Inverse optimal constrained input-to-state stabilization of nonlinear systems

A system with both control and disturbance inputs is constrained input-to-state stabilizable if there exist feedback control laws that render the closed-loop system constrained input-to-state stable (cISS) with respect to the disturbance. Based on a control Lyapunov function, Sontag type controllers to the constrained input-to-state stabilization problem is constructed for a class of nonlinear systems, and it is shown that input-to-state stabilizability implies (arbitrary) constrained input-to-state stabilizability, but the converse is not true. Moreover, the constrained input-to-state stabilizability is both necessary and sufficient for the solvability of an inverse optimal problem. The designed controllers remain cISS against a certain class of input uncertainties, even input unmodeled dynamics.