Optimal design of sensors for a damped wave equation

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

Optimal design of sensors for a damped wave equation

Yannick Privat ^1, and Emmanuel Trélat ^2,

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

Abstract
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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.

Keywords: shape optimization, Damped wave equation, observability..

Mathematics Subject Classification: Primary: 35P20, 93B07; Secondary: 58J51, 49K2.

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. 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. 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), 1024. Google Scholar
[4]	J.C. Bellido and A. Donoso, An optimal design problem in wave propagation,, J. Optim. Theory Appl., 134 (2007), 339. Google Scholar
[5]	N. Burq, Large-time dynamics for the one-dimensional Schrödinger equation,, Proc. Roy. Soc. Edinburgh Sect. A., 141 (2011), 227. 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), 749. Google Scholar
[7]	N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory,, Invent. Math., 173 (2008), 449. Google Scholar
[8]	N. Burq and N. Tzvetkov, Probabilistic well-posedness for the cubic wave equation,, J. Eur. Math. Soc. (JEMS), 16 (2014), 1. 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. 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), 113. 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. 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), 6133. 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. Google Scholar
[15]	Y. Privat, E. Trélat and E. Zuazua, Optimal observability of the one-dimensional wave equation,, J. Fourier Anal. Appl., 19 (2013), 514. 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), 921. 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), 1001. Google Scholar
[19]	M. Tucsnak and G. Weiss, Observation and control for operator semigroups,, Birkhäuser Advanced Texts: Basler Lehrbücher, (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), 459. 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. 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. 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. 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), 1024. Google Scholar
[4]	J.C. Bellido and A. Donoso, An optimal design problem in wave propagation,, J. Optim. Theory Appl., 134 (2007), 339. Google Scholar
[5]	N. Burq, Large-time dynamics for the one-dimensional Schrödinger equation,, Proc. Roy. Soc. Edinburgh Sect. A., 141 (2011), 227. 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), 749. Google Scholar
[7]	N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory,, Invent. Math., 173 (2008), 449. Google Scholar
[8]	N. Burq and N. Tzvetkov, Probabilistic well-posedness for the cubic wave equation,, J. Eur. Math. Soc. (JEMS), 16 (2014), 1. 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. 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), 113. 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. 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), 6133. 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. Google Scholar
[15]	Y. Privat, E. Trélat and E. Zuazua, Optimal observability of the one-dimensional wave equation,, J. Fourier Anal. Appl., 19 (2013), 514. 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), 921. 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), 1001. Google Scholar
[19]	M. Tucsnak and G. Weiss, Observation and control for operator semigroups,, Birkhäuser Advanced Texts: Basler Lehrbücher, (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), 459. 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. Google Scholar

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