Solution of a Singular Optimal Control Problem with State Delays: A Cheap Control Approach

An optimal control problem with quadratic cost functional for a linear system with point-wise and distributed time delays in the state variables is considered. The case where the cost functional does not contain a control cost is treated. The latter means that the problem under consideration is a singular optimal control problem. This control problem is associated with a new optimal control problem for the same equation of dynamics. The cost functional in this new problem is the sum of the original cost functional and an integral of the square of the control with a small positive weighting coefficient. Thus, the new problem is a cheap control problem. By using the singular perturbation technique, an asymptotically suboptimal state-feedback control is constructed for the cheap control problem. Then it is shown that this control constitutes a minimizing sequence for the original problem.