Stability of Dynamical Systems with Competitive Controls: The Degenerate Case

We study the stability of a dynamical system with two competitive controls to a general closed set. The stability is proven to be equivalent to the continuity on such set of the value function of a differential game with maximum cost functional. The existence of a suitably defined and continuous Lyapunov function is proven to imply the stability. Our Lyapunov function satisfies a differential inequality in the sense of the theory of viscosity solutions.