It is shown that every asymptotically controllable system can be globally stabilized by means of some (discontinuous) feedback law. The stabilizing strategy is based on pointwise optimization of a smoothed version of a control-Lyapunov function, iteratively sending trajectories into smaller and smaller neighborhoods of a desired equilibrium. A major technical problem, and one of the contributions of the present paper, concerns the precise meaning of "solution" when using a discontinuous controller.

Asymptotic controllability implies feedback stabilization

We study stability and stabilizability properties of systems with discontinuous righthand side (with solutions intended in Filippov's sense) by means of locally Lipschitz continuous and regular Lyapunov functions. The stability result is obtained in the more general context of differential inclusions. Concerning stabilizability, we focus on systems
 affine with respect to the input: we give some sufficient conditions for a system to be stabilized by means of a feedback law of the Jurdjevic-Quinn type.

/pdf/stability-and-stabilization-of-discontinuous-systems-and-1odv710k1a.pdf

Stability and Stabilization of Discontinuous Systems and Nonsmooth Lyapunov Functions

The paper surveys mathematical tools required for stability and convergence analysis of modern sliding mode control systems. Elements of Filippov theory of differential equations with discontinuous right-hand sides and its recent extensions are discussed. Stability notions (from Lyapunov stability (1982) to fixed-time stability (2012)) are observed. Concepts of generalized derivatives and non-smooth Lyapunov functions are considered. The generalized Lyapunov theorems for stability analysis and convergence time estimation are presented and supported by examples from sliding mode control theory.

https://hal.inria.fr/hal-00942319/document

Stability notions and Lyapunov functions for sliding mode control systems

http://directory.umm.ac.id/Data%20Elmu/jurnal/E/Ecological%20Economics/Vol36.Issue3.Mar2001/1180.pdf

A viability analysis for a bio-economic model

Viscosity solutions: A primer

We present a unified approach to a complex of related issues in control theory, one based to a great extent on the methods of nonsmooth analysis. The issues include invariance, stability, equilibria, monotonicity, the Hamilton-Jacobi equation, feedback synthesis, and necessary conditions.

P. H. Clarke

Papers

Qualitative properties of trajectories of control systems: A survey