Evolution Equations and Control Theory (EECT)

An Ingham--Müntz type theorem and simultaneous observation problems

Pages: 297 - 314, Volume 4, Issue 3, September 2015      doi:10.3934/eect.2015.4.297

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Vilmos Komornik - Département de mathématique, Université de Strasbourg, 7, rue René Descartes, 67084 Strasbourg Cedex, France (email)
Gérald Tenenbaum - Institut Élie Cartan, Université de Lorraine, BP 70239, 54506 Vandœuvre-lès-Nancy Cedex, France (email)

Abstract: 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.

Keywords:  Observability, non-harmonic Fourier series, Ingham's theorem, Müntz--Szász theorem, wave equation, heat equation, beam equation.
Mathematics Subject Classification:  Primary: 42C99; Secondary: 93B07.

Received: April 2015;      Revised: July 2015;      Available Online: September 2015.