Approximation and uniform polynomial stability of C$_{0}$-semigroups

Maniar, L.; Nafiri, S.

doi:10.1051/cocv/2015002

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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Résumé

Consider the classical solutions of the abstract approximate problems

$x_{n}^{'} (t) = A_{n} x_{n} (t), t \geq 0, x_{n} (0) = x_{0 n}, n \in ℕ,$

given by $x_{n} (t) = T_{n} (t) x_{0 n}, t \geq 0, x_{0 n} \in D (A_{n})$ , 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

$∥ T_{n} (t) x ∥ \leq C_{n} t^{- β} ∥ A_{n}^{α} x ∥, x \in D (A_{n}^{α}), t > 0, n \in ℕ,$

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.

Zbl

DOI : 10.1051/cocv/2015002

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

Affiliations des auteurs :

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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     author = {Maniar, L. and Nafiri, S.},
     title = {Approximation and uniform polynomial stability of {C}$_{0}$-semigroups},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {208--235},
     publisher = {EDP-Sciences},
     volume = {22},
     number = {1},
     year = {2016},
     doi = {10.1051/cocv/2015002},
     zbl = {1348.93227},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2015002/}
}

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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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