Integral input-to-state stabilization in different norms for a class of linear parabolic PDEs

In this paper, we study the problem of integral input-to-state stabilization in the spatial L
^p
-norm and W
^1,p
-norm for 1-D linear parabolic PDEs with spatially varying coefficients and L
^r
-inputs, whenever p ∈ [1, +∞] and r ∈ [p, +∞]. Based on the application of Volterra integral transformation, we design a boundary feedback controller by using the standard backstepping method, while not involving sliding mode control. To overcome the difficulty in finding an invertible transformation under an explicit form due to the appearance of external inputs, we apply the method of functional analysis and the theory of series to prove the existence of an invertible transformation, which is required in the assessment of well-posedness and stability of the closed-loop system. Meanwhile, we establish a prior estimates in different norms for the associated linear operators. Therefore, the existence, uniqueness and regularity of a solution to the closed-loop system are guaranteed. In addition, we apply the approximative Lyapunov method, as well as a prior estimates of the corresponding linear operators, to assess the stability in different norms for the closed-loop system with integrable inputs, which represents an improvement to known results based on Lyapnov stability analysis.