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 xneitωn |2 dt ∞n=p |xn|2/(1+bn), for all sequences {xn} with ∞n=1 |xn|2/(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. © 2006 Elsevier Inc. All rights reserved.