Optimal design of sensors for a damped wave equation

Yannick  Privat; Emmanuel Trélat

doi:10.3934/proc.2015.0936

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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

show all references

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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