Universal controllers for robust control problems

In this paper we discuss the construction of ''universal'' controllers for a class of robust stabilization problems. We give a general theorem on the construction of these controllers, which requires that a certain nonlinear inequality is solvable pointwisely or, equivalently, that a robust control Lyapunov function does exist. The constructive procedure produces almost smooth controllers. The robust control Lyapunov functions extend to uncertain systems the concept of control Lyapunov functions. If such a robust control Lyapunov function also satisfies a small control property, the resulting stabilizing controller is also continuous in the origin of the state space. Applications of our results range from optimal to robust control.