Global Stabilization of a Class of Nonlinear Reaction-Diffusion Partial Differential Equations by Boundary Feedback

Abstract

This paper provides global exponential stabilization results by means of boundary feedback control for 1-D nonlinear unstable reaction-diffusion Partial Differential Equations (PDEs) with nonlinearities of superlinear growth. The class of systems studied are parabolic PDEs with nonlinear reaction terms that provide "damping" when the norm of the state is large (the class includes reaction-diffusion PDEs with polynomial nonlinearities). The case of Dirichlet actuation at one end of the domain is considered and a Control-Lypunov Functional construction is applied in conjunction with Stampacchia's truncation method. The paper also provides several important auxiliary results; among which is an extension of Wirtinger's inequality, used here for the construction of the Control Lyapunov functional.

Feedback Stabilization of Tank-Liquid System with Robustness to Wall Friction

“…c) How large is the set X U (r) for which robust exponential stabilization is achieved? Proposition 2.5 and definition (20) guarantees that the set X U (r) for r > 0 contains a neighborhood of 0, 0,…”

“…, where X is defined by ( 28)). The size of the set X U (r) depends on r ∈ [0, R) that satisfies (49) and on β, γ, δ, q, k (recall definitions (17), (20), (32)). The dependence of X U (r) on q, k is clear: the larger q (or k) the smaller the set X U (r).…”

“…The dependence of X U (r) on q, k is clear: the larger q (or k) the smaller the set X U (r). However, the dependence of X U (r) on ω 1 ∈ (0, h * ) and ω 2 > 0 (through (45), (46), (49) and (50)), on β, γ, δ (through (17), (20), (32) and through the dependence of R on δ; see Remark 2.3) is not clear: it is a very complicated, non-monotonic dependence which cannot be described easily. When µ → 0 it holds that R → 0 (recall Remark 2.3) and consequently (since r ∈ [0, R)) r → 0.…”

“…Clearly, definition (20) implies that X U (r) ⊆ X V (r) for all r ≥ 0. Moreover, inequalities (27) imply that…”

Lyapunov-based boundary feedback design for parabolic PDEs

“…In many cases the stabilization results are local, guaranteeing exponential stability in specific spatial norms. The papers [10,11] presented methodologies for global feedback stabilization of boundary controlled nonlinear parabolic PDEs: a small-gain methodology is applied in [10], while a CLF methodology is used for PDEs with at most one unstable mode in [11].…”