Output-Feedback Control of the Semilinear Heat Equation via the $L^2$ Residue Separation and Harmonic Inequality

TL;DR

This paper extends the $L^2$ residue separation method to output-feedback control of the semilinear heat equation, introducing a harmonic inequality to handle residual modes and ensuring robust control against nonlinearities.

Contribution

It develops a novel output-feedback control approach for PDEs using $L^2$ residue separation and harmonic inequalities, addressing spillover effects in boundary control of nonlinear heat equations.

Findings

The method effectively eliminates spillover in boundary control.

Higher controller order increases admissible nonlinearities.

The approach connects $L^2$ residue separation with $H_$ theory.

Abstract

A popular approach to designing finite-dimensional boundary controllers for partial differential equations (PDEs) is to decompose the PDE into independent modes and focus on the dominant ones while neglecting highly damped residual modes. However, the neglected modes can adversely affect the overall system performance, causing spillover. The $L^2$ residue separation method was recently introduced to eliminate spillover in the state-feedback control design. In this paper, we extend this method to finite-dimensional output-feedback control, where the output is contaminated by the residual modes. To deal with the output residue, we introduce a new harmonic inequality that optimally bounds it. We develop the approach for a 1D heat equation with unknown nonlinearity, where boundary temperature measurements are used to control heat flux at the opposite boundary. By exploiting the connection between $L^2$ residue separation and $H_\infty$ theory, we show that the class of admissible nonlinearities can only increase with higher controller order.