Complete controllability of spectrum assignment in infinite dimensional spaces (Q1316451)

The authors address the problem of complete controllability in infinite dimensions: Let \(U\), \(X\) be Banach spaces and form with \(A\in {\mathcal L}(X)\), \(B\in {\mathcal L}(U, X)\) the initial value problem \[ x'(t)= Ax(t)+ Bu(t),\quad x(0)= x_ 0, \] where \(u\) is piecewise continuous. This problem is called completely controllable if, for each \(x_ 0\in X\), one can find \(u\) and \(T> 0\) such that the unique solution \(x\) satisfies \(x(T)= 0\). The authors consider six conditions for complete controllability, all of which were already discussed in the literature (at least in finite dimensions). The 6th condition is treated in infinite dimensions here for the first time but seems to clarify the picture somewhat. It is shown as the main result that all conditions are equivalent in Hilbert spaces (Theorem 10). The dual problem of observability is also discussed, and some relation with the theory of holomorphic operator functions is established.