Backstepping Control of a Class of Space-Time-Varying Linear Parabolic PDEs via Time Invariant Kernel Functions

This paper aims at addressing the problem of exponential stabilization and continuous dependence of solutions on initial data for a class of 1-D linear parabolic PDEs with space-time-varying coefficients under backstepping boundary control. More precisely, in order to stabilize the system without involving a Gevrey-like condition or the event-triggered scheme, a boundary feedback controller is designed via a time invariant kernel function. Thus, the complexity of control design and implementation can be significantly reduced. The well-posedness of the closed-loop system is assessed by using Schauder’s theory and the [Formula: see text] theory of parabolic PDEs. By using the approximative Lyapunov method and a priori estimates of certain linear operators associated with Volterra integral transformations, the exponential stability of the closed-loop system is established in the spatial [Formula: see text] norm whenever [Formula: see text]. Then, based on the obtained exponential stability, it is shown that the solution to the considered system depends continuously on the spatial [Formula: see text] norm of the initial data. Numerical simulations are conducted to illustrate the effectiveness of the proposed scheme.