Numerical approach to optimal control problems governed by delay differential equations

Abstrakt

This research presents a comprehensive framework for solving optimal control problems (OCPs) governed by delay differential equations (DDEs) with mixed constraints. For this more complex case of DDE-constrained problems, where analytical solutions are generally unavailable, a numerical methodology is developed. The approach involves discretizing the cost functional and the
DDE constraints using Simpson’s rule and the Crank-Nicolson method, respectively. The resulting finite-dimensional, constrained optimization problem is then transformed into an unconstrained form using the Augmented Lagrangian Method (ALM). This formulation gives rise to an associated operator that facilitates the application of the Extended Conjugate Gradient Method (ECGM) to efficiently solve the problem. Numerical experiments demonstrate that the solutions obtained from the proposed ALM-ECGM scheme are in strong agreement with analytical solutions for ODE cases and outperform existing methods for DDE cases. A rigorous convergence analysis confirms the scheme’s effectiveness, accuracy, and linear convergence rate, establishing it as a robust computational tool for this challenging class of OCPs.