A second-order maximum principle for discrete-time bilinear control systems with applications to discrete-time linear switched systems

A powerful approach for analyzing the stability of continuous–time switched systems is based on using optimal control theory to characterize the “most unstable” switching law. This reduces the problem of determining stability under arbitrary switching to analyzing stability for the specific “most unstable” switching law. For discrete–time switched systems, the variational approach received considerably less attention. This approach is based on using a first–order necessary optimality condition in the form of a maximum principle (MP), and typically this is not enough to completely characterize the “most unstable” switching law. In this paper, we provide a simple and self–contained derivation of a second–order necessary optimality condition for discrete–time bilinear control systems. This provides new information that cannot be derived using the first–order MP. We demonstrate several applications of this second–order MP to the stability analysis of discrete–time linear switched systems.