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Article Dans Une Revue Houston Journal of Mathematics Année : 2005

$\alpha$-admissibility of observation and control operators

Résumé

If $T(t) = e^{-tA}$ is a bounded strongly continuous semigroup on some Banach space $X$, and if $C\colon D(A^m)\to Y$ is a continuous mapping valued in some Banach space $Y$, we say that $C$ is $\alpha$-admissible if it satisfies an estimate of the form $\int_{0}^{\infty} t^{\alpha}\norm{CT(t)x}^2\, dt\leq M^2\norm{x}^2$. This extends the usual notion of admissibility, which corresponds to $\alpha=0$. In the case when $T(t)$ is a bounded analytic semigroup and $A$ has a `square function estimate', the second named author showed the validity of the so-called Weiss conjecture: $C$ is admissible if and only if $\{ t^{\frac{1}{2}} C(t+A)^{-1}\, :\, t>0 \}$ is a bounded set. In this paper, we extend that characterisation to our new setting. We show (under the same conditions on $T(t)$ and $A$) that $\alpha$-admissibility is equivalent to an appropriate resolvent estimate.
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Dates et versions

hal-00281617 , version 1 (23-05-2008)

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  • HAL Id : hal-00281617 , version 1

Citer

Bernhard H. Haak, Christian Le Merdy. $\alpha$-admissibility of observation and control operators. Houston Journal of Mathematics, 2005, 31 (4), pp.1153-1167. ⟨hal-00281617⟩
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