On the bounded and stabilizing solution of a generalized Riccati differential equation arising in connection with a zero‐sum linear quadratic stochastic differential game

Abstract

SummaryWe study a class of coupled nonlinear matrix differential equations arising in connection with the solution of a zero‐sum two‐player linear quadratic (LQ) differential game for a dynamical system modeled by an Itô differential equation subject to random switching of its coefficients. The system of differential equations under consideration contains as special cases the game‐theoretic Riccati differential equations arising in the solution of the H∞ control problem from the deterministic and stochastic cases. A set of sufficient conditions that guarantee the existence of the bounded and stabilizing solution of this kind of Riccati differential equations is provided. We show how such stabilizing solution is involved in the construction of the equilibrium strategy of a zero‐sum LQ stochastic differential game on an infinite‐time horizon and give as a byproduct the solution of such a control problem.