii operator under specific conditions in this paper.We describe a problem with the homogenized coefficients, corresponding to the initial optimal control problem.Under some known facts on G-convergence theory and assuming that it already admits optimal synthesis form, we ground approximate optimal control in the feedback form for the initial problem.
where y ε is a solution of the problem (1.1) with control u ε , y is a solution of the problem (2.2) with control u.
Let us assume that the following conditions hold: the problem (2.2), (2.3) has a unique solution {y, u} ; June 2018and together with weak convergence this provides a strong convergenceūε → u in L 2 (Q), ε → 0.(2.22)Finally, from (2.15) and (2.20) we get the statement of the theorem.