Controllability up to renormalization and small control operators
In this elaborated and well written paper the author presents results on various controllability types for systems described by the equation \(x'=Ax+Bu\), where \(A\) generates a continuous semigroup \(S(t)\) on a Banach space \(X\) and \(B\) is a bounded control operator. In the first part of the paper the author gives necessary or/and sufficient conditions for exact controllability and null controllability, involving \(B\) and \(S(t)\), by using results from linear operator theory. Although most of these results are not new (for example the lack of exact controllability with compact \(B\) is an old result of Triggiani), this part is a good background for the main part of the paper where the author goes deeply into comparison with the standard operator methods. Here \(X\) is a Hilbert space and \(A\) has a Riesz basis of eigenvectors, \(A\phi_n =-\lambda_n \phi_n\), \(n\geq 1\). The problem is to describe the space of reachable points, which is more general than that of exact or null controllability. In this context the author defines the notion of controllability up to renormalization (CR), i.e., the existence of weights \(w_n >0\) such that the restriction of \((A,B)\) on the space \[ H(w_n)= \left\{ \sum_{n\geq 1} a_n \phi_n ; \sum_{n\geq 1}| a_n | ^2 \| \phi_n \| ^2 w_n ^2 <\infty\right\} \] is well defined and exactly controllable. A spectral necessary condition for \(A\) in order to have (CR) is given and then, assuming that, necessary and sufficient conditions involving \(B\) are derived. The author obtains the following result: for every quasi-normed ideal of compact operators \({\mathcal I}\) (but not for the ideal of finite rank operators) there exists \(B\in {\mathcal I}\) such that the system \((A,B)\) enjoys the (CR) property. Necessary and sufficient conditions for null controllability involving \(\lambda_n\) and \(B\) are also presented and applications to both hyperbolic and parabolic equations are given. Finally, a link between control problems and Hardy space \(H^2\)-interpolation is discussed.
scientific article; zbMATH DE number 1187616