# American Institute of Mathematical Sciences

September  2015, 4(3): 297-314. doi: 10.3934/eect.2015.4.297

## An Ingham--Müntz type theorem and simultaneous observation problems

 1 Département de mathématique, Université de Strasbourg, 7, rue René Descartes, 67084 Strasbourg Cedex, France 2 Institut Élie Cartan, Université de Lorraine, BP 70239, 54506 Vandœuvre-lès-Nancy Cedex, France

Received  April 2015 Revised  July 2015 Published  September 2015

We establish a theorem combining the estimates of Ingham and Müntz--Szász. Moreover, we allow complex exponents instead of purely imaginary exponents for the Ingham type part or purely real exponents for the Müntz--Szász part. A very special case of this theorem allows us to prove the simultaneous observability of some string--heat and beam--heat systems.
Citation: Vilmos Komornik, Gérald Tenenbaum. An Ingham--Müntz type theorem and simultaneous observation problems. Evolution Equations & Control Theory, 2015, 4 (3) : 297-314. doi: 10.3934/eect.2015.4.297
##### References:
 [1] P. Albano and D. Tataru, Carleman estimates and boundary observability for a coupled parabolic-hyperbolic system,, Electron. J. Differential Equations, (2000). Google Scholar [2] C. Baiocchi, V. Komornik and P. Loreti, Ingham type theorems and applications to control theory,, Bol. Un. Mat. Ital. B, 2 (1999), 33. Google Scholar [3] C. Baiocchi, V. Komornik and P. Loreti, Ingham, Beurling type theorems with weakened gap conditions,, Acta Math. Hungar., 97 (2002), 55. doi: 10.1023/A:1020806811956. Google Scholar [4] 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 [5] A. Beurling, Interpolation for an interval in $\mathbbR^1$,, in The Collected Works of Arne Beurling, (1989). Google Scholar [6] J. A. Clarkson and P. Erdős, Approximation by polynomials,, Duke Math. J., 10 (1943), 5. doi: 10.1215/S0012-7094-43-01002-6. Google Scholar [7] J. Edwards, Ingham-type inequalities for complex frequencies and applications to control theory,, J. Math. Anal. Appl., 324 (2006), 941. doi: 10.1016/j.jmaa.2005.12.074. Google Scholar [8] H. O. Fattorini and D. L. Russell, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations,, Quart. Appl. Math., 32 (1974), 45. Google Scholar [9] K. D. Graham and D. L. Russell, Boundary value control of the wave equation in a spherical region,, SIAM J. Control, 13 (1975), 174. doi: 10.1137/0313011. Google Scholar [10] A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire,, J. Math. Pures Appl., 68 (1989), 457. Google Scholar [11] A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series,, Math. Z., 41 (1936), 367. doi: 10.1007/BF01180426. Google Scholar [12] S. Jaffard and S. Micu, Estimates of the constants in generalized Ingham's inequality and applications to the control of the wave equation,, Asymptotic Analysis, 28 (2001), 181. Google Scholar [13] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method,, Masson, (1994). Google Scholar [14] V. Komornik and P. Loreti, Fourier Series in Control Theory,, Springer-Verlag, (2005). Google Scholar [15] I. Lasiecka and R. Triggiani, Regularity of hyperbolic equations under $L_2(0,T; L_2(\Gamma ))$ boundary terms,, Appl. Math. and Optimiz., 10 (1983), 275. doi: 10.1007/BF01448390. Google Scholar [16] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories I,, Encyclopedia of Mathematics and Its Applications, (2000). Google Scholar [17] J.-L. Lions, Exact controllability, stabilization, and perturbations for distributed systems,, SIAM Rev., 30 (1988), 1. doi: 10.1137/1030001. Google Scholar [18] J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués I-II,, Masson, (1988). Google Scholar [19] Ch. H. Müntz, Über den Approximationssatz von Weierstrass,, in Mathematische Abhandlungen H. A. Schwarz gewidmet, (1914), 303. Google Scholar [20] J. Rauch, Xu Zhang and E. Zuazua, Polynomial decay for a hyperbolic-parabolic coupled system,, J. Math. Pures Appl. (9), 84 (2005), 407. doi: 10.1016/j.matpur.2004.09.006. Google Scholar [21] D. L. Russell, Controllability and stabilization theory for linear partial differential equations. Recent progress and open questions,, SIAM Rev., 20 (1978), 639. doi: 10.1137/1020095. Google Scholar [22] T. I. Seidman, Boundary control and observation for the heat equation,, in Calculus of Variations and Control Theory (ed. D. L. Russell), (1976), 321. Google Scholar [23] E. Sikolya, Simultaneous observability of networks of beams and strings,, Bol. Soc. Paran. Mat. (3), 21 (2003), 31. doi: 10.5269/bspm.v21i1-2.7505. Google Scholar [24] O. Szász, Über die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen,, Math. Ann., 77 (1916), 482. doi: 10.1007/BF01456964. Google Scholar [25] G. Tenenbaum and M. Tucsnak, Fast and strongly localized observation for the Schrödinger equation,, Trans. Amer. Math. Soc., 361 (2009), 951. doi: 10.1090/S0002-9947-08-04584-4. Google Scholar [26] Xu Zhang, E. Zuazua, Polynomial decay and control of a 1-d hyperbolic-parabolic coupled system,, J. Differential Equations, 204 (2004), 380. doi: 10.1016/j.jde.2004.02.004. Google Scholar [27] X. Zhang and E. Zuazua, Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction,, Arch. Ration. Mech. Anal., 184 (2007), 49. doi: 10.1007/s00205-006-0020-x. Google Scholar

