Ingham-type inequalities for complex frequencies and applications to control theory

For complex valued sequences {ωn}∞n=1 of the form ωn = an + ibn with an ∈ R and bn ≥ 0, we prove inequalities of the form ∫ T 0 | ∑∞ n=p xne itωn |2dt 3 ∞n=p |xn|/(1+bn), for all sequences {xn} with ∑∞ n=1 |xn|/(1 + bn) < ∞. We apply these to prove exact null-controllability for a class of hinged beam equations with mild internal damping with either boundary control or internal control.