Proof of Proposition 3.1 in the paper titled "Backstepping control of a class of space-time-varying linear parabolic PDEs via time invariant kernel functions''

TL;DR

This paper provides a detailed proof of a key proposition related to the exponential stabilization of space-time-varying linear parabolic PDEs using backstepping with time-invariant kernels, ensuring stability without Gevrey or event-triggered schemes.

Contribution

It offers a rigorous proof of exponential stability and continuous dependence for a class of PDEs controlled via time-invariant kernels, advancing theoretical understanding of backstepping control.

Findings

Exponential stability in L^p and W^{1,p} norms established.

Solution dependence on initial data shown to be continuous.

Backstepping control achieved without Gevrey or event-triggered schemes.

Abstract

We provide a detailed proof of Proposition 3.1 in the paper titled ``Backstepping control of a class of space-time-varying linear parabolic PDEs via time invariant kernel functions''. In the paper titled ``Backstepping control of a class of space-time-varying linear parabolic PDEs via time invariant kernel functions'', we addressed the problem of exponential stabilization and continuous dependence of solutions on initial data in different norms for a class of $1$-D linear parabolic PDEs with space-time-varying coefficients under backstepping boundary control. In order to stabilize the system without involving a Gevrey-like condition or the event-triggered scheme, a boundary feedback controller was designed via a time invariant kernel function. By using the approximative Lyapunov method, the exponential stability of the closed-loop system was established in the spatial $L^{p}$-norm and $W^{1,p}$-norm, respectively, whenever $p\in [1, +\infty]$. It was also shown that the solution to the considered system depends continuously on the spatial $L^{p}$-norm and $W^{1,p}$-norm, respectively, of the initial data.