Exponential stabilization of Volterra integrodifferential equations in Hilbert space (Q1097460)

This paper presents a feedback stabilization result for the abstract linear Volterra integrodifferential equation \[ (*)\quad u''(t)=-E_ 1Au(t)+E_ 2\int^{t}_{0}K(t-s)Au(s)ds+f(t). \] Here \(E_ 1\) and \(E_ 2\) are positive constants, A is a positive selfadjoint unbounded operator on a Hilbert space H,K is a bounded, nonnegative, convex, and exponentially decaying function, and the feedback control f is of the form \(f(t)=-C_ 0u(t)-C_ 1u'(t),\) where \(C_ 0\) and \(C_ 1\) are bounded linear operators of finite rank. A substantial part of the paper is devoted to a comparison of equation (*) and the equation \((**)\quad u''=-E_ 1u(t)-E_ 2K(0)/E_ 1u'(t)+f(t).\) Among others, it is shown that the resolvent operator for (*) has the same essential growth rate as the fundamental solution of (**). In addition, it is shown how modes that grow faster than the essential growth rate can be controlled by a suitable choice of \(C_ 0\) and \(C_ 1\), hence the equation can be exponentially stabilized. Two examples are given, where some fixed bodies with viscoelastic appendages are controlled.