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

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.