Rapid boundary stabilization of 1D nonlinear parabolic equations
Yu Xiao, Can Zhang
TL;DR
The paper addresses rapid boundary stabilization of 1D nonlinear parabolic PDEs by a modal-decomposition-based feedback that targets unstable spectral modes. It introduces a memory-boundary control $u(t)=\int_0^t e^{k(t-s)}\mathcal{K}(y(s))\,ds$ and proves local rapid stabilization under a local Lipschitz/global-growth condition (H) on the nonlinearity $f$, along with a globally rapid stabilization result for dissipative nonlinearities. The approach combines finite-dimensional pole-shifting for a set of unstable modes with Lyapunov analysis and an LMIs framework to control the infinite-dimensional tail, yielding an exponential decay $\|y(t)\|_{H^1}\le M e^{-\delta t}\|y_0\|_{H^1}$ for small data, and extends to global results in dissipative cases such as Burgers and Allen–Cahn. Applications demonstrate how the spectral-reduction strategy enables constructive and robust rapid stabilization for nonlinear PDEs, with potential impact on boundary control of diffusion-dominated systems in physics and engineering.
Abstract
In this paper, we focus on the rapid boundary stabilization of 1D nonlinear parabolic equations via the modal decomposition method. The nonlinear term is assumed to satisfy certain local Lipschitz continuity and global growth conditions. Through the modal decomposition, we construct a feedback control that modifies only the unstable eigenvalues to achieve spectral reduction. Under this control, we establish locally rapid stabilization by estimating the nonlinearity in Lyapunov stability analysis. Furthermore, utilizing the dissipative property, we derive a globally rapid stabilization result for dissipative systems such as the Burgers equation and the Allen-Cahn equation.