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Zbl 1049.93073
Karafyllis, I.; Tsinias, J.
A converse Lyapunov theorem for nonuniform in time global asymptotic stability and its application to feedback stabilization. (English)
[J] SIAM J. Control Optimization 42, No. 3, 936-965 (2003). ISSN 0363-0129; ISSN 1095-7138/e

The paper studies the notion of nonuniform in time robust global asymptotic stability (RGAS) for time-varying, nonlinear systems of the general form $$\dot x=f(t,x,d),\quad x\in\bbfR^n,\ d\in D,\ t\ge 0$$ where $D$ is a compact subset of $\bbfR^m$. 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.
[Andrea Bacciotti (Torino)]

MSC 2000:: *93D20 Asymptotic stability of control systems
93D30 Scalar and vector Lyapunov functions
37B55 Nonautonomous dynamical systems
93D15 Stabilization of systems by feedback
93D25 Input-output approaches to stability of control systems

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

Cited in: Zbl 1238.93079

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