Existence of Lipschitz and semiconcave control-Lyapunov functions (Q2706143)

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 \textit{E. Sontag} and \textit{H. Sussman}, Nonsmooth control Lyapunov function Proceedings of the IEEE Conference, 2799-2805 (1995) and \textit{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).NEWLINENEWLINENEWLINEAfter 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.