On Unimodular Transformations of Conservative L-systems

We study unimodular transformations of conservative L-systems. Classes \( {\mathfrak{M}}^{Q} ,\,\,{\mathfrak{M}}_{k}^{\,Q} ,\,\,{\mathfrak{M}}_{k}^{ - 1,\,Q} \, \) that are impedance functions of the corresponding L-systems are introduced. A unique unimodular transformation of a given L-system with impedance function from the mentioned above classes is found such that the impedance function of a new L-system belongs to \( {\mathfrak{M}}^{( - Q)} ,\,\,{\mathfrak{M}}_{k}^{( - Q)} ,\,\,{\mathfrak{M}}_{k}^{ - 1,\,( - Q)} \), respectively. As a result we get that considered classes (that are perturbations of the Donoghue classes of Herglotz–Nevanlinna functions with an arbitrary real constant Q) are invariant under the corresponding unimodular transformations of L-systems. We define a coupling of an L-system and a so-called F-system and on its basis obtain a multiplication theorem for their transfer functions. In particular, it is shown that any unimodular transformation of a given L-system is equivalent to a coupling of this system and the corresponding controller, an F-system with a constant unimodular transfer function. In addition, we derive an explicit form of a controller responsible for a corresponding unimodular transformation of an L-system. Examples that illustrate the developed approach are presented.