December  2015, 35(12): 6133-6153. doi: 10.3934/dcds.2015.35.6133

Complexity and regularity of maximal energy domains for the wave equation with fixed initial data

1. 

CNRS, Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France

2. 

Université Pierre et Marie Curie (Univ. Paris 6) and Institut Universitaire de France and Team GECO Inria Saclay, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris

3. 

BCAM - Basque Center for Applied Mathematics, Mazarredo, 14, E-48009 Bilbao-Basque Country

Received  September 2013 Revised  January 2014 Published  May 2015

We consider the homogeneous wave equation on a bounded open connected subset $\Omega$ of $\mathbb{R}^n$. Some initial data being specified, we consider the problem of determining a measurable subset $\omega$ of $\Omega$ maximizing the $L^2$-norm of the restriction of the corresponding solution to $\omega$ over a time interval $[0,T]$, over all possible subsets of $\Omega$ having a certain prescribed measure. We prove that this problem always has at least one solution and that, if the initial data satisfy some analyticity assumptions, then the optimal set is unique and moreover has a finite number of connected components. In contrast, we construct smooth but not analytic initial conditions for which the optimal set is of Cantor type and in particular has an infinite number of connected components.
Citation: Yannick Privat, Emmanuel Trélat, Enrique Zuazua. Complexity and regularity of maximal energy domains for the wave equation with fixed initial data. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 6133-6153. doi: 10.3934/dcds.2015.35.6133
References:
[1]

R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 1975.

[2]

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), 1024-1065. doi: 10.1137/0330055.

[3]

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), 749-752. doi: 10.1016/S0764-4442(97)80053-5.

[4]

R. M. Hardt, Stratification of real analytic mappings and images, Invent. Math., 28 (1975), 193-208. doi: 10.1007/BF01436073.

[5]

P. Hébrard and A. Henrot, A spillover phenomenon in the optimal location of actuators, SIAM J. Control Optim., 44 (2005), 349-366. doi: 10.1137/S0363012903436247.

[6]

A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006.

[7]

A. Henrot and M. Pierre, Variation et Optimisation de Formes (French) [Shape Variation and Optimization] Une Analyse Géométrique [A Geometric Analysis], Math. & Appl., 48, Springer, Berlin, 2005. doi: 10.1007/3-540-37689-5.

[8]

H. Hironaka, Subanalytic sets, in Number Theory, Algebraic Geometry and Commutative Algebra, In honor of Y. Akizuki, Tokyo, 1973, 453-493.

[9]

B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Math., 1150, Springer-Verlag, Berlin, 1985.

[10]

C. S. Kubrusly and H. Malebranche, Sensors and controllers location in distributed systems - a survey, Automatica, 21 (1985), 117-128. doi: 10.1016/0005-1098(85)90107-4.

[11]

S. Kumar and J. H. Seinfeld, Optimal location of measurements for distributed parameter estimation, IEEE Trans. Autom. Contr., 23 (1978), 690-698. doi: 10.1109/TAC.1978.1101803.

[12]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, 1988.

[13]

K. Morris, Linear-quadratic optimal actuator location, IEEE Trans. Automat. Control, 56 (2011), 113-124. doi: 10.1109/TAC.2010.2052151.

[14]

A. Münch, Optimal location of the support of the control for the 1-D wave equation: numerical investigations, Comput. Optim. Appl., 42 (2009), 443-470. doi: 10.1007/s10589-007-9133-x.

[15]

E. Nelson, Analytic vectors, Ann. Math., 70 (1959), 572-615. doi: 10.2307/1970331.

[16]

F. Periago, Optimal shape and position of the support for the internal exact control of a string, Syst. Cont. Letters, 58 (2009), 136-140. doi: 10.1016/j.sysconle.2008.08.007.

[17]

Y. Privat, E. Trélat and E. Zuazua, Optimal observation of the one-dimensional wave equation, J. Fourier Anal. Appl., 19 (2013), 514-544. doi: 10.1007/s00041-013-9267-4.

[18]

Y. Privat, E. Trélat and E. Zuazua, Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains, to appear in J. Europ. Math. Soc. (JEMS), preprint Hal, 2013.

