The paper deals with the optimal control of switched piece-wise linear autonomous systems, where the objective is that of minimizing a 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.