We consider an optimal control problem for a hybrid system whose continuous motion alternates with discrete variations (switchings) under which the dimension of the state space changes. The moments and the number of switchings are not specified in advance. They are determined as a result of minimizing a functional that incorporates the cost of each switching. The state space may change, for example, when the number of control objects varies, which is typical, in particular, of control problems for groups of a variable number of aircraft. We obtain sufficient optimality conditions for such systems and derive equations for the synthesis of optimal trajectories. The application of optimality conditions is demonstrated in academic examples.

中文翻译：

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变维混合系统的充分最优条件

我们考虑一个混合动力系统的最优控制问题，该混合动力系统的连续运动与离散的变化（切换）交替变化，状态空间的大小在变化下发生变化。力矩和开关次数没有预先指定。确定它们是因为将包含每个开关成本的功能降到最低。状态空间可以例如在控制对象的数量变化时改变，这尤其是对于可变数量的飞机的组的控制问题而言是典型的。我们为此类系统获得了充​​分的最优性条件，并推导了用于合成最优轨迹的方程式。最佳条件的应用在学术实例中得到了证明。