A Second-Order Optimality Condition for the Most Destabilizing Control of a Discrete-Time Bilinear Control System*

A powerful approach for analyzing the stability of continuous–time switched systems is based on using tools from 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 (DT) switched systems, the variational approach received considerably less attention. It 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 DT bilinear control systems. This provides new information on the optimal controls that cannot be derived using the first–order MP. We demonstrate several applications of this second–order MP to the stability analysis of DT linear switched systems.