The algebraic approach to stability analysis of differential-difference systems

A new approach to the exponential stability analysis of linear differential-difference systems with constant coefficients is suggested. The necessary and sufficient conditions of the exponential stability and instability of such systems, which are based on the derivation of the quadratic estimate for the quadratic functionals on some special set, are proved. These conditions allow to use Lyapunov second method for the analysis of exponential stability of systems with delay. On the base of proven statements the final constructive algorithm of control of the positive definiteness of the quadratic Lyapunov–Krasovskii functionals is constructed. Illustrative examples of the exponential stability analysis of the differential-difference equations using the introduced method are considered.