# American Institute of Mathematical Sciences

2015, 2015(special): 936-944. doi: 10.3934/proc.2015.0936

## Optimal design of sensors for a damped wave equation

 1 CNRS, Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris 2 Sorbonne Universités, UPMC Univ. Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Institut Universitaire de France, F-75005, Paris

Received  September 2014 Revised  January 2015 Published  November 2015

In this paper we model and solve the problem of shaping and placing in an optimal way sensors for a wave equation with constant damping in a bounded open connected subset $\Omega$ of $\mathbb{R}^n$. Sensors are modeled by subdomains of $\Omega$ of a given measure $L|\Omega|$, with $0 < L < 1$. We prove that, if $L$ is close enough to $1$, then the optimal design problem has a unique solution, which is characterized by a finite number of low frequency modes. In particular the maximizing sequence built from spectral approximations is stationary.
Citation: Yannick Privat, Emmanuel Trélat. Optimal design of sensors for a damped wave equation. Conference Publications, 2015, 2015 (special) : 936-944. doi: 10.3934/proc.2015.0936
##### References:
 [1] G. Allaire, S. Aubry and F. Jouve, Eigenfrequency optimization in optimal design, Comput. Methods Appl. Mech. Engrg., 190 (2001), 3565-3579. Google Scholar [2] A. Armaoua and M. Demetriou, Optimal actuator/sensor placement for linear parabolic PDEs using spatial $H^2$ norm, Chemical Engineering Science, 61 (2006), 7351-7367. Google Scholar [3] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), no. 5, 1024-1065. Google Scholar [4] J.C. Bellido and A. Donoso, An optimal design problem in wave propagation, J. Optim. Theory Appl., 134 (2007), 339-352. Google Scholar [5] N. Burq, Large-time dynamics for the one-dimensional Schrödinger equation, Proc. Roy. Soc. Edinburgh Sect. A., 141 (2011), no. 2, 227-251. Google Scholar [6] N. Burq and P. Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes (French) [A necessary and sufficient condition for the exact controllability of the wave equation], C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), no. 7, 749-752. Google Scholar [7] N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math., 173 (2008), no. 3, 449-475. Google Scholar [8] N. Burq and N. Tzvetkov, Probabilistic well-posedness for the cubic wave equation, J. Eur. Math. Soc. (JEMS), 16 (2014), no. 1, 1-30. Google Scholar [9] P. Hébrard and A. Henrot, Optimal shape and position of the actuators for the stabilization of a string, Syst. Cont. Letters, 48 (2003), 199-209. Google Scholar [10] P. Hébrard and A. Henrot, A spillover phenomenon in the optimal location of actuators,, SIAM J. Control Optim. \textbf{44} {2005}, 44 (): 349.   Google Scholar [11] K. Morris, Linear-quadratic optimal actuator location, IEEE Trans. Automat. Control, 56 (2011), no. 1, 113-124. Google Scholar [12] R.E.A.C. Paley and A. Zygmund, On some series of functions (1) (2) (3), Proc. Camb. Phil. Soc., 26, (1930) 337-357, 458-474, 28 (1932), 190-205. Google Scholar [13] Y. Privat, E. Trélat and E. Zuazua, Complexity and regularity of maximal energy domains for the wave equation with fixed initial data, Discrete Cont. Dynam. Syst., 35 (2015), no. 12, 6133-6153. Google Scholar [14] Y. Privat, E. Trélat and E. Zuazua, Optimal location of controllers for the one-dimensional wave equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 1097-1126. Google Scholar [15] Y. Privat, E. Trélat and E. Zuazua, Optimal observability of the one-dimensional wave equation, J. Fourier Anal. Appl., 19 (2013), no. 3, 514-544. Google Scholar [16] Y. Privat, E. Trélat and E. Zuazua, Optimal observability of the multi-dimensional wave and Schr\"odinger equations in quantum ergodic domains, to appear in J. Eur. Math. Soc., 2015. Google Scholar [17] Y. Privat, E. Trélat and E. Zuazua, Optimal shape and location of sensors for parabolic equations with random initial data, Arch. Ration. Mech. Anal., 216 (2015), no. 3, 921-981. Google Scholar [18] O. Sigmund and J.S. Jensen, Systematic design of phononic band-gap materials and structures by topology optimization, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 361 (2003), no. 1806, 1001-1019. Google Scholar [19] M. Tucsnak and G. Weiss, Observation and control for operator semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, Switzerland, 2009. Google Scholar [20] D. Ucinski and M. Patan, Sensor network design fo the estimation of spatially distributed processes, Int. J. Appl. Math. Comput. Sci., 20 (2010), no. 3, 459-481. Google Scholar [21] A. Vande Wouwer, N. Point, S. Porteman, M. Remy, An approach to the selection of optimal sensor locations in distributed parameter systems, J. Process Control, 10 (2000), 291-300. Google Scholar

