# 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 and 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), 15 pp. [2] C. Baiocchi, V. Komornik and P. Loreti, Ingham type theorems and applications to control theory, Bol. Un. Mat. Ital. B, 2 (1999), 33-63. [3] C. Baiocchi, V. Komornik and P. Loreti, Ingham, Beurling type theorems with weakened gap conditions, Acta Math. Hungar., 97 (2002), 55-95. doi: 10.1023/A:1020806811956. [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), 1024-1065. doi: 10.1137/0330055. [5] A. Beurling, Interpolation for an interval in $\mathbbR^1$, in The Collected Works of Arne Beurling, Vol. 2, Contemporary Mathematicians, Birkhäuser, Boston, 1989. [6] J. A. Clarkson and P. Erdős, Approximation by polynomials, Duke Math. J., 10 (1943), 5-11. doi: 10.1215/S0012-7094-43-01002-6. [7] J. Edwards, Ingham-type inequalities for complex frequencies and applications to control theory, J. Math. Anal. Appl., 324 (2006), 941-954. doi: 10.1016/j.jmaa.2005.12.074. [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), 45-69. [9] K. D. Graham and D. L. Russell, Boundary value control of the wave equation in a spherical region, SIAM J. Control, 13 (1975), 174-196. doi: 10.1137/0313011. [10] A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire, J. Math. Pures Appl., 68 (1989), 457-465. [11] A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series, Math. Z., 41 (1936), 367-379. doi: 10.1007/BF01180426. [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), 181-214. [13] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson, Paris; John Wiley & Sons, Chichester, 1994. [14] V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer-Verlag, New York, 2005. [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), 275-286. doi: 10.1007/BF01448390. [16] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories I, Encyclopedia of Mathematics and Its Applications, 74, Cambridge University Press, 2000. [17] J.-L. Lions, Exact controllability, stabilization, and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001. [18] J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués I-II, Masson, Paris, 1988. [19] Ch. H. Müntz, Über den Approximationssatz von Weierstrass, in Mathematische Abhandlungen H. A. Schwarz gewidmet, Berlin, 1914, 303-312. [20] J. Rauch, Xu Zhang and E. Zuazua, Polynomial decay for a hyperbolic-parabolic coupled system, J. Math. Pures Appl. (9), 84 (2005), 407-470. doi: 10.1016/j.matpur.2004.09.006. [21] D. L. Russell, Controllability and stabilization theory for linear partial differential equations. Recent progress and open questions, SIAM Rev., 20 (1978), 639-739. doi: 10.1137/1020095. [22] T. I. Seidman, Boundary control and observation for the heat equation, in Calculus of Variations and Control Theory (ed. D. L. Russell), Math. Res. Center, Univ. Wisconsin, Publ. No. 36, Academic Press, New York, 1976, 321-351. [23] E. Sikolya, Simultaneous observability of networks of beams and strings, Bol. Soc. Paran. Mat. (3), 21 (2003), 31-41. doi: 10.5269/bspm.v21i1-2.7505. [24] O. Szász, Über die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen, Math. Ann., 77 (1916), 482-496. doi: 10.1007/BF01456964. [25] G. Tenenbaum and M. Tucsnak, Fast and strongly localized observation for the Schrödinger equation, Trans. Amer. Math. Soc., 361 (2009), 951-977. doi: 10.1090/S0002-9947-08-04584-4. [26] Xu Zhang, E. Zuazua, Polynomial decay and control of a 1-d hyperbolic-parabolic coupled system, J. Differential Equations, 204 (2004), 380-438. doi: 10.1016/j.jde.2004.02.004. [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), 49-120. doi: 10.1007/s00205-006-0020-x.

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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), 15 pp. [2] C. Baiocchi, V. Komornik and P. Loreti, Ingham type theorems and applications to control theory, Bol. Un. Mat. Ital. B, 2 (1999), 33-63. [3] C. Baiocchi, V. Komornik and P. Loreti, Ingham, Beurling type theorems with weakened gap conditions, Acta Math. Hungar., 97 (2002), 55-95. doi: 10.1023/A:1020806811956. [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), 1024-1065. doi: 10.1137/0330055. [5] A. Beurling, Interpolation for an interval in $\mathbbR^1$, in The Collected Works of Arne Beurling, Vol. 2, Contemporary Mathematicians, Birkhäuser, Boston, 1989. [6] J. A. Clarkson and P. Erdős, Approximation by polynomials, Duke Math. J., 10 (1943), 5-11. doi: 10.1215/S0012-7094-43-01002-6. [7] J. Edwards, Ingham-type inequalities for complex frequencies and applications to control theory, J. Math. Anal. Appl., 324 (2006), 941-954. doi: 10.1016/j.jmaa.2005.12.074. [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), 45-69. [9] K. D. Graham and D. L. Russell, Boundary value control of the wave equation in a spherical region, SIAM J. Control, 13 (1975), 174-196. doi: 10.1137/0313011. [10] A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire, J. Math. Pures Appl., 68 (1989), 457-465. [11] A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series, Math. Z., 41 (1936), 367-379. doi: 10.1007/BF01180426. [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), 181-214. [13] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson, Paris; John Wiley & Sons, Chichester, 1994. [14] V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer-Verlag, New York, 2005. [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), 275-286. doi: 10.1007/BF01448390. [16] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories I, Encyclopedia of Mathematics and Its Applications, 74, Cambridge University Press, 2000. [17] J.-L. Lions, Exact controllability, stabilization, and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001. [18] J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués I-II, Masson, Paris, 1988. [19] Ch. H. Müntz, Über den Approximationssatz von Weierstrass, in Mathematische Abhandlungen H. A. Schwarz gewidmet, Berlin, 1914, 303-312. [20] J. Rauch, Xu Zhang and E. Zuazua, Polynomial decay for a hyperbolic-parabolic coupled system, J. Math. Pures Appl. (9), 84 (2005), 407-470. doi: 10.1016/j.matpur.2004.09.006. [21] D. L. Russell, Controllability and stabilization theory for linear partial differential equations. Recent progress and open questions, SIAM Rev., 20 (1978), 639-739. doi: 10.1137/1020095. [22] T. I. Seidman, Boundary control and observation for the heat equation, in Calculus of Variations and Control Theory (ed. D. L. Russell), Math. Res. Center, Univ. Wisconsin, Publ. No. 36, Academic Press, New York, 1976, 321-351. [23] E. Sikolya, Simultaneous observability of networks of beams and strings, Bol. Soc. Paran. Mat. (3), 21 (2003), 31-41. doi: 10.5269/bspm.v21i1-2.7505. [24] O. Szász, Über die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen, Math. Ann., 77 (1916), 482-496. doi: 10.1007/BF01456964. [25] G. Tenenbaum and M. Tucsnak, Fast and strongly localized observation for the Schrödinger equation, Trans. Amer. Math. Soc., 361 (2009), 951-977. doi: 10.1090/S0002-9947-08-04584-4. [26] Xu Zhang, E. Zuazua, Polynomial decay and control of a 1-d hyperbolic-parabolic coupled system, J. Differential Equations, 204 (2004), 380-438. doi: 10.1016/j.jde.2004.02.004. [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), 49-120. doi: 10.1007/s00205-006-0020-x.
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