Research paperNumerical solution of the state-delayed optimal control problems by a fast and accurate finite difference θ-method

- Linear and nonlinear state-delayed optimal control problem is solved by an accurate and fast finite difference θ-method.
- The proposed method satisfies the necessary conditions of nonlinear state-delayed optimal control problem.
- Error analysis and a matrix formulation of the proposed method are provided.
- To solve the infinite-time horizon time-delayed optimal control problems, a piecewise version of the θ-method is formulated.

Using the Pontryagin's maximum principle for a time-delayed optimal control problem results in a system of coupled two-point boundary-value problems (BVPs) involving both time-advance and time-delay arguments. The analytical solution of this advance-delay two-point BVP is extremely difficult, if not impossible. This paper provides a discrete general form of the numerical solution for the derived advance-delay system by applying a finite difference θ-method. This method is also implemented for the infinite-time horizon time-delayed optimal control problems by using a piecewise version of the θ-method. A matrix formulation and the error analysis of the suggested technique are provided. The new scheme is accurate, fast and very effective for the optimal control of linear and nonlinear time-delay systems. Various types of finite- and infinite-time horizon problems are included to demonstrate the accuracy, validity and applicability of the new technique.