Limit Behavior of a Linear-Quadratic Optimal Control Problem for Singularly Perturbed Systems with Small Delays

A finite horizon quadratic cost optimal control problem for a linear singularly perturbed differential system with small point-wise and distributed time delays in the state variable is considered. By a linear transformation, this problem is converted to an equivalent linear-quadratic optimal control problem for a much simpler system. For this new problem, an asymptotic solution of the Riccati-type matrix differential equation, arising in the control optimality condition, is constructed and justified. Based on this solution, a limit behavior of the optimal value of the cost functional in the transformed problem is obtained. Using this result, a limit behavior of the optimal value of the cost functional in the original problem is established.