Abstract
For a given bounded connected domain in \({{\mathbbm{R}}^n}\), the issue of computing the observability constant associated with a wave operator, an observation time T and a generic observation subdomain constitutes in general a hard task, even for one-dimensional problems. In this work, we introduce and describe two methods to provide precise (and even sharp in some cases) estimates of observability constants for general one-dimensional wave equations: the first one uses a spectral decomposition of the solution of the wave equation, whereas the second one is based on a propagation argument along the characteristics. Both methods are extensively described and we then comment on the advantages and drawbacks of each one. The discussion is illustrated by several examples and numerical simulations. As a by-product, we deduce from the main results estimates of the cost of control (resp. the decay rate of the energy) for several controlled (resp. damped) wave equations.
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References
Ball J. M., Slemrod M.: Nonharmonic Fourier series and the stabilization of distributed semilinear control systems. Comm. Pure Appl. Math. 32, no. 4, 555–587 (1979)
Bardos C., Lebeau G., Rauch J.: Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim. 30, no. 05, 1024–1065 (1992)
Chen G., Fulling S.A., Narcowich F.J., Sun S.: Exponential decay of energy of evolution equations with locally distributed damping. SIAM J. Appl. Math. 51, no. 1, 266–301 (1991)
Cox S., Zuazua E.: The rate at which energy decays in a damped string. Comm. in partial differential equations 19, 213–243 (1994)
S. Ervedoza and E. Zuazua, On the numerical approximation of controls for waves, Springer Briefs in Mathematics, Springer, New York (2013).
Haraux A.: A generalized internal control for the wave equation in a rectangle. J. Math. Anal. Appl. 153, 190–216 (1990)
A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d’une plaque rectangulaire. (French. English summary) [Lacunary series and semi-internal control of the vibrations of a rectangular plate], J. Math. Pures Appl. (9) 68 (1989), no. 4, 457–465 (1990).
A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Portugaliæ mathematica, 46 (1989), no. 3, 245–258.
A. Henrot, Extremum problems for eigenvalues of elliptic operators, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006.
Ingham A.E.: Some trigonometrical inequalities with applications to the theory of series. Math. Zeitschrift 41, 367–379 (1936)
Jaffard S., Micu S.: Estimates of the constants in generalized Ingham’s inequality and applications to the control of the wave equation. Asymptot. Anal. 28, no. 3-4, 181–214 (2001)
Jaffard S., Tucsnak M., Zuazua E.: On a theorem of Ingham. J. Fourier Anal. Appl. 3, 577–582 (1997)
B. Kawohl, Rearrangements and convexity of level sets in PDE, Springer Lecture Notes in Math. 1150, (1985), 1–134.
Komornik V., Loreti P.: Fourier Series in Control Theory. Springer-Verlag, New York (2005)
Komornik V., Miara B.: Cross-like internal observability of rectangular membranes. Evolution equ. control theory 3, no. 1, 135–146 (2014)
Lions J.-L.: Exact controllability, stabilizability and perturbations for distributed systems. SIAM Rev. 30, 1–68 (1988)
J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués, Tomes 1 & 2, Rech. Math. Appl. [Research in Applied Mathematics], Masson (1988).
Periago F.: Optimal shape and position of the support for the internal exact control of a string. Syst. Cont. Letters 58, no. 2, 136–140 (2009)
J. Pöschel, E. Trubowitz, Inverse spectral theory, Pure and Applied Mathematics, 130. Academic Press, Inc., Boston, MA, 1987. x+192 pp.
Privat Y., Trélat E., Zuazua E.: Optimal location of controllers for the one-dimensional wave equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 30, no. 6, 1097–1126 (2013)
Privat Y., Trélat E., Zuazua E.: Optimal observation of the one-dimensional wave equation. J. Fourier Anal. Appl. 19, no. 3, 514–544 (2013)
Y. Privat, E. Trélat, E. Zuazua, Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains, Preprint Hal (2012).
Russell D.L.: Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev. 20, no. 4, 639–739 (1978)
Zuazua E.: Exact controllability for semilinear wave equations in one space dimension. Ann. Inst. H. Poincaré Anal. Non Linéaire 10, no. 1, 109–129 (1993)
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This work is supported by ANR (AVENTURES - ANR-12-BLAN-BS01-0001-01).
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Haraux, A., Liard, T. & Privat, Y. How to estimate observability constants of one-dimensional wave equations? Propagation versus spectral methods. J. Evol. Equ. 16, 825–856 (2016). https://doi.org/10.1007/s00028-016-0321-y
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DOI: https://doi.org/10.1007/s00028-016-0321-y