Approximation and uniform polynomial stability of C₀-semigroups

Département de Mathématiques, Faculté des Sciences Semlalia, Laboratoire LMDP, UMMISCO (IRD-UPMC), Université Cadi Ayyad, B.P. 2390, 40000 Marrakesh, Morocco.
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Received: 23 April 2014
Revised: 28 July 2014

Abstract

Consider the classical solutions of the abstract approximate problems

        x'_n(t) = A_nx_n(t), t ≥ 0,      x_n(0) = x_0n, n ∈ ℕ,

given by x_n(t) = T_n(t)x_0n,t ≥ 0,x_0n ∈ D(A_n), where A_n generates a sequence of C₀-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_nt^−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.