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              <p>In this paper, we address the problem of nonlinear boundary control design for the 1D 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 analyze the wellposedness 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 to a linear one regarding closed-loop performance and the peaking effect.</p>
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