The stability cone for a delay differential matrix equation

We describe a surface in R3 which is called the stability cone. We prove necessary and sufficient stability conditions for the delay differential matrix equation ẋ+Ax+Bx(t−τ)=0. These conditions are formulated in terms of the location with respect to the stability cone of some points determined by the eigenvalues of matrices A,B and the delay value. We require that matrices A,B admit a simultaneous triangularization.