Optimization Theory and Dynamical Systems: Invariant Sets and Invariance Preserving Discretization Methods

Abstract

Invariant set is an important concept in the theory of dynamical systems and it has a wide range of applications in control and engineering. This thesis has four parts, each of which studies a fundamental problem arising in this field. In the first part, we propose a novel, simple, and unified approach to derive sufficient and necessary conditions, which are referred to as invariance conditions for simplicity, under which four classic families of convex sets, namely, polyhedral, polyhedral cones, ellipsoids, and Lorenz cones, are invariant sets for linear discrete or continuous dynamical systems. This novel method establishes a solid connection between optimization theory and dynamical systems. In the second part, we propose novel methods to compute valid or largest uniform steplength thresholds for invariance preserving of three classic types of discretization methods, i.e., forward Euler method, Taylor type approximation, and rational function type discretization methods. These methods enable us to find a pre-specified steplength threshold which preserves invariance of a set. The identification of such steplength threshold has a significant impact in practice. In the third part, we present a novel approach to ensure positive local and uniform steplength threshold for invariance preserving on a set when a discretization method is applied to a linear or nonlinear dynamical system. Our methodology not only applies to classic sets, discretization methods, and dynamical systems, but also extends to more general sets, discretization methods, and dynamical systems. In the fourth part, we derive invariance conditions for some classic sets for nonlinear dynamical systems. This part can be considered as an extension of the first part to a more general case