Approximation and uniform polynomial stability of C${}_0$-semigroups

Maniar, L.; Nafiri, S.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 1, pp. 208-235.

Consider the classical solutions of the abstract approximate problems

        $x_n^{\text{'}}\left(t\right)=A_n{x}_n\left(t\right),t\ge 0,x_n\left(0\right)=x_{0n},n\in \mathbb{N},$

given by $x_n\left(t\right)=T_n\left(t\right)x_{0n},t\ge 0,x_{0n}\in D\left(A_n\right)$, where $A_n$ generates a sequence of $C_0$-semigroups of operators $T_n\left(t\right)$ on the Hilbert spaces $H_n$. Classical solutions of this problem may converge to $0$ polynomially, but not exponentially, in the following sense

        $\parallel T_n\left(t\right)x\parallel \le C_n{t}^{-\beta }\parallel A_n^{\alpha }x\parallel ,x\in D\left(A_n^{\alpha }\right),t>0,n\in \mathbb{N},$

for some constants $C_n,\alpha$ and $\beta >0$. This paper has two objectives. First, necessary and sufficient conditions are given to characterize the uniform polynomial stability of the sequence $T_n\left(t\right)$ 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.

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