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

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.


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
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