On the existence of the stabilizing solution of generalized Riccati equations arising in zero sum stochastic difference games: The time-varying case

TL;DR

This paper investigates the existence and uniqueness of stabilizing solutions for time-varying Riccati equations in stochastic dynamic games, with applications to infinite-horizon zero-sum LQ games involving switching coefficients and noise.

Contribution

It establishes conditions for the existence of a unique stabilizing solution to generalized Riccati equations in a time-varying stochastic setting, extending previous results to more complex systems.

Findings

Proved existence and uniqueness of stabilizing solutions for a broad class of Riccati equations.

Applied results to infinite-horizon zero-sum LQ games with switching and noise.

Highlighted the role of these solutions in optimal control of stochastic systems.

Abstract

In this paper, a large class of time-varying Riccati equations arising in stochastic dynamic games is considered. The problem of the existence and uniqueness of some globally defined solution, namely the bounded and stabilizing solution, is investigated. As an application of the obtained existence results, we address in a second step the problem of infinite-horizon zero-sum two players linear quadratic (LQ) dynamic game for a stochastic discrete-time dynamical system subject to both random switching of its coefficients and multiplicative noise. We show that in the solution of such an optimal control problem, a crucial role is played by the unique bounded and stabilizing solution of the considered class of generalized Riccati equations.