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''

Qiaoling Chen; Jun Zheng; Guchuan Zhu

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arXiv · 2301.10415

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''

Qiaoling Chen·Jun Zheng·Guchuan Zhu

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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.

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Qiaoling Chen, Jun Zheng, Guchuan Zhu. 2023-01-25. 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''. https://arxiv.org/abs/2301.10415

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