Nice-Reachability Results for Discrete-Time Linear Switched Systems with Applications to Stability Under Arbitrary Switching Laws

A powerful approach for analyzing the stability under arbitrary switching of continuous-time switched systems is based on analyzing stability for the "most unstable" switching law. This approach has been successfully applied to derive nice-reachability-type results for both linear and nonlinear continuous-time switched systems. We develop an analogous approach for discrete-time linear switched systems. We first prove a necessary condition for the "most unstable" switching law in the form of a discrete-time maximum principle (MP). This MP is in fact weaker than its continuous-time counterpart. To overcome this, we introduce the auxiliary system of a discrete-time linear switched system, and show that regularity properties of time-optimal controls (TOCs) for the auxiliary system imply nice-reachability results for the original discrete-time linear switched system. We derive several new Lie-algebraic conditions guaranteeing nice-reachability results. These results, and their proofs, turn out to be quite different from their continuous-time counter parts.