A Gradient-based Kernel Optimization Approach for Parabolic Distributed Parameter Control Systems

TL;DR

This paper introduces a gradient-based kernel optimization method for parabolic distributed parameter systems that avoids solving complex PDEs, using polynomial parameterization and stability constraints, with demonstrated numerical effectiveness.

Contribution

A novel gradient-based optimization approach for feedback kernel design that bypasses traditional PDE solutions and ensures stability through coefficient constraints.

Findings

Effective kernel optimization demonstrated via numerical simulations.

Method avoids solving Riccati or Klein-Gorden PDEs.

Ensures closed-loop stability with coefficient constraints.

Abstract

This paper proposes a new gradient-based optimization approach for designing optimal feedback kernels for parabolic distributed parameter systems with boundary control. Unlike traditional kernel optimization methods for parabolic systems, our new method does not require solving non-standard Riccati-type or Klein-Gorden-type partial differential equations (PDEs). Instead, the feedback kernel is parameterized as a second-order polynomial whose coefficients are decision variables to be tuned via gradient-based dynamic optimization, where the gradient of the system cost functional (which penalizes both kernel and output magnitude) is computed by solving a so-called costate PDE instandard form. Special constraints are imposed on the kernel coefficients to ensure that, under mild conditions, the optimized kernel yields closed-loop stability. Numerical simulations demonstrate the effectiveness of the proposed approach.