The $R$-observability and $R$-controllability of linear algebraic-differential systems

We study the $R$-controllability (the controllability within the attainability set) and the $R$-observability of time-varying linear differential algebraic equations (DAE). We analyze DAE under assumptions guaranteeing the existence of a structural form (which is called “the equivalent form”) with separated “differential” and “algebraic” subsystems. We prove that the existence of this form guarantees the solvability of the corresponding conjugate system, and construct a corresponding equivalent form for the conjugate DAE. We obtain conditions for the $R$-controllability and $R$-observability, in particular, in terms of controllability and observability matrices. We prove theorems that establish certain connections between these properties.