On optimal periodic solution of differential equations


this paper dedicated to the construction of solution of a three time scale periodic singular perturbed non-linear quadratic optimal control problem by using the direct method. The algorithm of the method is the direct substitution of the postulated asymptotic expansion of the solution of the problem and then by the conditions of the problem we constructed of a series of problem and find terms of the asymptotic. We find the solution by using the Hamilton's function and maximum principle.