Invariance Conditions for Nonlinear Dynamical Systems

TL;DR

This paper extends invariance conditions from linear to nonlinear dynamical systems using advanced mathematical tools, and proposes an optimization framework for verification.

Contribution

It generalizes invariance conditions to nonlinear systems and introduces a computational method for their verification.

Findings

Derived invariance conditions for nonlinear systems using Farkas lemma, S-lemma, and Nagumo's Theorem.

Established an optimization framework for verifying invariance conditions.

Provided conditions for invariance of convex sets like polyhedral and ellipsoidal sets.

Abstract

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.