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

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 in standard 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.