Optimal control of autonomous linear systems switched with a pre-assigned finite sequence

The paper deals with the optimal control of switched piecewise linear autonomous systems, where the objective is to minimize the performance index over an infinite time horizon. We assume that the switching sequence has a finite length and is pre-assigned, while the unknown switching times are the optimization parameters. We also assume that at each switch a jump in the state space may occur and that a cost may be associated to each switch. The optimal control for this class of systems takes the form of a state feedback, i.e., it is possible to identify a region of the state space such that an optimal switch should occur if and only if the present state belongs to this region. We show how such a region can be computed with a numerical procedure and show that, in the particular case in which the switching costs is null, the region is homogeneous