Minimum projection method for asymptotic stabilization toward a set

We had proposed a nonsmooth control Lyapunov function (CLF) design method named the minimum projection method for stabilization on noncontractible manifolds. The CLF and the method consider stabilization at a single equilibrium. Whether the method can be applied for stabilization at the desired set is a natural question. Hence, we consider the problem of asymptotic stabilization at a compact set on a noncontractible manifold. In this paper, we extend the notion of CLF to be applied for stabilization toward the set. Moreover, we prove that the resulting function by the minimum projection method becomes a extended CLF on the original manifold.