Optimal control of a linear system with multiplicative noise at control parameter

TL;DR

This paper studies an optimal control problem for a linear system affected by multiplicative noise on the control parameter, deriving necessary conditions and transforming it into a deterministic problem for analysis.

Contribution

It introduces a novel approach to control systems with multiplicative noise by formulating necessary optimality conditions and transforming the stochastic problem into a deterministic one.

Findings

Derived Pontryagin maximum principle for the problem

Proved existence theorems for optimal controls

Transformed the stochastic problem into a deterministic auxiliary problem

Abstract

We investigate a control process described by a linear system of ordinary differential equations with a noise of special type acting to the control parameter. As the cost functional the probability of the final state vector to enter to a given set in the phase space is considered. Necessary conditions of optimality (of the Pontryagin maximum principle form) and existence theorems are developed. The initial control problem was trasformed to an auxiliary deterministic problem, the differentiability of the auxiliary functional was discussed.