Application of the Averaging Method to the Optimal Control Problem of Non-Linear Differential Inclusions on the Finite Interval

: In this paper, we use the averaging method to ﬁnd an approximate solution for the optimal control of non-linear differential inclusions with fast-oscillating coefﬁcients on a ﬁnite time interval.


Introduction
It is known that the averaging method is one of the most effective tools for solving various optimal control problems for differential equations [1][2][3][4] as well as for differential inclusions with fast-oscillating coefficients [5][6][7].Of the many published papers in which similar problems are considered (e.g., minimax, robust, and adaptive control), we mention [8][9][10].The Krasnoselski-Krein theorem [2] and its multi-valued analogue [11] play an essential role for the investigation of the above-mentioned problems.When dealing with multi-valued mappings, one faces specific problems; nevertheless, the application of the well-developed averaging method for the optimal control problems is possible in this case.
In the present paper, we consider the optimal control problem of a non-linear system of differential inclusions with fast-oscillating parameters.First, we prove the existence of solutions for the initial perturbed optimal control problem and the corresponding problem with averaged coefficients.Then, we prove that the optimal control of the problem with averaging coefficients can be considered as "approximately" optimal for the initial perturbed one.

Statement of the Problem
Let us consider an optimal control problem as follows.
Here ε > 0 is a small parameter, x : [0, T] → R is an unknown phase variable, u : [0, T] → R m is an unknown control function, X : R Assume that uniformly with respect to x for every u ∈ R m that we have the following: where the limits for multi-valued functions are considered in the sense of [12,13], dist H is the Hausdorff metric, Y : R n × R m → conv(R n ), and the integral of multi-valued function is considered in the sense of Aumann [14].We consider the following problem with an averaged right-hand side.
Under the natural assumptions on X, L, Φ, and U, we will show that problems (1) and ( 3) have solutions {x ε , u ε } and {y, u}, respectively: and up to a subsequence.
In what follows, we consider the problem of finding an approximate solution of (1) by transitions to the problem with averaged coefficients.We note that such transitions can essentially simplify the problem.

Assumptions and Notations
Let Q = {t ≥ 0, x ∈ R n , u ∈ R m } and assume that the following assumptions hold.Assumption 1. Mapping t, x, u → X(t, x, u) is continuous in Hausdorff metric.
We will consider the next multi-valued analogue of the Krasnoselsky-Krein theorem.
Theorem 1 ([2,5,11,16]).Suppose the following conditions are fulfilled for the differential inclusion with multi-valued mapping F(t, x, λ) taking values in conv(R n ) (that is, the subspace from conv(R n ), which consists of convex sets), defined for 0 ≤ t ≤ T; x ∈ D, D is a bounded domain in R n ; λ ∈ Λ, where Λ is a set of values for parameter λ for which λ 0 ∈ Λ is the limit point.
(1) Multi-valued mapping F(t, x, λ) is uniformly bounded, continuous on t, uniformly continuous on x with respect to t, and λ: ∀ε (2) Multi-valued mapping F(t, x, λ) is integrally continuous on λ at point λ 0 ; that is, for 0 ≤ t 1 ≤ t 2 ≤ T and for any x ∈ D, we have the following: where we consider integrals in the sense of Aumann [14].
(3) Solutions x(t, λ 0 ) of the inclusion satisfying the condition x(0, λ 0 ) = x 0 ∈ D 1 ⊂ D are defined for 0 ≤ t ≤ T and belong to domain D together with some ρ-neighborhood.
Remark 3. The concept of an integral continuity plays a key role in the investigation of the considered optimal control problem using an averaging method.It is known [17] that (2) is equivalent to the integral continuity.
Proof.Fix ε > 0 and suppress it in what follows.Under the conditions on L and Φ, the cost functional in (1) Let {x n , u n } n∈N be a minimizing sequence for problem (1) and J(x n , u n , ) ≤ J + 1 n .Due to (8), we have the uniform boundedness of sequence {x n } n∈N on every finite interval [0, T], i.e., ∃L > 0. sup and Thus, sequence {x n } n∈N is precompact in C([0, T]).Due to the Arzelà-Ascoli theorem, From [13] and ( 9), we deduce that x is absolutely continuous and ẋn → ẋ * -weakly as n → ∞ in L ∞ (0, T).Since ∀ε > 0 for a.e.t, there exists n 0 such that ∀n ≥ n 0 Then, by the Assumption 3, we have ẋn Taking into account the convergence theorem ( [18], p. 60) for a.e.t, we have ẋ(t) ∈ X t ε , x(t), u(t) .
Consider the following assumption.

Assumption 7.
Suppose that for all u(•) ∈ U, problem (5) has a unique solution.
Theorem 3. Suppose Assumptions 1-6 and (2) hold.Under Assumption 7, and up to a subsequence where {x ε n , u ε n } is the solution of (1), and {y, u} is the solution of (3). Proof.
(1) First, we prove that if u n → û in L 2 (0, T), x n is the solution of (4) with ε = ε n , u = u n ; then, where ŷ is the solution of ( 5) with u = û.Let ŷ be the unique solution of (5) with control u = û.Then, where It is known that every function in L 2 (0, T) can be approximated with continuous functions in L 2 -norm, and any continuous function can be approximated by a piecewise constant function in the continuous norm.Then, for any η > 0, let û ∈ C([0, T]) be such that the following is the case.
Let τ i = T•i m , i = 0, m and choose an m that is large enough so that ∀k ∈ 1, N at least one of {τ i } belongs to [t k , t k+1 ).By joining sets {t k } and {τ i } and denoting the resulting set as {t i } N i=0 with N ≤ N + m, we obtain the following. Then, n + I (2) n .
Now, we derive the upper bound for I n : and taking into account the boundedness of ŷ, we have the boundedness of the multi-valued function, Y, with constant C; therefore, we obtain Finally, I Using similar arguments, we derive the same upper bound for I n .Now, taking into account (2) and Theorem 1, we chose n 0 such that ∀n ≥ n 0 The assumptions of the Theorems 2 and 3 are fulfilled for problems (11) and (12).Therefore, y(T) = x 0 + T 0 u(t)dt and u(t) ≡ −1 is the approximate control.

Conclusions and Future Research
We sought to obtain a theoretical result that demonstrates the effectiveness of the averaging method of finding an approximate solution of the optimal control problem of a non-linear system of differential inclusions with fast-oscillating parameters.We proved that the optimal control of the problem with averaging coefficients can be considered as "approximately" optimal for the initial perturbed system.To demonstrate the effectiveness of the method, we plan to continue research focusing on the practical applications and simulation results using particular genetic algorithms.