Proportional feedback and optimal control of perturbed sine-Gordon kink solitons

Abstract

This study explores the stabilization of kink soliton solutions in the sine-Gordon equation against destabilizing perturbations. We compared two distinct control strategies: proportional feedback control, which offers computational efficiency, and an optimal control approach based on cost functional minimization, which balances control effort against tracking error. Using finite difference discretization and gradient descent optimization in our numerical simulations, we evaluated the effectiveness of each method. While both approaches maintain the topological integrity of the soliton, the optimal control method achieved superior perturbation suppression by selectively targeting unstable modes without disrupting soliton propagation. Additionally, we provided a quantitative assessment of the trade-offs based on stabilization time, energy consumption, and maximum deviation for each method. Our findings advanced previous research by establishing a framework for selecting control strategies based on precision requirements and available computational resources.