A step-type solution for the affine singularly perturbed problem of optimal control

The article deals with a dynamic analysis that provides a way of defining the location of boundary and internal transition layers for a certain singularly perturbed problem of optimal control. A direct diagram is used in the development of a formal asymptotic expansion for the optimal solution. Investigation of the problems of optimal control with the ‘step‐type’ contrast structure is extended from the scalar to the vector case and the approach to the study of similar problems is improved. The existence of the solution is proved and the estimate is given of the remainder terms of the asymptotic expansion of the ‘step‐type’ optimal solution. A numerical example is cited.