Differential games, partial-state stabilization, and model reference adaptive control

In this paper, we develop a unified framework to find state-feedback control laws that solve two-player zero-sum differential games over the infinite time horizon and guarantee partial-state asymptotic stability of the closed-loop system. Partial-state asymptotic stability is guaranteed by means of a Lyapunov function that is positive definite and decrescent with respect to part of the system state. The existence of a saddle point for the system׳s performance measure is guaranteed by the fact that this Lyapunov function satisfies a partial differential equation that corresponds to a steady-state form of the Hamilton-Jacobi-Isaacs equation. In the second part of this paper, we show how our differential game framework can be applied not only to solve pursuit-evasion and robust optimal control problems, but also to assess the effectiveness of a model reference adaptive control law. Specifically, the model reference adaptive control architecture is designed to guarantee satisfactory trajectory tracking for uncertain nonlinear dynamical systems, whose matched nonlinearities are captured by the regressor vector. By modeling matched and unmatched nonlinearities, which are not captured by the regressor vector, as the pursuer׳s and evader׳s control inputs in a differential game, we provide an explicit characterization of the system׳s uncertainties that do not disrupt the trajectory tracking capabilities of the adaptive controller. Two numerical examples illustrate the applicability of our results.