New meaning of exact controllability of linear systems in Hilbert spaces. (Q2712212)

In this note the authors announce the main results of a forthcoming paper. They consider the abstract differential equation \(\dot{x}(t) = A x(t) + Bu(t)\), where \(A\) generates a \(C_0\)-semigroup on a Hilbert space and \(B\) is a bounded operator. Exact controllability is defined as the property to steer from any initial state to any final state in final time. It is shown (under some additional conditions) that exact controllability is only possible when \(A\) generates a group. Furthermore, if the resolvent of \(A\) is compact, then \(A\) has a complete set of (generalized) eigenfunctions.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00050].