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 stable means that G and K do not have poles in the closed right half plane including infinity, and the Leech problem is to find a stable rational operator solution X such that G(s)X(s)=K(s)(s∈C+)andsup{‖X(s)‖:ℜs≥0}<1.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} G(s)X(s) = K(s) \\quad (s\\in \\mathbb {C}_+) \\quad \\hbox {and}\\quad \\sup \\{ \\Vert X(s) \\Vert :\\Re s \\ge 0 \\} < 1 . \\end{aligned}$$\\end{document}In the paper the solution of the Leech problem is given in the form of a state space realization. In this realization the finite dimensional operators involved are expressed in the operators of state space realizations of the functions G and K. The formulas are inspired by and based on ideas originating from commutant lifting techniques. However, the proof mainly uses the state space representations of the rational finite dimensional operator-valued functions involved. The solutions to the discrete time Leech problem on the unit circle are easier to develop and have been solved earlier; see, for example, Frazho et al. (Indagationes Math 25:250–274 2014).