Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains

Mokhtari Yacine

doi:10.3934/eect.2021004

Article Contents

2022, Volume 11, Issue 2: 373-397. Doi: 10.3934/eect.2021004

This issue Previous Article Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints Next Article An inverse problem for the pseudo-parabolic equation with p-Laplacian

Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains

Mokhtari Yacine^1,2, ,

1.
Laboratoire de Mathématiques UMR 6623, Université de Bourgogne Franche-Comté, 16, route de Gray, 25030 Besançon cedex, France
2.
University of Sciences and Technology Houari Boumedienne P.O.Box 32, El-Alia 16111, Bab Ezzouar, Algiers, Algeria

^* Corresponding author: Mokhtari Yacine
^* Corresponding author: Mokhtari Yacine

Received: July 1, 2020

Revised: October 1, 2020

Early access: January 7, 2021

Published online: January 7, 2021

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Abstract

In this paper, we deal with boundary controllability and boundary stabilizability of the 1D wave equation in non-cylindrical domains. By using the characteristics method, we prove under a natural assumption on the boundary functions that the 1D wave equation is controllable and stabilizable from one side of the boundary. Furthermore, the control function and the decay rate of the solution are given explicitly.

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Mathematics Subject Classification: Primary: 93D15, 93B05, 35L05.

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Figure 1. The curve $ (t,\alpha(t))_{t\geq0} $ in red and $ (t, \beta(t))_{t\geq0} $ in blue

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Figure 2. An example of a boundary curves $ (t,\alpha(t)))_{t\geq0} $ and $ (t,\beta(t)))_{t\geq0} $ that do not satisfy assumption (10). The values of the solution are not defined on the green part of the characteristic lines lying under or above these curves

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Figure 3. An example of a boundary curves $ (t,\alpha(t)))_{t\geq0} $ and $ (t,\beta(t)))_{t\geq0} $ that do not satisfy assumption (10). The values of the solution are not defined on the green part of the characteristic lines lying under or above these curves

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