The stability cone for a delay differential matrix equation

We describe a surface in R 3 which is called the stability cone. We prove necessary and sufficient stability conditions for the delay differential matrix equation x ̇ + A x + B x ( 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.