A Meshless Method for Optimal Control of Parabolic PDEs Using Rational Radial Basis Functions

This study introduces an innovative approach for addressing optimal control problems related to parabolic partial differential equations (PDEs) through the application of rational radial basis functions (RBFs). Parabolic PDEs, which are instrumental in modeling time-dependent processes such as heat transfer and diffusion, pose significant computational challenges in optimal control due to the requirement for precise approximations of both state and adjoint equations. The proposed approach exploits the adaptability and spectral accuracy of rational RBFs within a meshless framework, effectively addressing the limitations of traditional discretization methods. By enhancing the accuracy and efficiency of control strategies, this method significantly contributes to advancing the theory and application of optimal control in dynamic systems. The tunable shape parameters of rational RBFs allow for accurate representation of solution characteristics, including steep gradients and localized behaviors. Additionally, their meshless framework adeptly accommodates complex geometries and boundary conditions, ensuring computational efficiency through the generation of sparse and well-conditioned system matrices. This paper also introduces a novel hybrid rational RBF, termed the Gaussian rational hybrid RBF. The efficacy of the proposed approach is validated through a series of benchmark tests and practical applications, highlighting its ability to achieve high accuracy with reduced computational effort. The findings illustrate the potential of rational RBFs as a robust and versatile tool for solving optimal control problems governed by parabolic PDEs, paving the way for further exploration of advanced rational RBF-based techniques in the field of computational optimal control.

کلیدواژه ها:

Meshless methods‎ ، ‎Parabolic PDEs‎ ، ‎Rational RBFs‎ ، ‎Optimization‎ ، ‎Gaussian rational hybrid RBF