Homogeneous boundary control for a 1D reaction-diffusion PDE: modal decomposition approach

Overview & Synopsis

This paper addresses the problem of boundary stabilization for 1D linear reaction-diffusion PDE systems using the modal decomposition approach. We design nonlinear homogeneous full-state and output-feedback controllers to stabilize the unstable dynamics. We consider a linear infinite-dimensional observer for the design of the homogeneous output-feedback control. We investigate the well-posedness of the resulting closed-loop system and conduct a stability analysis by constructing a suitable Lyapunov function that accounts for the degree of homogeneity and a state-dependent switching condition based on the homogeneous norm. The properties of homogeneous-based controllers are exploited, and numerical simulations corroborate that the proposed controllers effectively mitigate peaking effects and control signal overshoot compared to classical linear boundary feedbacks designed via modal decomposition. Additional simulations are provided further to illustrate the performance of the proposed control laws.