Geodesic
Approximation and uniform polynomial stability of C 0 -semigroups
Maniar, L.1  ; Nafiri, S.1
1 Département de Mathématiques, Faculté des Sciences Semlalia, Laboratoire LMDP, UMMISCO (IRD-UPMC), Université Cadi Ayyad, B.P. 2390, 40000 Marrakesh, Morocco.
ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 1, pp. 208-235

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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 0n ,t≥0,x 0n ∈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.

DOI : 10.1051/cocv/2015002
Classification : 93C20, 93D20, 73C25, 65M06, 65M60, 65M70
Keywords: C0-semigroups, resolvent, uniform polynomial stability

Maniar, L. 1 ; Nafiri, S. 1

1 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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Maniar, L.; Nafiri, S. Approximation and uniform polynomial stability of C$_{0}$-semigroups. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 1, pp. 208-235. doi: 10.1051/cocv/2015002

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