Volume 22, Number 1, January-March 2016
|Page(s)||208 - 235|
|Published online||15 January 2016|
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
Revised: 28 July 2014
Consider the classical solutions of the abstract approximate problems
x'n(t) = Anxn(t), t ≥ 0, xn(0) = x0n, n ∈ ℕ,
given by xn(t) = Tn(t)x0n,t ≥ 0,x0n ∈ D(An), where An generates a sequence of C0-semigroups of operators Tn(t) on the Hilbert spaces Hn. Classical solutions of this problem may converge to 0 polynomially, but not exponentially, in the following sense
∥Tn(t)x∥ ≤ Cnt−β∥Anα x∥, x ∈ D(Anα), t > 0, n ∈ ℕ,
for some constants Cn,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 Tn(t) on Hilbert spaces Hn. 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.
Mathematics Subject Classification: 93C20 / 93D20 / 73C25 / 65M06 / 65M60 / 65M70
Key words: C0-semigroups / resolvent / uniform polynomial stability
© EDP Sciences, SMAI 2016
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