Asymptotic stabilization with locally semiconcave control Lyapunov functions on general manifolds

Asymptotic stabilization on noncontractible manifolds is a difficult control problem. If a configuration space is not a contractible manifold, we need to design a time-varying or discontinuous state feedback control for asymptotic stabilization at the desired equilibrium. For a system defined on Euclidean space, a discontinuous state feedback controller was proposed by Rifford with a semiconcave strict control Lyapunov function (CLF). However, it is difficult to apply Rifford's controller to stabilization on general manifolds. In this paper, we restrict the assumption of semiconcavity of the CLF to the "local" one, and introduce the disassembled differential of locally semiconcave functions as a generalized derivative of nonsmooth functions. Further, we propose a Rifford-Sontag-type discontinuous static state feedback controller for asymptotic stabilization with the disassembled differential of the locally semiconcave practical CLF (LS-PCLF) by means of sample stability. The controller does not need to calculate limiting subderivative of the LS-PCLF. Moreover, we show that the LS-PCLF, obtained by the minimum projection method, has a special advantage with which one can easily design a controller in the case of the minimum projection method. Finally, we confirm the effectiveness of the proposed method through an example. (C) 2013 Elsevier B.V. All rights reserved.