Counterexamples concerning observation operators for $C_0$-semigroups


Jacob, Birgit and Zwart, Hans (2004) Counterexamples concerning observation operators for $C_0$-semigroups. SIAM journal on control and optimization, 43 (1). pp. 137-153. ISSN 0363-0129

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Abstract:This paper concerns systems of the form $\dot{x}(t) = Ax(t)$, $y(t) = Cx(t)$, where $A$ generates a $C_0$-semigroup. Two conjectures which were posed in 1991 and 1994 are shown not to hold. The first conjecture (by G. Weiss) states that if the range of $C$ is one-dimensional, then $C$ is admissible if and only if a certain resolvent estimate holds. The second conjecture (by D. Russell and G. Weiss) states that a system is exactly observable if and only if a test similar to the Hautus test for finite-dimensional systems holds. The $C_0$-semigroup in both counterexamples is analytic and possesses a basis of eigenfunctions. Using the $(A, C)$-pair from the second counterexample, we construct a generator $A_e$ on a Hilbert space such that $(sI -A_e)$ is uniformly left-invertible, but its semigroup does not have this property.
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Copyright:© 2004 Society for Industrial and Applied Mathematics, SIAM
Electrical Engineering, Mathematics and Computer Science (EEMCS)
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