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##### References:
 [1] P. Albano and D. Tataru, Carleman estimates and boundary observability for a coupled parabolic-hyperbolic system,, Electron. J. Differential Equations, (2000). Google Scholar [2] C. Baiocchi, V. Komornik and P. Loreti, Ingham type theorems and applications to control theory,, Bol. Un. Mat. Ital. B, 2 (1999), 33. Google Scholar [3] C. Baiocchi, V. Komornik and P. Loreti, Ingham, Beurling type theorems with weakened gap conditions,, Acta Math. Hungar., 97 (2002), 55. doi: 10.1023/A:1020806811956. Google Scholar [4] 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 [5] A. Beurling, Interpolation for an interval in $\mathbbR^1$,, in The Collected Works of Arne Beurling, (1989). Google Scholar [6] J. A. Clarkson and P. Erdős, Approximation by polynomials,, Duke Math. J., 10 (1943), 5. doi: 10.1215/S0012-7094-43-01002-6. Google Scholar [7] J. Edwards, Ingham-type inequalities for complex frequencies and applications to control theory,, J. Math. Anal. Appl., 324 (2006), 941. doi: 10.1016/j.jmaa.2005.12.074. Google Scholar [8] H. O. Fattorini and D. L. Russell, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations,, Quart. Appl. Math., 32 (1974), 45. Google Scholar [9] K. D. Graham and D. L. Russell, Boundary value control of the wave equation in a spherical region,, SIAM J. Control, 13 (1975), 174. doi: 10.1137/0313011. Google Scholar [10] A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire,, J. Math. Pures Appl., 68 (1989), 457. Google Scholar [11] A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series,, Math. Z., 41 (1936), 367. doi: 10.1007/BF01180426. Google Scholar [12] S. Jaffard and S. Micu, Estimates of the constants in generalized Ingham's inequality and applications to the control of the wave equation,, Asymptotic Analysis, 28 (2001), 181. Google Scholar [13] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method,, Masson, (1994). Google Scholar [14] V. Komornik and P. Loreti, Fourier Series in Control Theory,, Springer-Verlag, (2005). Google Scholar [15] I. Lasiecka and R. Triggiani, Regularity of hyperbolic equations under $L_2(0,T; L_2(\Gamma ))$ boundary terms,, Appl. Math. and Optimiz., 10 (1983), 275. doi: 10.1007/BF01448390. Google Scholar [16] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories I,, Encyclopedia of Mathematics and Its Applications, (2000). Google Scholar [17] J.-L. Lions, Exact controllability, stabilization, and perturbations for distributed systems,, SIAM Rev., 30 (1988), 1. doi: 10.1137/1030001. Google Scholar [18] J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués I-II,, Masson, (1988). Google Scholar [19] Ch. H. Müntz, Über den Approximationssatz von Weierstrass,, in Mathematische Abhandlungen H. A. Schwarz gewidmet, (1914), 303. Google Scholar [20] J. Rauch, Xu Zhang and E. Zuazua, Polynomial decay for a hyperbolic-parabolic coupled system,, J. Math. Pures Appl. (9), 84 (2005), 407. doi: 10.1016/j.matpur.2004.09.006. Google Scholar [21] D. L. Russell, Controllability and stabilization theory for linear partial differential equations. Recent progress and open questions,, SIAM Rev., 20 (1978), 639. doi: 10.1137/1020095. Google Scholar [22] T. I. Seidman, Boundary control and observation for the heat equation,, in Calculus of Variations and Control Theory (ed. D. L. Russell), (1976), 321. Google Scholar [23] E. Sikolya, Simultaneous observability of networks of beams and strings,, Bol. Soc. Paran. Mat. (3), 21 (2003), 31. doi: 10.5269/bspm.v21i1-2.7505. Google Scholar [24] O. Szász, Über die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen,, Math. Ann., 77 (1916), 482. doi: 10.1007/BF01456964. Google Scholar [25] G. Tenenbaum and M. Tucsnak, Fast and strongly localized observation for the Schrödinger equation,, Trans. Amer. Math. Soc., 361 (2009), 951. doi: 10.1090/S0002-9947-08-04584-4. Google Scholar [26] Xu Zhang, E. Zuazua, Polynomial decay and control of a 1-d hyperbolic-parabolic coupled system,, J. Differential Equations, 204 (2004), 380. doi: 10.1016/j.jde.2004.02.004. Google Scholar [27] X. Zhang and E. Zuazua, Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction,, Arch. Ration. Mech. Anal., 184 (2007), 49. doi: 10.1007/s00205-006-0020-x. Google Scholar
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