Exponential stabilization and continuous dependence of solutions on initial data in different norms for space-time-varying linear parabolic PDEs

TL;DR

This paper develops a method to achieve exponential stabilization and continuous dependence of solutions on initial data in various norms for space-time-varying linear parabolic PDEs without relying on Gevrey-like conditions, using a combination of backstepping and ALFs.

Contribution

It introduces a novel approach combining backstepping and approximation of Lyapunov functionals to stabilize PDEs and analyze solution dependence without Gevrey conditions.

Findings

Achieved exponential stabilization in L^p and W^{1,p} norms.

Established continuous dependence of solutions on initial data in different norms.

Developed a method to construct time-independent kernel functions without Gevrey conditions.

Abstract

For an arbitrary parameter $p\in [1,+\infty]$, we consider the problem of exponential stabilization in the spatial $L^{p}$-norm, and $W^{1,p}$-norm, respectively, for a class of anti-stable linear parabolic PDEs with space-time-varying coefficients in the absence of a Gevrey-like condition, which is often imposed on time-varying coefficients of PDEs and used to guarantee the existence of smooth (w.r.t. the time variable) kernel functions in the literature. Then, based on the obtained exponential stabilities, we show that the solution of the considered system depends continuously on the $L^{p}$-norm, and $W^{1,p}$-norm, respectively, of the initial data. In order to obtain time-independent (and thus sufficiently smooth) kernel functions without a Gevrey-like condition and deal with singularities arising in the case of $p\in[1,2)$, we apply a combinatorial method, i.e., the combination of backstepping and approximation of Lyapunov functionals (ALFs), to stabilize the considered system and establish the continuous dependence of solutions on initial data in different norms.