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Global feedback stabilization for a class of nonlocal transport equations: the continuous and discrete case. (English) Zbl 1368.35177

Summary: 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 the third author, SIAM J. Math. Anal. 45, No. 5, 2646–2665 (2013; Zbl 1323.93060)] 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.

MSC:

35L65 Hyperbolic conservation laws
93C20 Control/observation systems governed by partial differential equations
93D15 Stabilization of systems by feedback
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs

Citations:

Zbl 1323.93060
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References:

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