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.

中文翻译：

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Borel空间中的离散时间混合控制

介绍了具有Borel状态和作用空间的离散时间混合控制模型。在这种类型的模型中，系统的动力学是由两个影响状态演化的子动力学组成的。一种是几乎每次运行的标准类型，另一种是在特殊情况下处于活动状态的特殊类型。控制器能够使用两种不同类型的动作，它们分别应用于两个子动力学中的每一个，并且根据激活规则可以激活这些子动力学。可以由控制器处理。控制器的目的是找到一种控制策略，其中包含（标准或特殊类型的）行为混合，以最小化其折扣因子取决于状态行为的无限水平折现成本准则。历史，在某些阶段可能等于一。为了保证（i）成本准则的有限性，（ii）最优值函数的特征以及（iii）最优控制策略的存在，提出了两种不同的条件。为此，我们采用了动态编程方法。还提供了有用的表征，该表征用所谓的接触集来表示子动力学变化之间的准确时间。最后，