A practical approach for optimal control problems under partial differential equations

Abstract

This paper introduces a novel pseudo-spectral (PS) method tailored for the numerical solution of optimal control problems (OCPs) governed by partial differential equations (PDEs). The proposed technique leverages two-dimensional interpolating polynomials constructed on shifted Legendre-Gauss-Lobatto nodes to approximate the state and control variables. This spectral discretization transforms the original infinite-dimensional control problem into a finite-dimensional nonlinear programming (NLP) formulation, enabling efficient numerical treatment. To solve the resulting NLP, we derive the Karush-Kuhn-Tucker (KKT) optimality conditions in full detail, leading to a system of algebraic equations that encapsulate the necessary conditions for optimality. The Levenberg-Marquardt algorithm, known for its robustness in solving nonlinear algebraic systems, is then employed to solve this system iteratively, yielding an accurate approximation of the optimal solution. The effectiveness and reliability of the proposed method are demonstrated through a series of benchmark numerical examples involving PDE-constrained OCPs. Comparative analyses reveal that our approach not only achieves superior accuracy and computational efficiency compared to existing methods but also offers notable advantages in terms of implementation simplicity and scalability. These features make it a compelling alternative for tackling complex OCPs in various scientific and engineering domains.