# Lyapunov-Based Boundary Feedback Design For Parabolic PDEs

## TL;DR

This paper introduces a Lyapunov-based boundary feedback design method for parabolic PDEs, constructing simple control functionals that enable global exponential stabilization with nonlinearities.

## Contribution

It presents a novel, simple CLF construction and a boundary feedback design methodology that achieves stabilization of nonlinear parabolic PDEs.

## Key findings

- Constructed simple, almost diagonal CLFs with only single integrals.
- Designed boundary feedback controllers with internal dynamics.
- Achieved global exponential stabilization for certain nonlinear PDEs.

## Abstract

This paper presents a methodology for the construction of simple Control Lyapunov Functionals (CLFs) for boundary controlled parabolic Partial Differential Equations (PDEs). The proposed methodology provides functionals that contain only simple (and not double or triple) integrals of the state. Moreover, the constructed CLF is "almost diagonal" in the sense that it contains only a finite number of cross-products of the (generalized) Fourier coefficients of the state. The methodology for the construction of a CLF is combined with a novel methodology for boundary feedback design in parabolic PDEs. The proposed feedback design methodology is Lyapunov-based and the feedback controller is an "integral" controller with internal dynamics. It is also shown that the obtained simple CLFs can provide nonlinear boundary feedback laws which achieve global exponential stabilization of semilinear parabolic PDEs with nonlinearities that satisfy a linear growth condition.

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Source: https://tomesphere.com/paper/1905.01701