Optimal Control on the Doubly Infinite Continuous Time Axis and Coprime Factorizations

We study the problem of existence of weak right or left or strong coprime factorizations in H-infinity over the right half-plane of an analytic function defined in some subset of the right half-plane. We give necessary and sufficient conditions for the existence of such coprime factorizations in terms of an optimal control problem over the doubly infinite continuous-time axis. In particular, we show that an equivalent condition for the existence of a strong coprime factorization is that both the control and the filter algebraic Riccati equation (of an arbitrary realization that need not be well-posed) have a solution (in general unbounded and not even densely defined) and that a coupling condition involving these two solutions is satisfied. The proofs that we give are partly based on corresponding discrete time results which we have recently obtained.