We consider the issue of asymptotic stability of a linear continuous-discrete system of functional differential equations with constant coefficients. Such systems consist of two subsystems, continuous and discrete, and are often called hybrid. The continuous subsystem is a system of differential equations. The feature of the hybrid system under consideration is that its continuous part is a system of differential equations with concentrated delay, while the overwhelming majority of papers consider hybrid systems whose continuous part is a system of ordinary differential equations. The standard approach to studying the stability of the latter systems is integration on each finite interval and the construction of the monodromy matrix. However, this approach is generally inapplicable to the problem of studying the stability of hybrid systems such that the continuous part is a system of differential equations with deviating argument. In the present paper, the method of generating functions and the analysis of the spectrum of shift operator along the trajectory of the solution of a hybrid system are applied to study the stability of hybrid systems. Construction of the generating function for the Cauchy matrix and for the fundamental solution allows us to reduce the problem of asymptotic stability of a hybrid system to the problem of studying the location of the roots of a certain function in the complex plane. For this function, it is natural to introduce the term «the characteristic function of the hybrid system», which was done. In addition, it is proved that for these hybrid systems, the asymptotic stability is equivalent to the uniform exponential stability. This approach is compatible with the D-partition method, which allows us to use it to obtain new effective coefficient conditions of asymptotic stability for hybrid systems: in particular, to construct the stability region. In this article, a new simple necessary criterion for the asymptotic stability of a hybrid system is constructed, which is reduced to checking two elementary numerical inequalities.

hybrid systems, continuous-discrete systems, asymptotic stability, Cauchy matrix, generating function.