(In-)Stability of Differential Inclusions: Notions, Equivalences, and Lyapunov-like Characterizations

Lyapunov methods have been and still are one of the main tools to analyze stability properties of dynamical systems. In this monograph Lyapunov results characterizing stability and stabilizability 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, invariance of stability and instability properties of equilibria of differential equations with respect to a 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.