Asymptotic analysis of classic optimal control problems by means of direct scheme

The report is dedicated to the so-called direct scheme of asymptotic analysis of the solutions of optimal control problems with a small parameter. This scheme consists of the immediate substitution of the postulated asymptotic expansion of the solution into problem condition and the construction of the optimal control problems series for finding the expansion terms. The asymptotics of the solution is obtained by the direct scheme for the following optimal control problems: the problems with singular perturbation by the matrix in the state equation and performance index; the control problems in which the solutions of the contrast structures type arise under some conditions; the continuous and discrete problems for weakly controllable systems. The closeness estimates between approximate and exact solution are given; the lack of increase of the performance index is established when the new terms of the control asymptotics are used.