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''

2023/01/25 by Chen, Qiaoling, Zheng, Jun, Zhu, Guchuan
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2301.10415

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∈ [1, +∞]. 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.

Related

• Exponential stabilization and continuous dependence of solutions on initial data in different norms for space-time-varying linear parabolic PDEs
• Input-to-State Stabilization of 1-D Parabolic PDEs under Output Feedback Control
• Output Feedback Stabilization of Semilinear Parabolic PDEs using\n Backstepping
• Sampled-Data Boundary Feedback Control of 1-D Parabolic PDEs
• Boundary Control of Coupled Reaction-Advection-Diffusion Systems with Spatially-Varying Coefficients