# Optimal Stabilization Control for Discrete-time Markov Jump Linear System with Control Input Delay

## TL;DR

This paper develops necessary and sufficient conditions for stabilizing and controlling Markov jump linear systems with input delay, using coupled algebraic Riccati equations, advancing the theoretical understanding of such systems.

## Contribution

It introduces explicit stabilization conditions and designs an optimal controller for MJLS with input delay, using a novel Lyapunov equation and Riccati framework.

## Key findings

- Stabilization conditions are both necessary and sufficient.
- Optimal controller is derived via coupled algebraic Riccati equations.
- Stabilizability is characterized by positive definite solutions under observability.

## Abstract

This paper will investigate the infinite horizon optimal control and stabilization problems for the Markov jump linear system (MJLS) subject to control input delay. Different from previous works, for the first time, the necessary and sufficient stabilization conditions are explored under explicit expressions, and the optimal controller for infinite horizon is designed with a coupled algebraic Riccati equation. By introducing a new type of Lyapunov equation, we show that under the exact observability assumption, the MJLS with control input delay is stabilizable in the mean square sense with the optimal controller if and only if a coupled algebraic Riccati equation has a unique positive definite solution. The presented results are parallel to the optimal control and stabilization for standard system with input delay.