Stabilizability, detectability, and spectrum assignment for linear autonomous systems with general time delays

A general linear autonomous system with both discrete and distributed delays in state and control variables is considered and an open-loop stabilizability problem is posed. It is proven that a simple algebraic rank condition, similar to the well-known Hautus condition, is necessary for open-loop stabilizability. This condition is also shown to be sufficient by constructing a proper stabilizing state feedback. The detectability problem for systems with general state and output delays is proven to be dual to state-feedback stabilizability of a transposed system with state and control delays. If the delays appear in control variables only the state-feedback spectrum assignability is equivalent to formal controllability of a certain pair of real matrices and, equivalently, to system state controllability.