High order asymptotic solution of linear-quadratic optimal control problems under cheap controls with two different costs

The paper deals with linear-quadratic optimal control problems the performance index of which contains small parameters of two different orders of smallness at quadratic forms with respect to a control. Such problems can be considered as a result of applying the convolution method to problems with three performance indices where the cost of one cheap control is negligible compared with another one. Asymptotic approximations of a solution of arbitrary orders are constructed using the direct scheme method, which consists of an immediate substitution of a postulated asymptotic expansion of a solution into the problem condition and determining a series of optimal control problems for finding terms of an asymptotic expansion. At first, using the variables change, the original problem is transformed to a singularly perturbed optimal control problem with three-tempo state variables. The constructed asymptotic solution contains regular and boundary functions of four types.