Separation Theorem in Control Problems for Bundles of Trajectories of Deterministic Linear Switchable Systems

We consider optimal (on average) control problems and guaranteeing optimal control problems for bundles of trajectories of deterministic linear switchable systems such that the continuous variation of their state is described by differential equations, while the instant discrete changes of their state (switchings) are described by recurrent equations. The switching times and the number of switchings are not given. For one trajectory, the quality of the control is characterized by a quadratic functional taking into account the expenditure for each switching as well. Exact information about the initial state of the system is not known; therefore, control problems for bundles of trajectories are investigated. For linear-quadratic control problems for switching systems, the classical separation principle is not fulfilled. It turns out that its modification called the conditional separation principle holds. We provide academic examples of the synthesis of the optimal control on average and guaranteeing optimal control in which the separation principle is not fulfilled, while the conditional separation principle holds.