Application of the averaging method to the problems of optimal control of the impulse systems

Abstract

The problem of optimal control at finite time interval for a system of differential equations with impulse action at fixed moments of time as well as the corresponding averaged system of ordinary differential equations are considered.
 It is proved the existence of optimal control of exact and averaged problems. Also, it is established that optimal control of averaged problem realize the approximate optimal synthesis of exact problem. The main result of the article is a theorem, where it is proved that optimal contol of an averaged problem is almost optimal for exact problem. Substantiation of proximity of solutions of exact and averaged problems is obtained.

Application of the method of averaging to boundary value problems for differential equations with non-fixed moments of impulse

“…Clearly, problem (71) satisfies all conditions of Theorem 1, except for (7), since, in this case, F 0 (x 0 ) = x 0 − x 0 ≡ 0. The boundary value problem (71) has a solution if and only if the initial value x 0 = x(0) satisfies the condition x 0 = (x 0 + ε)e ε sin T ε .…”

“…For example, in [20] it was applied to the study of the existence of bounded on the entire axis solutions of non-autonomous systems of differential equations. The method was employed to the study of optimal control problems [7,8,10,23,25] and invariant sets of stochastic systems [21,22].…”