Abstract This paper discusses a special kind of hybrid system, named hybrid parametric system, which subsystem is characterized by parameters of model function. The parameters of model function represent the batch operation (discrete control). The joint optimization of both continuous and discrete control variables is named hybrid parametric optimal control problem. The corresponding necessary optimality conditions are established in the form of the hybrid parametric minimum principles, which are proposed and proved in this paper. The first principle, i.e. single stage hybrid parametric minimum principle, is proved based on the similar technique used by the proof of the classical Pontryagin maximum principle, and the second principle, multistage hybrid parametric minimum principle, is proved by reducing it into a nominal case of first principle. Moreover, in the virtue of the continuity of parameter space, the hybrid parametric minimum principle only requires a prescribed number of switching times.

中文翻译：

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混合参数最小值原理

摘要 本文讨论了一种特殊的混合系统，称为混合参数系统，其子系统以模型函数的参数为​​特征。模型函数的参数代表批量操作（离散控制）。连续和离散控制变量的联合优化称为混合参数优化控制问题。以混合参数极小原理的形式建立了相应的必要优化条件，本文提出并证明了这些原理。第一个原理，即单级混合参数最小值原理，是基于经典庞特里亚金最大值原理证明所使用的类似技术得到的证明，第二个原理，多级混合参数最小值原理，通过将其简化为标称情况来证明的第一原则。