# Optimal Control Problem for Discrete-Time Systems with Colored Multiplicative Noise

## TL;DR

This paper addresses the optimal control of discrete-time systems affected by colored multiplicative noise, providing necessary and sufficient conditions for solvability using stochastic difference equations.

## Contribution

It introduces a comprehensive analysis of control problems with colored noise and derives solvability conditions for systems with and without input delay.

## Key findings

- Derived necessary and sufficient conditions for control problem solvability.
- Analyzed systems with colored multiplicative noise and delay.
- Provided a framework for solving stochastic difference equations.

## Abstract

The optimal control problem for discrete-time systems with colored multiplicative noise is discussed in this paper. The problem will be more difficult to deal with than the case of white noise due to the correlation of the adjoining state. By solving the forward and backward stochastic difference equations (FBSDEs), the necessary and sufficient conditions for the solvability of the optimal control problems in both delay-free and one-step input delay case are given.