Shooting continuous Runge--Kutta method for delay optimal control problems

In this paper, we present an efficient method to solve linear time-delay optimal control problems with a quadratic cost function.  In this regard, first, by employing the Pontryagin maximum principle to time-delay systems, the original problem is converted into a sequence of two-point boundary value problems (TPBVPs) that have both advance and delay terms. Then, using the continuous Runge--Kutta (CRK) method, the resulting sequences are recursively solved by the shooting method to obtain an optimal control law. This obtained optimal control consists of a linear feedback term, which is obtained by solving a Riccati matrix differential equation, and a forward term, which is an infinite sum of adjoint vectors, that can be obtained by solving sequences of delay TPBVPs by the shooting CRK method.Finally, numerical results and their comparison with other available results illustrate the high accuracy and efficiency of our proposed method.