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

We are considering Ingham's inequality (see [8]) for a family of exponential functions {e^iλ_nt}_n≥1, in the case in which the distance |λ_n+1−λ_n| 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.