Solving Continuous Time Leech Problems for Rational Operator Functions

Abstract The main continuous time Leech problems considered in this paper are based on stable rational finite dimensional operator-valued functions G and K . Here means that do not have poles the closed right half plane including infinity, problem is to find a operator solution X such $$\begin{aligned} G(s)X(s) = K(s) \quad (s\in \mathbb {C}_+) \hbox {and}\quad \sup \{ \Vert X(s) :\Re s \ge 0 \} < 1 \end{aligned}$$ G ( s ) X = K ∈ C + and sup { ‖ : ℜ ≥ 0 } < 1 . In of given form state space realization. realization operators involved expressed realizations formulas inspired by ideas originating from commutant lifting techniques. However, proof mainly uses representations involved. solutions discrete unit circle easier develop been solved earlier; see, for example, Frazho et al. (Indagationes Math 25:250–274 2014).