# Numerical solution of Lyapunov equations related to Markov jump linear systems

## TL;DR

This paper compares various numerical methods for solving Lyapunov equations in Markov jump linear systems, highlighting the effectiveness of fixed-point and trust-region approaches for large-scale problems.

## Contribution

It introduces and evaluates multiple numerical algorithms, including fixed point, Krylov subspace, and optimization-based methods, for solving Lyapunov equations in Markov jump systems.

## Key findings

- Fixed-point approach outperforms optimization methods.
- Trust-region method is most effective for large-scale problems.
- Application to networked control systems with wireless-induced Markov jumps.

## Abstract

We suggest and compare different methods for the numerical solution of Lyapunov like equations with application to control of Markovian jump linear systems. First, we consider fixed point iterations and associated Krylov subspace formulations. Second, we reformulate the equation as an optimization problem and consider steepest descent, conjugate gradient, and a trust-region method. Numerical experiments illustrate that for large-scale problems the trust-region method is more effective than the steepest descent and the conjugate gradient methods. The fixed-point approach, however, is superior to the optimization methods. As an application we consider a networked control system, where the Markov jumps are induced by the wireless communication protocol.