Necessary and sufficient condition for locally semiconcave practical control Lyapunov functions*

A semiconcave function plays an important role in nonlinear control theory. Nakamura et al. proposed a static discontinuous global feedback controller that is based on a locally semiconcave practical control Lyapunov function (LS-PCLF) for asymptotic stabilization of a nonlinear system defined on a manifold. Further, when comparing related researches, it is important to analyze the relationship between generalized differentials of the LS-PCLF. For the LS-PCLF, equivalent definitions using a directional subderivative and a disassembled differential have been proposed. However, an equivalent definition of the LS-PCLF using a reachable differential has not yet been established, and the original definition for a disassembled differential is not adequately formal. In this paper, we present a formal definition for a disassembled differential, review its basic properties, and present an equivalent definition for the LS-PCLF using a reachable differential. Through analysis, we confirm that a disassembled differential is a useful tool for analyzing the relationship between generalized differentials of the LS-PCLF.