Positive solution to a generalized Lyapunov equation via a coupled fixed point theorem in a metric space endowed with a partial order

Abstract

We consider the generalized continuous-time Lyapunov equation:
$$
A^*XB + B^*XA =-Q,
$$
where $Q$ is an $N\times N$ Hermitian positive definite matrix and $A,B$ are arbitrary $N\times N$ matrices. Under certain conditions, using a coupled fixed point theorem du to Bhaskar and Lakshmikantham combined with the Schauder fixed point theorem, we establish an existence and uniqueness result of Hermitian positive definite solution to such equation. Moreover, we provide an iteration method to find convergent sequences which converge to the solution if one exists. Numerical experiments are presented to illustrate our theoretical results.