Invariance Conditions for Nonlinear Dynamical Systems

Recently, Horv\'ath, Song, and Terlaky [\emph{A novel unified approach to invariance condition of dynamical system, submitted to Applied Mathematics and Computation}] proposed a novel unified approach to study, i.e., invariance conditions, sufficient and necessary conditions, under which some convex sets are invariant sets for linear dynamical systems. In this paper, by utilizing analogous methodology, we generalize the results for nonlinear dynamical systems. First, the Theorems of Alternatives, i.e., the nonlinear Farkas lemma and the \emph{S}-lemma, together with Nagumo's Theorem are utilized to derive invariance conditions for discrete and continuous systems. Only standard assumptions are needed to establish invariance of broadly used convex sets, including polyhedral and ellipsoidal sets. Second, we establish an optimization framework to computationally verify the derived invariance conditions. Finally, we derive analogous invariance conditions without any conditions

Robust Non-Zenoness of Piecewise Affine Systems with Applications to Linear Complementarity Systems

Abstract Piecewise affine systems (PASs) constitute an important class of nonsmooth switching dynamical systems subject to state dependent mode transitions arising from control and dynamic optimization. A fundamental issue in dynamics analysis of switching systems pertains to the possible occurrence of infinitely many switchings in finite time, referred to as the Zeno behavior. There has been a growing interest in characterization of Zeno free switching systems. Different from the recent non-Zeno analysis of switching systems, the present paper studies non-Zeno properties of PASs subject to system parameter and/or initial state perturbations, inspired by sensitivity and uncertainty analysis of PASs. Specifically, by exploiting the geometry of polyhedral subdivisions and dynamical system techniques, this paper establishes a uniform bound on the number of mode switchings for a family of Lipschitz PASs under mild uniform conditions on system parameters and associated polyhedral subdivisions. This result is employed to show robust non-Zenoness of several classes of Lipschitz linear complementarity systems in different switching notions. The paper also develops partial results for robust non-Zenoness of non-Lipschitz PASs, particularly well-posed bimodal non-Lipschitz PASs