Averaging in optimal control problems for systems of difference equations

In this paper we consider optimal control problem for systems of difference equations in the standard Bogolyubov form. The presence of small parameter allow us to average such systems and reduce the initial nonautonomous problem to more simple autonomous one. The averaging method was studied by many authors. N. Bogolyubov [3] developed a general averaging approach for system of ordinary differential equations. Further this method was applied to systems of functional-differential equations, difference equations, stochastic systems [2,9]. Optimal control problems for systems of differential and difference equations are particularly important for applied goals. Methods have been developed to investigate them. For more details see [4,5,6,7,10,11]. Fundamental results in application of averaging method to optimal control problems were obtained by V. Plotnikov [11]. In work [10] authors propose a new scheme of averaging for optimal control problem. This work is devoted to the application of averaging method to optimal control problems for systems of difference equations. We investigate relationship between optimal controls of the averaged and the original systems and prove that the optimal control for the averaged system is ε-optimal for the original problem.


Introduction
In this paper we consider optimal control problem for systems of difference equations in the standard Bogolyubov form.The presence of small parameter allow us to average such systems and reduce the initial nonautonomous problem to more simple autonomous one.
The averaging method was studied by many authors.N. Bogolyubov [3] developed a general averaging approach for system of ordinary differential equations.Further this method was applied to systems of functional-differential equations, difference equations, stochastic systems [2,9].
Optimal control problems for systems of differential and difference equations are particularly important for applied goals.Methods have been developed to investigate them.For more details see [4,5,6,7,10,11].Fundamental results in application of averaging method to optimal control problems were obtained by V. Plotnikov [11].In work [10] authors propose a new scheme of averaging for optimal control problem.
This work is devoted to the application of averaging method to optimal control problems for systems of difference equations.We investigate relationship between optimal controls of the averaged and the original systems and prove that the optimal control for the averaged system is ε-optimal for the original problem.

Iryna Volodymyrivna Komashynska
Statement of the problem.
Let us consider the optimal control problem for system of difference equations: with a given initial condition x 0 ∈ D.
Here ε > 0 is a small parameter, Controls u n are called admissible if the following conditions are satisfied: 2) for every u n there exists a constant u 0 ∈ D such that | u n − u 0 |≤ ϕ n , where We introduce F to denote the set of all admissible controls.For every admissible control u n we denote the solution of system (1) by x n (u n ).
Our aim is to find an admissible control u = u n which minimizes the functional , where Φ(x) is a given function, T > 0 is a certain constant, and [.] is an integer part of a number.Denote We associate the system (1) on 0, T ε with averaged system where and Let ū * n (ε) be an optimal control for averaged problem (2).In this work we prove that the control ū * n (ε) is η−optimal for system (1), i.e., for any η > 0 there exists ε 0 > 0 such that, for all 0 < ε < ε 0 the inequality

Preliminaries
To obtain the main result we need two lemmas.The first one is a discrete version of the known Gronwall-Bellman inequality [1,8], the second one is a generalization of averaging method for difference equations in the case where right sides depend on functional parameters.Lemma 2.2.Suppose that the following conditions are satisfied in the domain ) is bounded and satisfies the Lipschitz condition with respect to x and u with a constant M ; 2) a solution y = y n (u n ), y 0 (u 0 ) = x 0 of the averaged system is defined for all admissible u n and belongs to the domain D together with a some ρ−neighborhood; 3) the limit (3) exists uniformly in x ∈ D and u ∈ U .
Proof: First we find a sequence {ψ N } such that for all x ∈ D and admissible control u n the estimate holds.Note that lim Indeed, condition (3) implies the existence of sequence {a N }, that converges to 0, such that for all x ∈ D, u ∈ U it follows Iryna Volodymyrivna Komashynska Thus we have It remains to denote Hence for any admissible u n in system (1), we can consider the next system as the averaged one.Now for solutions x n (u n ) and y n (u 0 ) of systems (1) and (8) we apply the analog of the first Bogolyubov theorem for difference equations.From estimate (6) it follows that for any η > 0 and T > 0 there exists ε 0 = ε 0 (η, T ) > 0 such that for all 0 < ε < ε 0 and n ∈ 0, T ε the estimate holds.Here ε 0 does not depend on u n .Next, we evaluate the norm of the difference between solutions of systems (8) and (2) for n ∈ 0, T ε .Represent systems (2) and (8) in the form Subtracting (11) from (10) and adding to and subtracting from the right side of the equality the function f 0 (y k (u 0 ), u k ), we obtain: Hence, using Lipschitz condition, we get According to the Lemma 2.1, we observe that for all integers n ∈ 0, T ε the following estimate is true Now, we choose ε 1 ≤ ε 0 , such that for all ε ≤ ε 1 as n ∈ 0, T ε the inequality holds.
Finally from (9) and (13), we obtain estimate (5) in the lemma.This completes the proof of the Lemma.✷

Main result
Let us now state and prove the main result of this work.
Theorem 3.1.Assume that in the domain 1) the function f n (x, u) is bounded by a constant K. Furthermore, it is Lipschitz with respect to x and u with the constant M ; 2) the solution y = y n (u n ), y(u 0 ) = x 0 of the averaged system (2) is defined for all admissible u n and it belongs to the domain D together with some ρ−neighborhood; 3) the limit (3) exists uniformly in x ∈ D and u ∈ U ;

Lemma 2 . 1 .
(Discrete version of Gronwal -Bellman inequality).Let {y n } and {a n } be non-negative sequences and C > 0 is a constant.If y n ≤ C + n k=0 a k y k , then y n ≤ C exp( n k=0 a k ).