The strict bounded real lemma for linear time-varying systems (Q1973949)

The main results of this paper are connected to the system \[ \dot x= A(t)x+ B(t) w(t),\quad x(0)= 0,\quad t\geq 0, \] \[ z(t)= C(t)x(t)+ D(t)w(t). \] Let \(X_A(t,\cdot)\) be the state transition matrix associated to \(A(t)\) which defines an exponentially stable evolution; define the bounded input/output operator \(T_{zw}: L^2(\mathbb{R}_+, \mathbb{R}^q)\to L^2(\mathbb{R}_+, \mathbb{R}^p)\) as \[ (T_{zw}(w))(t)= \int^t_0 C(t) X_A(t, \tau)B(\tau) w(\tau) d\tau+ D(t)w(t) \] it is proved that \(\|T_{zw}\|< 1\) is equivalent to the existence of a positive definite stabilizing solution of a certain matrix Riccati equation. This results hold both for \(D(t)\equiv 0\) and \(D(t)\not\equiv 0\). Application to time-varying \(H_\infty\) problems is sketched.