Differential inclusions with mean derivatives having extreme right-hand sides and optimal control

Abstract

First we prove the existence of solutions of some special stochastic differential inclusion with mean derivatives having lower semi-continuous right-hand sides that may not be convex. Then we show that among those solutions there is a solution that minimizes a certain cost criterion. After that this result is applied to investigation of controlled stochastic differential equations with feed back, whose right-hand sides take values in extreme sets of Hausdorff continuous set-valued vector field with bounded convex images. By reducing the equation to the inclusion of above-mentioned sort we prove that there exists a control that realizes the optimal solution of the inclusion as an optimal solution of the equation (an analogue of Filippov's theorem).

Keywords:

• Mean derivatives
• stochastic differential inclusions
• decomposable set-valued mappings
• extreme points
• controlled stochastic differential equations
• optimal solutions

AMS Subject Classifications:

• 60H10
• 47H04
• 34A60
• 60H30
• 60K30

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research is supported in part by RFBR Grant 18–01–00048.