Direct Scheme for the Asymptotic Solution of Linear-Quadratic Problems with Cheap Controls of Different Costs

Abstract

For linear-quadratic problems whose performance criteria contain a sum of two quadratic forms with respect to the control with different powers of a small parameter, an algorithm for constructing asymptotic approximations of arbitrary order to the solution with boundary functions of four types is justified. The proposed algorithm is based on the direct substitution of the postulated asymptotic expansion of the solution into the condition of the transformed problem and the construction of a series of optimal control problems for determining the terms of the asymptotics of the solution of the transformed problem which is a singularly perturbed three-rate optimal control problem in the critical case. The estimates of the proximity between the asymptotic and exact solutions are proved, as well as the fact that the values of the functional to be minimized do not increase when they are used in the next approximation to the optimal control.