Exact controllability in "arbitrarily short time" of the semilinear wave equation

Previous Article
Simple umbilic points on surfaces immersed in $\R^4$
DCDS Home
This Issue
Next Article
Existence of traveling wavefronts of delayed reaction diffusion systems without monotonicity

July 2003, 9(4): 901-924. doi: 10.3934/dcds.2003.9.901

Exact controllability in "arbitrarily short time" of the semilinear wave equation

Patrick Martinez ^1, and Judith Vancostenoble ^1,

1.	Laboratoire de Mathématiques MIP, UMR CNRS 5640, Université Paul Sabatier Toulouse III, 118 route de Narbonne, 31 062 Toulouse cedex 4, France, France

Received October 2001 Revised November 2002 Published April 2003

Abstract
Related Papers

We study the exact controllability of the one dimensional semilinear wave equation by a control acting on an open subset $(a,b)$ of the domain $(0,1)$. With the aid of d'Alembert's formula and sidewise energy estimates, we obtain sharp conditions in the space-time support of the control, that coincide with the by now well-known geometric control condition.
More precisely, by classical results of J. Lagnese, A. Haraux and E. Zuazua, exact controllability holds in time $T > T_0 $:$= 2 max (a , 1-b)$ and fails if $T < T_0$. We weaken strongly their results: given $T>T_0$, we prove that the control can be chosen so that it is supported only on some special time intervals: they are parts of $(0,T)$, in finite number (depending on $a$ and $b$), and their total length can be arbitrarily small. The only condition is that they have to be "close enough" from each other. If this condition holds, we study the observability cost. If it fails, we prove that exact controllability in time $T$ does not hold, but can still be true in time $T'$ large enough.

Keywords: exact controllability., distributed parameter control, Wave equation, damping.

Mathematics Subject Classification: 35L05, 93B05, 93C2.

Citation: Patrick Martinez, Judith Vancostenoble. Exact controllability in "arbitrarily short time" of the semilinear wave equation. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 901-924. doi: 10.3934/dcds.2003.9.901

[1]	Ngoc Minh Trang Vu, Laurent Lefèvre. Finite rank distributed control for the resistive diffusion equation using damping assignment. Evolution Equations & Control Theory, 2015, 4 (2) : 205-220. doi: 10.3934/eect.2015.4.205
[2]	Arnaud Heibig, Mohand Moussaoui. Exact controllability of the wave equation for domains with slits and for mixed boundary conditions. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 367-386. doi: 10.3934/dcds.1996.2.367
[3]	Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control & Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305
[4]	Umberto De Maio, Akamabadath K. Nandakumaran, Carmen Perugia. Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition. Evolution Equations & Control Theory, 2015, 4 (3) : 325-346. doi: 10.3934/eect.2015.4.325
[5]	Manuel González-Burgos, Sergio Guerrero, Jean Pierre Puel. Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation. Communications on Pure & Applied Analysis, 2009, 8 (1) : 311-333. doi: 10.3934/cpaa.2009.8.311
[6]	Klaus-Jochen Engel, Marjeta Kramar FijavŽ. Exact and positive controllability of boundary control systems. Networks & Heterogeneous Media, 2017, 12 (2) : 319-337. doi: 10.3934/nhm.2017014
[7]	Jamel Ben Amara, Hedi Bouzidi. Exact boundary controllability for the Boussinesq equation with variable coefficients. Evolution Equations & Control Theory, 2018, 7 (3) : 403-415. doi: 10.3934/eect.2018020
[8]	Irena Lasiecka, Roberto Triggiani. Global exact controllability of semilinear wave equations by a double compactness/uniqueness argument. Conference Publications, 2005, 2005 (Special) : 556-565. doi: 10.3934/proc.2005.2005.556
[9]	Dalibor Pražák. On the dimension of the attractor for the wave equation with nonlinear damping. Communications on Pure & Applied Analysis, 2005, 4 (1) : 165-174. doi: 10.3934/cpaa.2005.4.165
[10]	Belhassen Dehman, Jean-Pierre Raymond. Exact controllability for the Lamé system. Mathematical Control & Related Fields, 2015, 5 (4) : 743-760. doi: 10.3934/mcrf.2015.5.743
[11]	Kareem T. Elgindy. Optimal control of a parabolic distributed parameter system using a fully exponentially convergent barycentric shifted gegenbauer integral pseudospectral method. Journal of Industrial & Management Optimization, 2018, 14 (2) : 473-496. doi: 10.3934/jimo.2017056
[12]	Ning-An Lai, Jinglei Zhao. Potential well and exact boundary controllability for radial semilinear wave equations on Schwarzschild spacetime. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1317-1325. doi: 10.3934/cpaa.2014.13.1317
[13]	Kim Dang Phung. Decay of solutions of the wave equation with localized nonlinear damping and trapped rays. Mathematical Control & Related Fields, 2011, 1 (2) : 251-265. doi: 10.3934/mcrf.2011.1.251
[14]	Nicolas Fourrier, Irena Lasiecka. Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions. Evolution Equations & Control Theory, 2013, 2 (4) : 631-667. doi: 10.3934/eect.2013.2.631
[15]	Jiayun Lin, Kenji Nishihara, Jian Zhai. Critical exponent for the semilinear wave equation with time-dependent damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4307-4320. doi: 10.3934/dcds.2012.32.4307
[16]	Aníbal Rodríguez-Bernal, Enrique Zuazua. Parabolic singular limit of a wave equation with localized boundary damping. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 303-346. doi: 10.3934/dcds.1995.1.303
[17]	Tae Gab Ha. On viscoelastic wave equation with nonlinear boundary damping and source term. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1543-1576. doi: 10.3934/cpaa.2010.9.1543
[18]	Moez Daoulatli. Energy decay rates for solutions of the wave equation with linear damping in exterior domain. Evolution Equations & Control Theory, 2016, 5 (1) : 37-59. doi: 10.3934/eect.2016.5.37
[19]	Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361
[20]	Mohammed Aassila. Exact boundary controllability of a coupled system. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 665-672. doi: 10.3934/dcds.2000.6.665

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences