Well-posedness and admissible stabilizability for Pritchard-Salamon systems
This paper can be seen as a direct follow-up of the paper by \textit{R. F. Curtain}, \textit{H. Logemann}, \textit{S. Townley} and \textit{H. Zwart} [J. Math. Syst. Estim. Control 7, 439-476 (1997; Zbl 0815.93046)]. Both papers study the Pritchard-Salamon class of infinite-dimensional systems, i.e., abstract systems of the form \(\dot x(t)= Ax(t)+ Bu(t)\), \(y(t)= Cx(t)\), where \(A\) generates a \(C_0\)-semigroup on \(W\) and \(V\), two Hilbert spaces, with \(W\) densely contained in \(V\). The input to state map is bounded from \(L^2(0,t;U)\) to \(W\) and the state to output map is bounded from \(V\) to \(L^2(0,t;Y)\), where \(U\) and \(Y\) are the input and output spaces, respectively. A more or less standard assumption for this class of systems is that the domain of the generator \(A\) on \(V\) is densely contained in \(W\). In [loc. cit.] this assumption was shown not to be needed for many properties. However, for some feedback results it was still needed. In this short note, it is shown that even for these results this smoothness assumption is not needed. Hence it is very natural to conjecture that this smoothness condition is unnecessary, as is stated in the last line of this paper.
scientific article; zbMATH DE number 1878929
scientific article