A generalization of Zubov's method to perturbed systems

F. Camilli, L. Grüne, F. Wirth: A generalization of Zubov's method to perturbed systems
in: Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, Nevada, 2002, 3518 - 3523

DOI: 10.1109/CDC.2002.1184420
Keywords: asymptotic stability; Zubov's method; robust stability; domain of attraction; viscosity solutions
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We present a generalization of Zubov's method to perturbed differential equations. The goal is to characterize the domain of attraction of a set which is uniformly locally asymptotically stable under all admissible time varying deterministic perturbations with values in some given compact set of perturbation values. We show that in this general setting a straightforward generalization of the classical Zubov equation has a unique viscosity solution which characterizes the robust domain of attraction as a suitable sublevel set. In addition, we give several properties of this unique viscosity solution (which will not be differentiable in general) and discuss the existence of smooth solutions.

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