The Infinite-Dimensional Continuous Time Kalman - Yakubovich - Popov Inequality

We study the set Mσof all generalized solutions (that may be unbounded and have an unbounded inverse) of the KYP (Kalman–Yakubovich–Popov) inequality for a infinite-dimensional linear time-invariant systemσin continuous time with scattering supply rate. It is shown that if Mσis nonempty, then the transfer function of σ coincides with a Schur class function in some right half-plane. For a minimal system σ the converse is also true. In this case the set of all H ∈ Mσwith the property that the system is still minimal when the original norm in the state space is replaced by the norm induced by H is shown to have a minimal and a maximal solution, which correspond to the available storage and the required supply, respectively. We show by an example that the stability of the system with respect to the norm induced by some H ∈ Mσdepends crucially on the particular choice of H. In this example, depending on the choice of the original realization, some or all H ∈ Mσwill be unbounded and/or have an unbounded inverse.