A converse Lyapunov theorem for nonuniform in time global asymptotic stability and its application to feedback stabilization.

*(English)*Zbl 1049.93073The paper studies the notion of nonuniform in time robust global asymptotic stability (RGAS) for time-varying, nonlinear systems of the general form

$$\dot{x}=f(t,x,d),\phantom{\rule{1.em}{0ex}}x\in {\mathbb{R}}^{n},\phantom{\rule{4pt}{0ex}}d\in D,\phantom{\rule{4pt}{0ex}}t\ge 0$$

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 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 |