Some new results on the global uniform asymptotic stability of time-varying dynamical systems

2017 ◽  
Vol 35 (3) ◽  
pp. 901-922
Author(s):  
Nizar Hadj Taieb ◽  
Mohamed Ali Hammami
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Stability ◽  
Time Varying ◽  
Uniform Asymptotic Stability

Uniform Global Asymptotic Stability of Adaptively Controlled Nonlinear Systems via Strict Lyapunov Functions

2008 ◽  
Author(s):  
Fre´de´ric Mazenc ◽  
Marcio de Queiroz ◽  
Michael Malisoff
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Functions ◽  
Unknown Parameter ◽  
Time Varying ◽  
Excitation Condition ◽  
Parameter Vector ◽  
Uniform Asymptotic Stability ◽  
Asymptotic Tracking ◽  
Persistency Of Excitation ◽  
Unknown Parameter Vector

We prove global uniform asymptotic stability of adaptively controlled dynamics by constructing explicit global strict Lyapunov functions. We assume a persistency of excitation condition that implies both asymptotic tracking and parameter identification. We also construct input-to-state stable Lyapunov functions under an added growth assumption on the regressor, assuming that the unknown parameter vector is subject to suitably bounded time-varying uncertainties. This quantifies the effects of uncertainties on the tracking and parameter estimation. We demonstrate our results using the Ro¨ssler system.

scholarly journals New approach to asymptotic stability: time-varying nonlinear systems

1997 ◽  
Vol 20 (2) ◽  
pp. 347-366 ◽  
Author(s):  
L. T. Grujić
Keyword(s):  
Nonlinear Systems ◽  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Sufficient Conditions ◽  
Stability Domain ◽  
Time Varying ◽  
Uniform Asymptotic Stability ◽  
Stability Time ◽  
Necessary And Sufficient

The results of the paper concern a broad family of time-varying nonlinear systems with differentiable motions. The solutions are established in a form of the necessary and sufficient conditions for: 1) uniform asymptotic stability of the zero state, 2) for an exact single construction of a system Lyapunov function and 3) for an accurate single determination of the (uniform) asymptotic stability domain. They permit arbitrary selection of a functionp(⋅)from a defined functional family to determine a Lyapunov functionv(⋅),[v(⋅)], by solvingv′(⋅)=−p(⋅){or equivalently,v′(⋅)=−p(⋅)[1−v(⋅)]}, respectively. Illstrative examples are worked out.

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