The asymptotic expansion for a class of non-linear singularly perturbed problems with optimal control

In this article, we discuss a class of three-dimensional non-linear singularly perturbed systems with optimal control. Firstly, we confirm the existence of heteroclinic orbits connecting two equilibrium points about their associated systems by necessary conditions of optimal control and functional theory. Secondly, we study the asymptotic solutions of the singularly perturbed optimal control problems by the methods of boundary layer functions and prove the existence of the smooth solutions and the uniform validity of the asymptotic expansion. Finally, we cite an example to illustrate the result.