Finite spectrum assignment problem in linear systems with state delay by static output feedback

We consider a control system defined by a linear time-invariant system of differential equations with delay
\begin{equation} \dot x(t)=Ax(t)+A_1x(t-h)+Bu(t),\quad y(t)=C^*x(t),\quad t>0. \tag{1} \end{equation}
We construct the controller for the system $(1)$ as linear output feedback $u(t)=Q_0 y(t)+Q_1 y(t-h)$. We study a finite spectrum assignment problem for the closed-loop system. One needs to construct gain matrices $Q_0$, $Q_1$ such that the characteristic quasipolynomial of the closed-loop system becomes a polynomial with arbitrary preassigned coefficients. We obtain conditions on coefficients of the system $(1)$ under which the criterion was found for solvability of the finite spectrum assignment problem. The obtained result extends to systems with several delays. Corollaries on stabilization by linear static output feedback with delay are obtained for system $(1)$ as well as for systems of type $(1)$ with several delays.