# American Institute of Mathematical Sciences

March  2020, 9(1): 1-25. doi: 10.3934/eect.2020014

## Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints

 Laboratory of Functional Analysis and Geometry of Spaces, Faculty of Mathematics and Computer Sciences, University of M'sila, 28000, Algeria

Received  November 2017 Revised  May 2018 Published  March 2020 Early access  October 2019

Fund Project: The author is supported by the Algerian Ministry of Higher Education and Scientific Research, under a PRFU Project No. C00L03UN280120180007.

We study the wave equation in an interval with two linearly moving endpoints. We give the exact solution by a series formula, then we show that the energy of the solution decays at the rate $1/t$. We also establish observability results, at one or at both endpoints, in a sharp time. Moreover, using the Hilbert uniqueness method, we derive exact boundary controllability results.

Citation: Abdelmouhcene Sengouga. Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints. Evolution Equations & Control Theory, 2020, 9 (1) : 1-25. doi: 10.3934/eect.2020014
##### References:
 [1] N. L. Balazs, On the solution of the wave equation with moving boundaries, J. Math. Anal. Appl., 3 (1961), 472-484.  doi: 10.1016/0022-247X(61)90071-3.  Google Scholar [2] C. Bardos and G. Chen, Control and stabilization for the wave equation. Ⅲ: Domain with moving boundary, SIAM J. Control Optim., 19 (1981), 123-138.  doi: 10.1137/0319010.  Google Scholar [3] G. Birkhoff and G.-C. Rota, Ordinary Differential Equations, 4$^{th}$ edition, John Wiley & Sons, Inc., New York, 1989.  Google Scholar [4] J. Cooper and C. Bardos, A nonlinear wave equation in a time dependent domain, J. Math. Anal. Appl., 42 (1973), 29-60.  doi: 10.1016/0022-247X(73)90120-0.  Google Scholar [5] L. Z. Cui, X. Liu and H. Gao, Exact controllability for a one-dimensional wave equation in non-cylindrical domains, J. Math. Anal. Appl., 402 (2013), 612-625.  doi: 10.1016/j.jmaa.2013.01.062.  Google Scholar [6] V. V. Dodonov, A. B. Klimov and D. E. Nikonov, Quantum phenomena in resonators with moving walls, J. Math. Phys., 34 (1993), 2742-2756.  doi: 10.1063/1.530093.  Google Scholar [7] E. Knobloch and R. Krechetnikov, Problems on time-varying domains: Formulation, dynamics, and challenges, Acta Appl. Math., 137 (2015), 123-157.  doi: 10.1007/s10440-014-9993-x.  Google Scholar [8] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics, Masson, Paris, John Wiley & Sons, Ltd., Chichester, 1994.  Google Scholar [9] V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics, Springer-Verlag, New York, 2005. doi: 10.1007/b139040.  Google Scholar [10] J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod-Gautier Villars, Paris, 1969.  Google Scholar [11] J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systemes Distribués. Tome 1. Contrôlabilité Exacte, Recherches en Mathématiques Appliquées, 8. Masson, Paris, 1988.  Google Scholar [12] L. Q. Lu, S. J. Li, G. Chen and P. F. Yao, Control and stabilization for the wave equation with variable coefficients in domains with moving boundary, Systems & Control Lett., 80 (2015), 30-41.  doi: 10.1016/j.sysconle.2015.04.003.  Google Scholar [13] L. A. Medeiros, J. Limaco and S. B. Menezes, Vibrations of elastic strings: Mathematical aspects, part two, J. Comput. Anal. Appl., 4 (2002), 211-263.  doi: 10.1023/A:1013151525487.  Google Scholar [14] M. Milla Miranda, Exact controllability for the wave equation in domains with variable boundary, Rev. Mat. Complut. Madrid, 9 (1996), 435-457.  doi: 10.5209/rev_rema.1996.v9.n2.17595.  Google Scholar [15] G. T. Moore, Quantum theory of electromagnetic field in a variable-length one-dimensional cavity, J. Math. Phys., 11 (1970), 2679-2691.  doi: 10.1063/1.1665432.  Google Scholar [16] M. A. Pinsky, Introduction to Fourier Analysis and Wavelets, Graduate Studies in Mathematics, 102. American Mathematical Society, Providence, RI, 2009. doi: 10.1090/gsm/102.  Google Scholar [17] D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 639-739.  doi: 10.1137/1020095.  Google Scholar [18] A. Sengouga, Observability and controllability of the 1-D wave equation in domains with moving boundary, Acta Appli. Math., 157 (2018), 117-128.  doi: 10.1007/s10440-018-0166-1.  Google Scholar [19] A. Sengouga, Observability of the 1-D wave equation with mixed boundary conditions in a non-cylindrical domain, Mediterr. J. Math., 15 (2018), Art. 62, 22 pp. doi: 10.1007/s00009-018-1107-y.  Google Scholar [20] H. C. Sun, H. F. Li and L. Q. Lu, Exact controllability for a string equation in domains with moving boundary in one dimension, Electron. J. Diff. Equations, 2015 (2015), 7 pp.  Google Scholar [21] P. K. C. Wang, Stabilization and control of distributed systems with time-dependent spatial domains, J. Optim. Theory Appl., 65 (1990), 331-362.  doi: 10.1007/BF01102351.  Google Scholar [22] E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods, SIAM Rev., 47 (2005), 197-243.  doi: 10.1137/S0036144503432862.  Google Scholar

