MODELING STABILITY OF DIFFERENTIAL-DIFFERENCE EQUATIONS WITH DELAY

Abstract

Differential-difference and differential-functional equations are mathematical models of ma\-ny applied problems in automatic control and management systems, chemical, biological, technical, economic and other processes whose evolution depends on prehistory. In the study of the problems of stability, oscillation, bifurcation, control, and stabilization of solutions of linear differential-difference equations, the location of the roots of the corresponding characteristic equations is very important. Note that there are currently no effective algorithms for finding the zeros of quasipolynomials. When studying the approximation of a system of linear differential-difference equations, it was found that the approximation of nonsymptotic roots of their quasi-polynomials can be found with the help of characteristic polynomials of the corresponding approximating systems of ordinary differential equations . This paper investigates the application of approximation schemes for differential-difference equations to construct algorithms for the approximate finding of nonsymptotic roots of quasipolynomials and their application to study the stability of solutions of systems of linear differential equations with many delays. The equivalence of the exponential stability of systems with delay and of the proposed system of ordinary differential equations is established. This allowed us to build an algorithm for studying the location of non-asymptotic roots of quasi-polynomials, which are implemented on a computer. Computational experiments on special test examples showed the high efficiency of the proposed algorithms for studying the stability of linear differential-difference equations.