Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems

Abstract

In this article, we obtain explicit bounds on the norms of biorthogonal functions to sets of complex exponentials {exp(− λ k t)}, where the { λ k } belong to a sector of the positive real axis. The bounds obtained are uniform within a class of sequences which exhibit similar growth and separation properties. The dependence of the bounds upon the growth and separation parameters is carefully examined. The main theorem then leads to some results regarding the solvability of moment problems of the form ∝ 0 T f( t)exp(− λ k t) dt = c k , which arise in control problems associated with damped or dissipative systems. As an application, we establish an exact null controllability result for boundary control of a rectangular structurally damped plate.