Stability analysis in continuous and discrete time, using the Cayley transform (Q616751)

Let \((e^{At})_{t\geq 0}\) and \((e^{\tilde{A}t})_{t\geq 0}\) be two strongly continuous semigroups on a separable Hilbert space \(X\). If, for all \(x_0\in X\), \[ \int^\infty_0\left\|\left(e^{At}-e^{\tilde{A}t}\right)x_0\right\|^2\tfrac{1}{t}\,dt, \int^\infty_0\left\|\left(e^{A^\ast t}-e^{\tilde{A^\ast}t}\right)x_0\right\|^2\tfrac{1}{t}\,dt<\infty, \] then one says that the two semigroups \((e^{At})_{t\geq 0}\) and \((e^{\tilde{A}t})_{t\geq 0}\) have finite Bergman distance. It is shown that two semigroups with finite Bergman distance have the same stability properties, and the same conclusion is also true for the discrete-time case.