Lyapunov and feedback characterizations of state constrained controllability and stabilization

A standard control system is considered, in conjunction with a state constraint S and a target set Σ. The properties of open loop S-constrained control to Σ and practical closed loop S-constrained control to Σ are shown to be equivalent, and to be characterizable in terms of the existence of certain types of control Lyapunov functions. Feedback S-constrained stabilizability to Σ can be added to the list of equivalences, when a small time controllability property is posited.