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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