Discrete-Time Hybrid Control in Borel Spaces

A discrete-time hybrid control model with Borel state and action spaces is introduced. In this type of models, the dynamic of the system is composed by two sub-dynamics affecting the evolution of the state; one is of a standard-type that runs almost every time and another is of a special-type that is active under special circumstances. The controller is able to use two different type of actions, each of them is applied to each of the two sub-dynamics, and the activations of these sub-dynamics are possible according to an activation rule that can be handled by the controller. The aim for the controller is to find a control policy, containing a mix of actions (of either standard- or special-type), with the purpose of minimizing an infinite-horizon discounted cost criterion whose discount factor is dependent on the state-action history and may be equal to one at some stages. Two different sets of conditions are proposed to guarantee (i) the finiteness of the cost criterion, (ii) the characterization of the optimal value function and (iii) the existence of optimal control policies; to do so, we employ the dynamic programming approach. A useful characterization that signalizes the accurate times between changes of sub-dynamics in terms of the so-named contact set is also provided. Finally, we introduce two examples that illustrate our results and also show that control models such as discrete-time impulse control models and discrete-time switching control models become special cases of our present hybrid model