EXISTENCE OF NASH STRATEGIES AND SOLUTIONS TO COUPLED RICCATI EQUATIONS IN LINEAR-QUADRATIC GAMES

The existence of linear Nash strategies for the linear-quadratic game is considered. The solvability of the coupled Riccati matrix equations and the stability of the closed-loop matrix are investigated by using Brower's fixed-point theorem. The conditions derived state that the linear closed-loop Nash strategies exist, if the open loop matrix A has a sufficient degree of stability which is determined in terms of the norms of the weighting matrices. When A is not necessarily stable, sufficient conditions for existence are given in terms of the solutions of auxiliary problems using the same procedure. © 1979 Plenum Publishing Corporation.