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

Ben Mansour Dia (), Mouhamadou Samsidy Goudiaby and Oliver Dorn
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Ben Mansour Dia: College of Petroleum Engineering and Geosciences (CPG), King Fahd University of Petroleum and Minerals (KFUPM), Dhahran 31261, Saudi Arabia
Mouhamadou Samsidy Goudiaby: Département de Mathématiques, UFR des Sciences et Technologies, Université Assane Seck de Ziguinchor, Ziguinchor BP 523, Senegal
Oliver Dorn: Department of Mathematics, Alan Turing Building, The University of Manchester, Oxford Rd., Manchester M13 9PL, UK

Mathematics, 2022, vol. 10, issue 21, 1-21

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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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