Summary: The established theory of nonzero sum games is used to solve a mixed ${H}_{2}/{H}_{\infty}$ control problem. Our idea is to use the two play-off functions associated with a two-player Nash game to represent the ${H}_{2}$ and ${H}_{\infty}$ criteria separately. We treat the state-feedback problem and we find necessary and sufficient conditions for the existence of a solution. Both the finite and infinite time problems are considered. In the infinite horizon case we present a full stability analysis. The resulting controller is a constant state-feedback law, characterized by the solution to a pair of cross-coupled Riccati equations, which may be solved using a standard numerical integration procedure.

We begin our development by considering strategy sets containing linear controllers only. At the end of the paper we broaden the strategy sets to include a class of nonlinear controls. It turns out that this extension has no effect on the necessary and sufficient conditions for the existence of a solution or on the nature of the controllers.

##### MSC:

93B36 | ${H}^{\infty}$-control |