Some Unsolved Problems in Stability and Optimal Control Theory of Stochastic Systems

Leonid Shaikhet

doi:10.3390/math10030474

Abstract

In spite of the fact that the theory of stability and optimal control for different types of stochastic systems is well developed and very popular in research, there are some simply and clearly formulated problems, solutions of which have not been found so far. To the readers’ attention six open stability problems for stochastic differential equations with delay, for stochastic difference equations with discrete and continuous time and one open optimal control problem for stochastic hyperbolic equation with two-parameter white noise are offered.

Highlights

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in
Some unsolved problems of stability and optimal control in the theory of stochastic systems are offered to the attention of the readers
To explain the idea of the proposed open problem note that the second moment y(t) = Ex2 (t) of the solution x (t) of the scalar linear stochastic differential equation dx (t) = − ax (t)dt + σ (t) x (t)dw(t) is a solution of the deterministic differential equation [1]

Summary

Introduction

Some unsolved problems of stability and optimal control in the theory of stochastic systems are offered to the attention of the readers. Among the proposed problems there are both long-existing problems and those that have arisen quite recently, both already published before and published for the first time. All these problems are united by the absence of their solutions. The stability problem for the stochastic difference equation with continuous time (Problem 6) appeared and were published more than 10 years ago, but have not been solved yet. The optimal control problem (Problem 7) arose more than 40 years ago, but has not yet been solved too. The author hopes that combining both previously published and new unsolved problems in one paper will help readers to take a fresh look at these problems and get their rigorous and beautiful solutions

Stochastic Differential Equations

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Stochastic Hyperbolic Equation

Conclusions