CONSTRUCTION OF STABILITY DOMAINS FOR LINEAR DIFFERENTIAL EQUATIONS WITH SEVERAL DELAYS

Abstract

The aim of the present article is to investigate of solutions stability of linear autonomous differential equations with retarded argument. The investigation of stability can be reduced to the root location problem for the characteristic equation. For the linear differential equation with several delays it is obtained the necessary and sufficient conditions, for all the roots of the characteristic equation equation to have negative real part (and hence the zero solution to be asymptotically stable). For the scalar delay differential equation $$ \frac{dz}{dt}=c z(t) + a_1 z(t-1) + a_2 z(t-2) + ... + a_n z(t-n), $$ with fixed $c$, $c \in \mathbb{R}$, $a_k \in \mathbb{R}$, $1 \leq k \leq n$, stability domains in the parameter plane are obtained. We investigate the boundedness conditions and construct a domain of stability for linear autonomous differential equation with several delays. We use D-partition method, argument principle and numerical methods to construct of stability domains.