Exponential stabilization of Volterra integrodifferential equations in Hilbert space

Abstract

We consider exponential stabilization of an abstract linear Volterra integrodifferential equation in a Hilbert space u ′=−E 1Au(t)+E 2∫ 0 1k(t−s)Au(s)ds+f(t) by a feedback control f( t) = − C 0u( t) − C 1u′( t) with bounded linear operators C 0 and C 1 of finite rank. A is a positive semidefinite, self-adjoint unbounded operator, k is nonnegative, convex, and decreasing with exponential decay, k(0) being finite. We introduce a concept of essential growth rate for resolvent operators, similar to that for C 0 -semigroups and prove that the resolvent for the Volterra equation has the same essential growth as the solution semigroup to u′'(t)= − E 1 Au(t) − E 2k(0) E 1 u′(t) + f(t) . Stability can be checked by the location of poles, and it is shown that exponential stabilization is possible with a decay rate of at most E 2k(0) 2E 1 . We apply our results to two problems concerning stabilization of motion of viscoelastic bodies.