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Existence of Lipschitz and semiconcave control-Lyapunov functions. (English) Zbl 0982.93068
The author constructs a Lyapunov function for globally asymptotically controllable nonlinear control systems. The controllability to the origin is studied via Lyapunov functions satisfying a differential inequality with proximal subdifferentials. This is one possible way for studying nonsmooth Lyapunov functions (see also E. Sontag and H. Sussman, Nonsmooth control Lyapunov function Proceedings of the IEEE Conference, 2799-2805 (1995) and J. P. Aubin, Viability Theory, Birkhäuser (1992; Zbl 0755.93003)]). The main point consists in noticing that the epigraph of the Lyapunov function has some viability – or weak-invariance – property. This property can be equivalently characterized through proximal subdifferentials, or the contingent derivative, or viscosity supersolutions, or Dini derivatives (in the Lipschitz case).
After constructing a Lipschitz control-Lyapunov function, the author proves that there is also another control-Lyapunov function that is semiconcave. This fact is proved by using inf-convolutions and nonsmooth analysis.

MSC:
93D30 Lyapunov and storage functions
93D15 Stabilization of systems by feedback
93B05 Controllability
49J52 Nonsmooth analysis
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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