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 & Continuous Dynamical Systems - A, 2015, 35 (12) : 6133-6153. doi: 10.3934/dcds.2015.35.6133
References:
[1]

R. A. Adams, Sobolev Spaces,, Pure and Applied Mathematics, (1975).   Google Scholar

[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.  doi: 10.1137/0330055.  Google Scholar

[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.  doi: 10.1016/S0764-4442(97)80053-5.  Google Scholar

[4]

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

[5]

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

[6]

A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators,, Frontiers in Mathematics, (2006).   Google Scholar

[7]

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

[8]

H. Hironaka, Subanalytic sets,, in Number Theory, (1973), 453.   Google Scholar

[9]

B. Kawohl, Rearrangements and Convexity of Level Sets in PDE,, Lecture Notes in Math., (1150).   Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[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.  doi: 10.1007/s10589-007-9133-x.  Google Scholar

[15]

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

[16]

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

[17]

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

[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), (2013).   Google Scholar

[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.  doi: 10.1016/j.anihpc.2012.11.005.  Google Scholar

[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.  doi: 10.1007/s00205-014-0823-0.  Google Scholar

[21]

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

[22]

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

[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.  doi: 10.2478/v10006-010-0034-2.  Google Scholar

show all references

References:
[1]

R. A. Adams, Sobolev Spaces,, Pure and Applied Mathematics, (1975).   Google Scholar

[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.  doi: 10.1137/0330055.  Google Scholar

[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.  doi: 10.1016/S0764-4442(97)80053-5.  Google Scholar

[4]

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

[5]

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

[6]

A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators,, Frontiers in Mathematics, (2006).   Google Scholar

[7]

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

[8]

H. Hironaka, Subanalytic sets,, in Number Theory, (1973), 453.   Google Scholar

[9]

B. Kawohl, Rearrangements and Convexity of Level Sets in PDE,, Lecture Notes in Math., (1150).   Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[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.  doi: 10.1007/s10589-007-9133-x.  Google Scholar

[15]

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

[16]

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

[17]

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

[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), (2013).   Google Scholar

[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.  doi: 10.1016/j.anihpc.2012.11.005.  Google Scholar

[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.  doi: 10.1007/s00205-014-0823-0.  Google Scholar

[21]

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

[22]

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

[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.  doi: 10.2478/v10006-010-0034-2.  Google Scholar

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