Multilayer Minimum Projection Method with Infinite Layers

Abstract

Asymptotic stabilization on noncontractible manifolds is a difficult control problem. To address the problem, we had proposed the multilayer minimum projection method to design nonsmooth control Lyapunov functions. The method considers many simple-structured smooth manifolds associated with the original manifold by local diffeomorphisms. However, the case of infinitely many manifolds was not considered.Then, we consider infinitely many simple-structured smooth manifolds associated with the original manifold by local diffeomorphisms in the paper. First of all, we show that the function obtained by “infimum projection” of functions does not become a control Lyapunov function. Then, we suppose that the parameter space of mappings is a compact Hausdorff space. In this case, we prove the fact that the function obtained by the minimum projection becomes a control Lyapunov function. Finally, the advantage of consideration of the infinitely many manifolds is confirmed through an example.