Global Feedback Stabilization for a Class of Nonlocal Transport Equations: The Continuous and Discrete Case
Abstract
In this paper, we prove the global output feedback stabilization for a class of nonlinear transport equations with nonlocal velocity. It models a highly re-entrant system which is widely encountered in semiconductor manufacturing. The exponential stability of the solution to a constant equilibrium is proved by a Lyapunov function method under a natural feedback law. The smallness restriction on the initial data in [J.-M. Coron and Z. Wang, SIAM J. Math. Anal., 45 (2013), pp. 2646--2665] is removed by using the special feature of the velocity function. The exponential stabilization results for the related discretized system with an upwind scheme are obtained by the eigenvalue decomposition method and by a Lyapunov function method. Numerical simulations are provided to supplement the theoretical results.