Asymptotic solution of linear-quadratic control problem with intermediate points and small parameter in performance index

The asymptotic expansion of the solution of a linear-quadratic optimal control problem for a descriptor system with intermediate points and a small parameter in a performance index has been constructed as series of non-negative integer powers of a small parameter. The estimates have been obtained for the proximity of the asymptotic approximate solutions to the exact one. The nice property is proved, namely, the values of the minimized functional do not increase when higher-order approximations to the optimal control are used. The numerical example is given in order to illustrate the proposed method.