一类线性奇异摄动最优控制问题的渐近解

Asymptotic Solution for a Class of Linear Singularly Perturbed Optimal Control Problem

研究了一类线性奇异摄动最优控制问题的空间对照结构,讨论了初始点固定,终端自由的情形.首先根据变分法得到了一阶最优性条件,其次运用退化最优控制问题的解证明了异宿轨道的存在性,从而结合奇异摄动理论证明了原问题空间对照结构解的存在性.进一步根据解的结构,利用边界层函数法构造了奇异摄动最优控制问题一致有效的形式渐近解.最后,通过例子验证了结果的可行性.

In this paper,we consider the contrast structure for a class of linear singularly perturbed optimal control problem with fixed initial and free terminal states.Firstly,the variational method is used to obtain the optimality condition.Secondly,we prove the existence of heteroclinic orbit by the solution of reduced problem,then,the existence of contrast structure solution for the original problem is obtained by the singularly perturbed theory.Moreover,based on the structure of the solution,we construct the uniformly valid formal asymptotic solution for the singularly perturbed optimal control problem by the boundary layer function method.Finally,an example is given to show the main result.