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##### References:
 [1] N. L. Balazs, On the solution of the wave equation with moving boundaries, J. Math. Anal. Appl., 3 (1961), 472-484.  doi: 10.1016/0022-247X(61)90071-3.  Google Scholar [2] C. Bardos and G. Chen, Control and stabilization for the wave equation. Ⅲ: Domain with moving boundary, SIAM J. Control Optim., 19 (1981), 123-138.  doi: 10.1137/0319010.  Google Scholar [3] G. Birkhoff and G.-C. Rota, Ordinary Differential Equations, 4$^{th}$ edition, John Wiley & Sons, Inc., New York, 1989.  Google Scholar [4] J. Cooper and C. Bardos, A nonlinear wave equation in a time dependent domain, J. Math. Anal. Appl., 42 (1973), 29-60.  doi: 10.1016/0022-247X(73)90120-0.  Google Scholar [5] L. Z. Cui, X. Liu and H. Gao, Exact controllability for a one-dimensional wave equation in non-cylindrical domains, J. Math. Anal. Appl., 402 (2013), 612-625.  doi: 10.1016/j.jmaa.2013.01.062.  Google Scholar [6] V. V. Dodonov, A. B. Klimov and D. E. Nikonov, Quantum phenomena in resonators with moving walls, J. Math. Phys., 34 (1993), 2742-2756.  doi: 10.1063/1.530093.  Google Scholar [7] E. Knobloch and R. Krechetnikov, Problems on time-varying domains: Formulation, dynamics, and challenges, Acta Appl. Math., 137 (2015), 123-157.  doi: 10.1007/s10440-014-9993-x.  Google Scholar [8] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics, Masson, Paris, John Wiley & Sons, Ltd., Chichester, 1994.  Google Scholar [9] V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics, Springer-Verlag, New York, 2005. doi: 10.1007/b139040.  Google Scholar [10] J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod-Gautier Villars, Paris, 1969.  Google Scholar [11] J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systemes Distribués. Tome 1. Contrôlabilité Exacte, Recherches en Mathématiques Appliquées, 8. Masson, Paris, 1988.  Google Scholar [12] L. Q. Lu, S. J. Li, G. Chen and P. F. Yao, Control and stabilization for the wave equation with variable coefficients in domains with moving boundary, Systems & Control Lett., 80 (2015), 30-41.  doi: 10.1016/j.sysconle.2015.04.003.  Google Scholar [13] L. A. Medeiros, J. Limaco and S. B. Menezes, Vibrations of elastic strings: Mathematical aspects, part two, J. Comput. Anal. Appl., 4 (2002), 211-263.  doi: 10.1023/A:1013151525487.  Google Scholar [14] M. Milla Miranda, Exact controllability for the wave equation in domains with variable boundary, Rev. Mat. Complut. Madrid, 9 (1996), 435-457.  doi: 10.5209/rev_rema.1996.v9.n2.17595.  Google Scholar [15] G. T. Moore, Quantum theory of electromagnetic field in a variable-length one-dimensional cavity, J. Math. Phys., 11 (1970), 2679-2691.  doi: 10.1063/1.1665432.  Google Scholar [16] M. A. Pinsky, Introduction to Fourier Analysis and Wavelets, Graduate Studies in Mathematics, 102. American Mathematical Society, Providence, RI, 2009. doi: 10.1090/gsm/102.  Google Scholar [17] D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 639-739.  doi: 10.1137/1020095.  Google Scholar [18] A. Sengouga, Observability and controllability of the 1-D wave equation in domains with moving boundary, Acta Appli. Math., 157 (2018), 117-128.  doi: 10.1007/s10440-018-0166-1.  Google Scholar [19] A. Sengouga, Observability of the 1-D wave equation with mixed boundary conditions in a non-cylindrical domain, Mediterr. J. Math., 15 (2018), Art. 62, 22 pp. doi: 10.1007/s00009-018-1107-y.  Google Scholar [20] H. C. Sun, H. F. Li and L. Q. Lu, Exact controllability for a string equation in domains with moving boundary in one dimension, Electron. J. Diff. Equations, 2015 (2015), 7 pp.  Google Scholar [21] P. K. C. Wang, Stabilization and control of distributed systems with time-dependent spatial domains, J. Optim. Theory Appl., 65 (1990), 331-362.  doi: 10.1007/BF01102351.  Google Scholar [22] E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods, SIAM Rev., 47 (2005), 197-243.  doi: 10.1137/S0036144503432862.  Google Scholar
Extension of an initial data $\phi ^{0}$ when $v _{1}<v _{2}$
Propagation of a wave with a small support near an endpoint $(v _{2}<v _{1})$
Propagation of small disturbances with supports near one or two ends $(v _{1}<v _{2})$
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