  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.
