We consider the problem of average optimal control for a linear hybrid system whose continuous motion alternates with discrete changes (switchings) that change the state space. The initial system state is random. The control quality is characterized by the mean value of a quadratic functional. Switching times and their number are not known in advance. They are determined by minimizing the functional. For the problem under consideration, the classical separation principle does not hold. We prove the so-called conditional separation principle. We also show sample applications of conditional and classical separation principles.

中文翻译：

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变维混合系统平均最优控制的分离定理

我们考虑一个线性混合系统的平均最优控制问题，该系统的连续运动与改变状态空间的离散变化（开关）交替出现。初始系统状态是随机的。控制质量的特征在于二次函数的平均值。事先不知道切换时间及其数量。通过最小化功能来确定它们。对于所考虑的问题，经典分离原理不成立。我们证明了所谓的条件分离原理。我们还展示了条件分离和经典分离原理的示例应用。