Some local properties of spectrum of linear dynamical systems in Hilbert space (Q1773046)

The author considers the abstract linear system \(\dot{x}(t) = A x(t) + Bu(t), y(t) = C x(t) + D u(t)\), where \(A\) is the infinitesimal generator of a strongly continuous semigroup on the state space \(X\), and \(B\) and \(C\) are unbounded operators. Furthermore, it is assumed that the above differential equation has for every initial condition and every (locally) square integrable input, \(u\), a weak solution with \(x\) continuous and \(y\) (locally) square integrable. Let \(s_0\) be an isolated point of the spectrum of \(A\), and let \(P(s_0)\) be the associated spectral projection. The author derives characterizations for the controllability/observability of the system restricted to \(P(s_0)X\). One of these characterization is the well-known Hautus test. Furthermore, he shows that if the system restricted to \(P(s_0)X\) is controllable and observable, then \(s_0\) is a pole of order \(m\) for \((sI-A)^{-1}\) if and only if it is a pole of the same order for the transfer function \(G(s) = C(sI-A)^{-1}B + D\). He uses these results to study pole-zero cancellation between two transfer functions in a cascade connection.