一类二阶常微分方程的利普希茨控制问题研究

Lipschitz controls for a class of second-order ordinary differential equations

研究了一类由二阶常微分方程所支配的具有状态约束的控制问题.通过状态方程的外部函数来实现系统的控制.在适当的假设条件下,证明了状态方程解的存在惟一性,并证明了当控制函数在利普希茨连续函数类中时,最优控制的存在性.在光滑最优控制情形下,导出了一个最优化必要条件.

The control problem with pointwise state constraints governed by a class of second-order ordinary differential equations is studied.The control of the system is realized through the external function of the state equation.Under appropriate assumptions,it is proved that the existence and uniqueness of the solution of the state equation is existing and unique,and it is also proved that optimal control is existing when the control function is Lipschitz continuous function.A necessary optimality condition is deduced under the smooth optimal control function.