Small-Gain-Based Boundary Feedback Design for Global Exponential Stabilization of One-Dimensional Semilinear Parabolic PDEs

Abstract

This paper presents a novel methodology for the design of boundary feedback stabilizers for 1-D, semilinear, parabolic PDEs. The methodology is based on the use of small-gain arguments and can be applied to parabolic PDEs with nonlinearities that satisfy a linear growth condition. The nonlinearities may contain nonlocal terms. Two different types of boundary feedback stabilizers are constructed: a linear static boundary feedback and a nonlinear dynamic boundary feedback. It is also shown that there are fundamental limitations for feedback design in the parabolic case: arbitrary gain assignment is not possible by means of boundary feedback. An example with a nonlocal nonlinear term illustrates the applicability of the proposed methodology.

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

“…where 22 , : (0,1) (0,1) K G L L  are the continuous linear operators defined by the following equations for all…”

“…Consider the Volterra operator 22 : (0,1) (0,1) , the time derivative of the control action would not be differentiable if the feedback law is nonlinear. That would not allow the homogenization of the boundary conditions (as performed in the proof of Theorem 3.1), which is necessary for the development of existence/uniqueness results.…”

“…The recent paper [22] provided global exponential stabilization results in the 2 L spatial norm for 1-D nonlinear parabolic PDEs by means of boundary feedback. A small-gain methodology (based on the results in [20]) was applied to parabolic PDEs with nonlinearities that satisfy a linear growth condition.…”

“…We consider the case of Dirichlet actuation at one end of the domain. Our methodology is different from the small-gain methodology in [22]: here we construct Control Lyapunov Functionals (CLFs) which provide exponential stability estimates in the 2 L spatial norm. Lyapunov functionals for reaction-diffusion PDEs were recently proposed in [27,28].…”

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].…”