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.

We consider the optimal control problem of a non-linear system of differential inclusions with fast-oscillating parameters on semi-axes. Using the averaging method, we find an approximate solution for the optimal control of non-linear differential inclusions with fast-oscillating coefficients on a semi-axes. Thus, we prove the convergence of the optimal controls of the initial control problem to the optimal process of the averaged problem and the convergence of the corresponding cost functionals.

In this paper we investigate the optimal control problem for a parabolic differential inclusion with rapidly oscillating variables in the finite interval. There are many approaches intended for the investigation of control problems for differential equations and inclusions. Thus, in particular, the asymptotic methods are used fairly extensively. Among these methods, we can especially mention the averaging method, which was mathematically rigorously substantiated by Krylov M.M. and Bogolyubov M.M. The well-known Krasnoselski–Krein theorem and its multi-valued analogue play an essential role for the investigation of the above-mentioned problems. The averaging method was substantiated, in particular, for ordinary differential inclusions, inclusions with partial derivatives, and inclusions with the Hukuhara derivative. When dealing with multi-valued mappings one faces specific problems, such as closedness, convexity of the family of solutions, existence of limit solutions, selection of solutions with given properties, etc. However, the well-developed apparatus of mathematical analysis applied to the study of multi-valued functions makes it possible to apply the averaging method to the optimal control problem described above. Thus, using the averaging method the convergence of optimal controls and optimal trajectories of solutions of the exact problem to optimal control and the trajectory of the averaged problem is proved in the paper.

Optimal Control Problem for Some Degenerate Variation Inequality: Attainability Problem

In this paper, we use the averaging method to find an approximate solution for the optimal control of parabolic differential inclusion with fast-oscillating coefficients on a semi-axis.

We investigate the optimal control problem with respect to coefficients of the degenerate parabolic variational inequality. Since problems of this type can have the Lavrentieff effect, we consider the optimal control problem in a class of so-called Hadmissible solutions. We substantiate the attainability of H-optimal pairs via optimal solutions of some nondegenerate perturbed optimal control problems.