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Vol. 51 (2015) No. 12 p. 803-813




We proposed a Sontag-type feedback controller by using a locally semiconcave practical control Lyapunov function (LS-PCLF) for asymptotic stabilization of a nonlinear system defined on a non-contractible manifold. We also proposed a design method of a nonsmooth control Lyapunov function called the multilayer minimum projection method. However, whether the method can be used as an LS-PCLF design method is not discussed, and we cannot design a controller by using the method and the controller in this situation. In this paper, we study a disassembled differential of a locally semiconcave function to handle an LS-PCLF. The minimal and maximal disassemble differentials, which are special types of a disassembled differential, are introduced. We elucidate that a set of reachable gradients and the Fréchet differential are equivalent to the minimal and maximal disassembled differential, respectively. Based on this fact, we prove that this method can be used as an LS-PCLF design method.

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