Estimates of the constants in generalized Ingham's inequality and applications to the control of the wave equation (Q2774103)

It is well known that, if the exponential family \(\{ e^{i\lambda_n t}\}_{n\geq 1}\) has the property \(|\lambda_{n+1}-\lambda_n|>2\pi/T\) for all \(n\geq 1\), then the following Ingham's inequality holds true: NEWLINE\[NEWLINE C_1 \sum_{n\geq 1} |a_n |^2 \leq \int_0^T \Bigl|\sum_{n\geq 1} a_n e^{i\lambda_n t}\Bigr|^2 dt \leq C_2 \sum_{n\geq 1} |a_n|^2NEWLINE\]NEWLINE for all \((a_n)\in l^2\). It is proved in [\textit{A. Haraux}, J. Math. Pures Appl. (9) 68, 457-465 (1989; Zbl 0685.93039)] that the above inequality still holds if we add any finite number \(N\) of different exponentials. In a first result of this paper, the authors obtain explicit estimates for the constants \(C_1= C_1(N,T)\) and \(C_2=C_2(N,T)\). The main result of this paper shows that there exist a constant \(c_1\) not depending on \(N\) and a range \(I_1(N)\) such thatNEWLINE\[NEWLINE C_1(N,T) \sum_{n\leq I_1(N)} |a_n |^2 + c_1 \sum_{n>I_1(N)} |a_n|^2 \leq \int_0^T \Bigl|\sum_{n\geq 1} a_n e^{i\lambda_n t}\Bigr|^2 dt.NEWLINE\]NEWLINE The previous results are then applied to the boundary controllability problem of the wave equation in the case in which the geometric condition for controllability considered in [\textit{C. Bardos, G. Lebeau} and \textit{J. Rauch}, SIAM J. Control Optimization 30, 1024-1065 (1992; Zbl 0786.93009)] is not satisfied.