A converse Lyapunov theorem for nonuniform in time global asymptotic stability and its application to feedback stabilization. (English) Zbl 1049.93073

Karafyllis, I.; Tsinias, J.

SIAM J. Control Optimization 42, No. 3, 936-965 (2003).

The paper studies the notion of nonuniform in time robust global asymptotic stability (RGAS) for time-varying, nonlinear systems of the general form \[ \dot x=f(t,x,d),\quad x\in\mathbb{R}^n,\;d\in D,\;t\geq 0 \] where \(D\) is a compact subset of \(\mathbb{R}^m\). The authors present some equivalent definitions of RGAS and provide a Lyapunov characterization. These results are applied to derive necessary and sufficient conditions for ISS-feedback stabilization of input time-varying, nonlinear systems: this actually constitutes an extension of the well-known Artstein-Sontag theorem. For systems which exhibit an affine structure, an explicit formula of the stabilizing feedback is given. Finally, ISS-stabilization is also considered for certain cascade systems.

Reviewer: Andrea Bacciotti (Torino)

MSC:

93D20	Asymptotic stability in control theory
93D30	Lyapunov and storage functions
37B55	Topological dynamics of nonautonomous systems
93D15	Stabilization of systems by feedback
93D25	Input-output approaches in control theory

Keywords:

nonuniform in time asymptotic stability; input-to-state stability; Lyapunov functions; feedback stabilization; time-varying systems; nonlinear systems; Artstein-Sontag theorem; cascade systems