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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Ben Mansour Dia & Mouhamadou Samsidy Goudiaby & Oliver Dorn, 2022. "Boundary Feedback Stabilization of Two-Dimensional Shallow Water Equations with Viscosity Term," Mathematics, MDPI, vol. 10(21), pages 1-21, October.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:21:p:4036-:d:958490
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