Time-Optimal Control for Semilinear Stochastic Functional Differential Equations with Delays

Yong Han Kang,Jin-Mun Jeong

doi:10.3390/math9161956

Abstract

The purpose of this paper is to find the time-optimal control to a target set for semilinear stochastic functional differential equations involving time delays or memories under general conditions on a target set and nonlinear terms even though the equations contain unbounded principal operators. Our research approach is to construct a fundamental solution for corresponding linear systems and establish variations of a constant formula of solutions for given stochastic equations. The existence result of time-optimal controls for one point target set governed by the given semilinear stochastic equation is also established.

Highlights

This paper deals with the existence of optimal control to reach the target set governed by semilinear stochastic differential equations: (R0 x (t) = Ax (t) + −h a1 (s) A1 x (t + s)ds + f (t, xt )dω + Bu(t), t > 0, (1)x (0) = φ0 ∈ L2 (Ω, H ), x (s) = φ1 (s), s ∈ [−h, 0], where A is an elliptic differential operator of the second order induced by the sesquilinear form, A1 is a closed linear operator with domain D ( A1 ) containing the D ( A), h > 0, and the function a(·) is real-valued and Hölder-continuous
Little study has been done of the existence of time-optimal control for control to a target set for nonlinear stochastic functional differential equations with unbounded operators
Motivated by the above-mentioned works [13,21], we deal with the time-optimal control problem to a bounded target set for semilinear stochastic control equations involving time delays or memories even though the equations contain unbounded principal operators and nonlinear terms by using an easy consequence of real interpolation spaces

Summary

Introduction

This paper deals with the existence of optimal control to reach the target set governed by semilinear stochastic differential equations:. Little study has been done of the existence of time-optimal control for control to a target set for nonlinear stochastic functional differential equations with unbounded operators. Motivated by the above-mentioned works [13,21], we deal with the time-optimal control problem to a bounded target set for semilinear stochastic control equations involving time delays or memories even though the equations contain unbounded principal operators and nonlinear terms by using an easy consequence of real interpolation spaces. As the time-optimal control theory for the standard results, we refer to the linear case as in [13] (or semilinear equation [21]) and extend the results in our semilinear stochastic functional differential equations with delays. The existence of the optimal control to a singleton target is derived from the convergence of optimal controls to decreasing target sets containing the singleton

Retarded Linear Equations

Semilinear Stochastic Differential Equations

Time-Optimal Control Problems

Conclusions