[19]

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. doi: 10.1016/j.anihpc.2012.11.005.

[20]

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), 921-981. doi: 10.1007/s00205-014-0823-0.

[21]

J.-M. Rakotoson, Réarrangement Relatif, Math. & Appl. (Berlin) [Mathematics & Applications], Vol. 64, Springer, Berlin, 2008. doi: 10.1007/978-3-540-69118-1.

[22]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, Switzerland, 2009. doi: 10.1007/978-3-7643-8994-9.

[23]

D. Ucinski and M. Patan, Sensor network design fo the estimation of spatially distributed processes, Int. J. Appl. Math. Comput. Sci., 20 (2010), 459-481. doi: 10.2478/v10006-010-0034-2.

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 1975.

[2]

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), 1024-1065. doi: 10.1137/0330055.

[3]

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), 749-752. doi: 10.1016/S0764-4442(97)80053-5.

[4]

R. M. Hardt, Stratification of real analytic mappings and images, Invent. Math., 28 (1975), 193-208. doi: 10.1007/BF01436073.

[5]

P. Hébrard and A. Henrot, A spillover phenomenon in the optimal location of actuators, SIAM J. Control Optim., 44 (2005), 349-366. doi: 10.1137/S0363012903436247.

[6]

A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006.

[7]

A. Henrot and M. Pierre, Variation et Optimisation de Formes (French) [Shape Variation and Optimization] Une Analyse Géométrique [A Geometric Analysis], Math. & Appl., 48, Springer, Berlin, 2005. doi: 10.1007/3-540-37689-5.

[8]

H. Hironaka, Subanalytic sets, in Number Theory, Algebraic Geometry and Commutative Algebra, In honor of Y. Akizuki, Tokyo, 1973, 453-493.

[9]

B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Math., 1150, Springer-Verlag, Berlin, 1985.

[10]

C. S. Kubrusly and H. Malebranche, Sensors and controllers location in distributed systems - a survey, Automatica, 21 (1985), 117-128. doi: 10.1016/0005-1098(85)90107-4.

[11]

S. Kumar and J. H. Seinfeld, Optimal location of measurements for distributed parameter estimation, IEEE Trans. Autom. Contr., 23 (1978), 690-698. doi: 10.1109/TAC.1978.1101803.

[12]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, 1988.

[13]

K. Morris, Linear-quadratic optimal actuator location, IEEE Trans. Automat. Control, 56 (2011), 113-124. doi: 10.1109/TAC.2010.2052151.

[14]

A. Münch, Optimal location of the support of the control for the 1-D wave equation: numerical investigations, Comput. Optim. Appl., 42 (2009), 443-470. doi: 10.1007/s10589-007-9133-x.

[15]

E. Nelson, Analytic vectors, Ann. Math., 70 (1959), 572-615. doi: 10.2307/1970331.

[16]

F. Periago, Optimal shape and position of the support for the internal exact control of a string, Syst. Cont. Letters, 58 (2009), 136-140. doi: 10.1016/j.sysconle.2008.08.007.

[17]

Y. Privat, E. Trélat and E. Zuazua, Optimal observation of the one-dimensional wave equation, J. Fourier Anal. Appl., 19 (2013), 514-544. doi: 10.1007/s00041-013-9267-4.

[18]

Y. Privat, E. Trélat and E. Zuazua, Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains, to appear in J. Europ. Math. Soc. (JEMS), preprint Hal, 2013.

[19]

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. doi: 10.1016/j.anihpc.2012.11.005.

[20]

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), 921-981. doi: 10.1007/s00205-014-0823-0.

[21]

J.-M. Rakotoson, Réarrangement Relatif, Math. & Appl. (Berlin) [Mathematics & Applications], Vol. 64, Springer, Berlin, 2008. doi: 10.1007/978-3-540-69118-1.

[22]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, Switzerland, 2009. doi: 10.1007/978-3-7643-8994-9.

[23]

D. Ucinski and M. Patan, Sensor network design fo the estimation of spatially distributed processes, Int. J. Appl. Math. Comput. Sci., 20 (2010), 459-481. doi: 10.2478/v10006-010-0034-2.

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