Results on robust stability and feedback stabilization for systems with a continuum of equilibria

A discrete-time dynamical system, with a continuum of equilibria and nonlinear, multivalued dynamics is discussed. An asymptotic stability property, its robustness, and necessary and sufficient Lyapunov-like conditions are presented. The conditions involve a set-valued Lyapunov function. Then a control system is studied, in which the stability property can be achieved by open-loop controls, and a feedback control construction is presented. Set-valued control Lyapunov functions are introduced, for the purpose of robust feedback design.