Asymptotics of the solution to a liner-quadratic periodic problem with a singular matrix perturbation in the performance index

A liner-quadratic periodic optimal control problem with a singular matrix perturbation in the performance index is considered. An asymptotic expansion of its solution in nonnegative integer powers of is constructed by applying a direct scheme. The proximity of the exact solution to the approximate one is estimated for the control, trajectory, and functional. It is shown that the functional minimized does not increase when a higher order asymptotic approximation to the optimal control is used.