On the observability of linear differential-algebraic systems with delays

The problem of Ropfⁿ-observability is considered for the simplest linear time- delay differential-algebraic system consisting of differential and difference equations. A determining equation system is introduced and a number of algebraic properties of the determining equation solutions is established, in particular, the well-known Hamilton-Cayley matrix theorem is generalized to the solutions of determining equation. As a result, an effective parametric rank criterion for the Ropfⁿ-observability is given. A dual controllability result is also formulated