Approximation and uniform polynomial stability ofC0-semigroups

Abstract

Consider the classical solutions of the abstract approximate problems x' n ( t ) = A n x n ( t ), t ≥ 0, x n (0) = x 0n , n ∈ ℕ, given by x n ( t ) = T n ( t ) x 0 n ,t ≥ 0, x 0 n ∈ D ( A n ), where A n generates a sequence of C 0 -semigroups of operators T n ( t ) on the Hilbert spaces H n . Classical solutions of this problem may converge to 0 polynomially, but not exponentially, in the following sense ∥ T n ( t ) x ∥ ≤ C n t − β ∥ A n α x ∥, x ∈ D ( A n α ), t > 0, n ∈ ℕ, for some constants C n ,α and β > 0. This paper has two objectives. First, necessary and sufficient conditions are given to characterize the uniform polynomial stability of the sequence T n ( t ) on Hilbert spaces H n . Secondly, approximation in control of a one-dimensional hyperbolic-parabolic coupled system subject to Dirichlet−Dirichlet boundary conditions, is considered. The uniform polynomial stability of corresponding semigroups associated with approximation schemes is proved. Numerical experimental results are also presented.