We study a class of discrete-time stochastic systems composed of a large number of N interacting objects, which are classified in a finite number of classes. The behavior of the objects is controlled by a central decision-maker as follows. At each stage, once the configuration of the system is observed, the controller takes a decision; then a cost is incurred and there is a positive probability the process stops, otherwise the objects move randomly among the classes according to a transition probability. That is, with positive probability, the system is absorbed by a configuration that represents the death of the system, and there it will remain without incurring cost. Due to the large number of objects, the control problem is studied according to the mean field theory. Thus, instead of analyzing a single object, we focus on the proportions of objects occupying each class, and then we study the limit as N goes to infinity.

我们研究了一类由大量N组成的离散时间随机系统交互对象，它们被划分为有限数量的类。对象的行为由中央决策者控制，如下所示。在每个阶段，一旦观察到系统的配置，控制器就会做出决定；则将产生成本，并且过程停止的概率为正，否则，对象将根据转移概率在类之间随机移动。就是说，以正概率，系统会被代表系统死亡的配置所吸收，并且将在那里保留而不会产生成本。由于对象数量众多，因此根据平均场理论研究了控制问题。因此，我们不分析单个对象，而是关注占据每个类的对象的比例，然后将极限研究为N 达到无穷大。