(In-)Stability of Differential Inclusions
Notions, Equivalences, and Lyapunov-like Characterizations
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Beschreibung
Lyapunov methods have been and are still one of the main tools to analyze the stability properties of dynamical systems. In this monograph, Lyapunov results characterizing the stability and stability of the origin of differential inclusions are reviewed. To characterize instability and destabilizability, Lyapunov-like functions, called Chetaev and control Chetaev functions in the monograph, are introduced. Based on their definition and by mirroring existing results on stability, analogue results for instability are derived. Moreover, by looking at the dynamics of a differential inclusion in backward time, similarities and differences between stability of the origin in forward time and instability in backward time, and vice versa, are discussed. Similarly, the invariance of the stability and instability properties of the equilibria of differential equations with respect to scaling are summarized. As a final result, ideas combining control Lyapunov and control Chetaev functions to simultaneously guarantee stability, i.e., convergence, and instability, i.e., avoidance, are outlined. The work is addressed at researchers working in control as well as graduate students in control engineering and applied mathematics.
Lyapunov methods have been and are still one of the main tools to analyze the stability properties of dynamical systems. In this monograph, Lyapunov results characterizing the stability and stability of the origin of differential inclusions are reviewed. To characterize instability and destabilizability, Lyapunov-like functions, called Chetaev and control Chetaev functions in the monograph, are introduced. Based on their definition and by mirroring existing results on stability, analogue results for instability are derived. Moreover, by looking at the dynamics of a differential inclusion in backward time, similarities and differences between stability of the origin in forward time and instability in backward time, and vice versa, are discussed. Similarly, the invariance of the stability and instability properties of the equilibria of differential equations with respect to scaling are summarized. As a final result, ideas combining control Lyapunov and control Chetaev functions to simultaneously guarantee stability, i.e., convergence, and instability, i.e., avoidance, are outlined. The work is addressed at researchers working in control as well as graduate students in control engineering and applied mathematics.
Offers a unified presentation of stability results for dynamical systems using Lyapunov-like characterizations Provides derivation of strong/weak complete instability results for systems in terms of Lyapunov-like and comparison functions Discusses combined stability and avoidance problem for control systems from the perspective of Lyapunov functions
Autor*in
Philipp Braun
Themen in »(In-)Stability of Differential Inclusions«
Lyapunov methods differential inclusions stability of nonlinear systems instability of nonlinear systems stabilization/destabilization of nonlinear systems stabilizability and destabilizability
Stimmen zu »(In-)Stability of Differential Inclusions«
“The book is written in a clear, rigorous and alive style. The results are illustrated by numerous examples.” (Aurelian Cernea, zbMATH 1482.34002, 2022)
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