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

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The fundamental theory that emerged from Aleksandr Mikhailovich Lyapunovs doctoral thesis [50] more than 100 years ago has been and still is the main tool to analyze stability properties of dynamical systems. Lyapunov or Lyapunov-like functions are monotone functions when evaluated along the solution of a dynamical system. Based on the monotonicity property, stability or instability of invariant sets can be concluded without the need to derive explicit solutions of the system dynamics.

In this monograph, existing results characterizing stability and stabilizability of the origin of differential inclusions through Lyapunov and control Lyapunov functions are reviewed and new characterizations for instability and destabilization characterized through Lyapunov-like arguments are derived. To distinguish between stability and instability, stability results are characterized through Lyapunov and control Lyapunov functions whereas instability is characterized through Chetaev and control Chetaev functions. In addition, similarities and differences between stability and instability and stabilizability and destabilizability of the origin of a differential inclusion are summarized. These connections are established by considering dynamics in forward time, in backward time, or by considering a scaled version of the differential inclusion. In total, the diagram shown in Fig. 1.1 is obtained, unifying new and existing results in a consistent notation.

As a last contribution of the monograph, ideas combining control Lyapunov and control Chetaev functions into a single framework are discussed. Through this approach, convergence (i.e., stability) and avoidance (i.e., instability) are guaranteed simultaneously.

The genesis of this monograph emerged from the preliminary results in [11], published as a conference paper in the proceedings of the 57th IEEE Conference on Decision and Control. Additionally, the ideas combining properties of control Lyapunov and control Chetaev functions rely on conference papers [13] and [14].