Estimates of the constants in generalized Ingham's inequality and applications to the control of the wave equation

Article type: Research Article

Authors: Jaffard, Stéphane^; | Micu, Sorin^{; ;}

Affiliations: Université Paris 12 Créteil, 61 avenue du Général de Gaulle, 94010 Créteil cedex, France E‐mail: jaffard@univ‐paris12.fr | Institut Universitaire de France | Departamento de Matemática Aplicada, Facultad de Ciencias Matemáticas, Universidad Complutense, 28040 Madrid, Spain | Facultatea de Matematica‐Informatica, Universitatea din Craiova, 1100, Romania E‐mail: [email protected]

Note: [] Partially supported by Grant PB96‐0663 of DGES (Spain) and Grant 10/1998 of CNCSU (Romania).

Abstract: We are considering Ingham's inequality (see [8]) for a family of exponential functions {eiλnt}n≥1, in the case in which the distance |λn+1−λn| between two consecutive exponents becomes smaller and smaller for |n|≤N but there still exists an asymptotic gap sufficiently large. We give explicit estimates for the two constants appearing in the inequality and we analyze how does the small gap between the first exponents affect the constants. These results are applied to a control problem for the wave equation in a case in which the geometric condition for controllability, deduced in [4], are not satisfied.

Journal: Asymptotic Analysis, vol. 28, no. 3-4, pp. 181-214, 2001