Homogeneous boundary control design for a one-dimensional viscous Burgers equation: modal decomposition approach

Abstract In this paper, we address the problem of non-linear boundary control design for the one-dimensional viscous Burgers equation. Using the modal decomposition approach, we design a finite-dimensional homogeneous controller for the controlled subsystem containing unstable modes. We prove that the closed-loop system with the homogeneous state-feedback control is locally asymptotically stable by constructing an appropriate Lyapunov functional depending on the canonical homogeneous norm and the degree of homogeneity. Next, we construct a linear infinite-dimensional observer and design the homogeneous output-feedback controller to achieve the local stability of the closed-loop system. We analyse the well-posedness of the closed-loop system with the finite-dimensional homogeneous state/output feedback control by using the Arzela–Ascoli theorem. Finally, we present numerical simulations to support the analytical results and compare our proposed controller with a linear one regarding closed-loop performance and the peaking effect.