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

For complex valued sequences {omega(n)}(infinity)(n=1) of the form omega(n) = a(n) + ib(n) with a(n) is an element of R and b(n) >= 0, we prove inequalities of the form integral(T)(0)vertical bar Sigma(infinity)(n=p) x(n)e(it omega n)vertical bar(2) dt asymptotic to Sigma(infinity)(n=p) vertical bar x(n)vertical bar(2)/(1+b(n)), for all sequences {x(n)} with Sigma(infinity)(n=1)vertical bar x(n)vertical bar(2)/(1+b(n)) < infinity. 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. (c) 2006 Elsevier Inc. All rights reserved.