On finite spectrum assignment problem in bilinear systems with state delay

We consider a bilinear control system defined by a linear time-invariant system of differential equations with delay in the state variable. We study an arbitrary finite spectrum assignment problem by stationary control. One needs to construct constant control vector such that the characteristic quasi-polynomial of the closed-loop system becomes a polynomial with arbitrary preassigned coefficients. We obtain conditions on coefficients of the system under which the criterion was found for solvability of this finite spectrum assignment problem. This criterion is expressed in terms of rank conditions for matrices of the special form. Interconnection of these rank conditions with the property of consistency for truncated system without delay is shown. Corollaries on stabilization of a bilinear system with delay are obtained. The results extend the previously obtained results on spectrum assignment for linear systems with static output feedback with delay and for bilinear systems without delay. The results obtained are transferred to discrete-time bilinear systems with delay. An illustrative example is considered.