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##### References:
 [1] G. Allaire, S. Aubry and F. Jouve, Eigenfrequency optimization in optimal design, Comput. Methods Appl. Mech. Engrg., 190 (2001), 3565-3579. Google Scholar [2] A. Armaoua and M. Demetriou, Optimal actuator/sensor placement for linear parabolic PDEs using spatial $H^2$ norm, Chemical Engineering Science, 61 (2006), 7351-7367. Google Scholar [3] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), no. 5, 1024-1065. Google Scholar [4] J.C. Bellido and A. Donoso, An optimal design problem in wave propagation, J. Optim. Theory Appl., 134 (2007), 339-352. Google Scholar [5] N. Burq, Large-time dynamics for the one-dimensional Schrödinger equation, Proc. Roy. Soc. Edinburgh Sect. A., 141 (2011), no. 2, 227-251. Google Scholar [6] N. Burq and P. Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes (French) [A necessary and sufficient condition for the exact controllability of the wave equation], C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), no. 7, 749-752. Google Scholar [7] N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math., 173 (2008), no. 3, 449-475. Google Scholar [8] N. Burq and N. Tzvetkov, Probabilistic well-posedness for the cubic wave equation, J. Eur. Math. Soc. (JEMS), 16 (2014), no. 1, 1-30. Google Scholar [9] P. Hébrard and A. Henrot, Optimal shape and position of the actuators for the stabilization of a string, Syst. Cont. Letters, 48 (2003), 199-209. Google Scholar [10] P. Hébrard and A. Henrot, A spillover phenomenon in the optimal location of actuators,, SIAM J. Control Optim. \textbf{44} {2005}, 44 (): 349.   Google Scholar [11] K. Morris, Linear-quadratic optimal actuator location, IEEE Trans. Automat. Control, 56 (2011), no. 1, 113-124. Google Scholar [12] R.E.A.C. Paley and A. Zygmund, On some series of functions (1) (2) (3), Proc. Camb. Phil. Soc., 26, (1930) 337-357, 458-474, 28 (1932), 190-205. Google Scholar [13] Y. Privat, E. Trélat and E. Zuazua, Complexity and regularity of maximal energy domains for the wave equation with fixed initial data, Discrete Cont. Dynam. Syst., 35 (2015), no. 12, 6133-6153. Google Scholar [14] Y. Privat, E. Trélat and E. Zuazua, Optimal location of controllers for the one-dimensional wave equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 1097-1126. Google Scholar [15] Y. Privat, E. Trélat and E. Zuazua, Optimal observability of the one-dimensional wave equation, J. Fourier Anal. Appl., 19 (2013), no. 3, 514-544. Google Scholar [16] Y. Privat, E. Trélat and E. Zuazua, Optimal observability of the multi-dimensional wave and Schr\"odinger equations in quantum ergodic domains, to appear in J. Eur. Math. Soc., 2015. Google Scholar [17] Y. Privat, E. Trélat and E. Zuazua, Optimal shape and location of sensors for parabolic equations with random initial data, Arch. Ration. Mech. Anal., 216 (2015), no. 3, 921-981. Google Scholar [18] O. Sigmund and J.S. Jensen, Systematic design of phononic band-gap materials and structures by topology optimization, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 361 (2003), no. 1806, 1001-1019. Google Scholar [19] M. Tucsnak and G. Weiss, Observation and control for operator semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, Switzerland, 2009. Google Scholar [20] D. Ucinski and M. Patan, Sensor network design fo the estimation of spatially distributed processes, Int. J. Appl. Math. Comput. Sci., 20 (2010), no. 3, 459-481. Google Scholar [21] A. Vande Wouwer, N. Point, S. Porteman, M. Remy, An approach to the selection of optimal sensor locations in distributed parameter systems, J. Process Control, 10 (2000), 291-300. Google Scholar
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