In-stability of Differential Inclusions : Notions, Equivalences, and Lyapunov-like Characterizations

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.

Philipp Braun received his Ph.D. in Applied Mathematics from the University of Bayreuth, Bayreuth, Germany, in 2016. He is a Research Fellow in the School of Engineering at the Australian National University, Canberra, Australia. His main research interests include control theory with a focus on stability analysis of nonlinear systems using Lyapunov methods and model predictive control.

Lars Grüne received the Ph.D. in Mathematics from the University of Augsburg, Augsburg, Germany, in 1996 and the Habilitation from Goethe University Frankfurt, Frankfurt, Germany, in 2001. He is currently a Professor of Applied Mathematics with the University of Bayreuth, Bayreuth, Germany. His research interests include mathematical systems and control theory with a focus on numerical and optimization-based methods for nonlinear systems.

Christopher M. Kellett received his Ph.D. in electrical and computer engineering from the University of California, Santa Barbara, in 2002. He is currently a Professor and the Director of the School of Engineering at Australian National University. His research interests are in the general area of systems and control theory with an emphasis on the stability, robustness, and performance of nonlinear systems with applications in both social and technological systems.