Approximate observability of abstract evolution equation with unbounded observation operator
The standard assumptions in this paper are: (i) \(A\) is the infinitesimal generator of a strongly continuous semigroup on the Banach space \(X\), (ii) \(A\) has a complete basis of (generalized) eigenfunctions, and (iii) the sequence \(\{f_{jk}(t):= t^k e^{\lambda_j t}\), \(j\in\mathbb{N}\), \(k\in\{0,1,\dots, \alpha_j- 1\}\}\) is minimal on \(L^2(0, t_1)\) for some \(t_1> 0\), where \(\lambda_j\) is the \(j\)th eigenvalue of \(A\) with multiplicity \(\alpha_j\). Under these assumptions the author shows that the system \(\dot x(t)= Ax(t)\), \(y(t)= Cx(t)\) is approximately observable if and only if there exists no eigenvector of \(A\) in the kernel of \(C\). Unfortunately, the author seems to be unaware of the results obtained by \textit{S. A. Avdonin} and \textit{S. A. Ivanov} as presented in their book [Families of exponentials. The method of moments in controllability problems for distributed parameter systems, Cambridge Univ. Press (1995; Zbl 0866.93001)].
scientific article; zbMATH DE number 1992507