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Volume 10
Issue 21
10.3390/math10214036
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Article

Boundary Feedback Stabilization of Two-Dimensional Shallow Water Equations with Viscosity Term

by
Ben Mansour Dia
1,*,
Mouhamadou Samsidy Goudiaby
2 and
Oliver Dorn
3
1
College of Petroleum Engineering and Geosciences (CPG), King Fahd University of Petroleum and Minerals (KFUPM), Dhahran 31261, Saudi Arabia
2
Département de Mathématiques, UFR des Sciences et Technologies, Université Assane Seck de Ziguinchor, Ziguinchor BP 523, Senegal
3
Department of Mathematics, Alan Turing Building, The University of Manchester, Oxford Rd., Manchester M13 9PL, UK
*
Author to whom correspondence should be addressed.
Mathematics 2022, 10(21), 4036; https://doi.org/10.3390/math10214036
Submission received: 26 August 2022 / Revised: 11 October 2022 / Accepted: 15 October 2022 / Published: 31 October 2022
(This article belongs to the Special Issue Applications of Partial Differential Equations)
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Abstract

This paper treats a water flow regularization problem by means of local boundary conditions for the two-dimensional viscous shallow water equations. Using an a-priori energy estimate of the perturbation state and the Faedo–Galerkin method, we build a stabilizing boundary feedback control law for the volumetric flow in a finite time that is prescribed by the solvability of the associated Cauchy problem. We iterate the same approach to build by cascade a stabilizing feedback control law for infinite time. Thanks to a positive arbitrary time-dependent stabilization function, the control law provides an exponential decay of the energy.
Keywords: shallow water flow; Faedo–Galerkin method; feedback control; PDE’s stabilization shallow water flow; Faedo–Galerkin method; feedback control; PDE’s stabilization

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MDPI and ACS Style

Dia, B.M.; Goudiaby, M.S.; Dorn, O. Boundary Feedback Stabilization of Two-Dimensional Shallow Water Equations with Viscosity Term. Mathematics 2022, 10, 4036. https://doi.org/10.3390/math10214036

AMA Style

Dia BM, Goudiaby MS, Dorn O. Boundary Feedback Stabilization of Two-Dimensional Shallow Water Equations with Viscosity Term. Mathematics. 2022; 10(21):4036. https://doi.org/10.3390/math10214036

Chicago/Turabian Style

Dia, Ben Mansour, Mouhamadou Samsidy Goudiaby, and Oliver Dorn. 2022. "Boundary Feedback Stabilization of Two-Dimensional Shallow Water Equations with Viscosity Term" Mathematics 10, no. 21: 4036. https://doi.org/10.3390/math10214036

APA Style

Dia, B. M., Goudiaby, M. S., & Dorn, O. (2022). Boundary Feedback Stabilization of Two-Dimensional Shallow Water Equations with Viscosity Term. Mathematics, 10(21), 4036. https://doi.org/10.3390/math10214036

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