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 and uniformly bounded on some 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 well-posed realization) have a solution (in general unbounded and not even densely defined) and that a coupling condition involving these two solutions is satisfied.
Opmeer, M., & Staffans, O. (2019). Optimal Control on the Doubly Infinite Time Axis for Well-Posed Linear Systems. SIAM Journal on Control and Optimization, 57(3), 1985-2015. https://doi.org/10.1137/18M1181304