Passive linear stationary dynamical scattering systems with continuous time

In this paper we consider systems with the separable Hilbert inner, input and output spacesX, $mathfrak{N}^ - $ , $mathfrak{N}^ + $ of the form $$frac{{dx(t)}}{{dt}} = hat Bx(t) + Lvarphi ^ - (t),varphi ^ + (t) = N(x(t),varphi ^ - (t)),x(0) = a$$ with some natural restrictions on the coefficients which have been proposed by Yu.L. Shmuljan. For each system the concepts of simple, minimal, passive scattering, conservative scattering, optimal passive scattering ones are introduced. We realize any $[mathfrak{N}^ - ,mathfrak{N}^ + ]$ valued function θ(p) which is holomorphic with contractive values in the right half plane as the transfer function (t.f.) of a simple conservative scattering system and also as the t.f. of a minimal optimal passive scattering system. Both these realizations are defined by θ(p) uniquely up to unitary similarity. Reduction of the problem to the corresponding problems for systems with discrete time via Cayley transform is used.