Controllability and Observability of Infinite-Dimensional Descriptor Systems

In this paper, the classical systems theoretic concepts of controllability and observability are considered for descriptor systems with an infinite-dimensional state space. These are systems of the form $E\dot{x}(t)=Ax(t)+Bu(t)$, $y(t)=Cx(t)$, where $x(\cdot)$ , $u(\cdot)$, and $y(\cdot)$ are functions with values in separable Hilbert spaces $X, U$, and $Y$. For the operators, we assume that $E:X{\rightarrow} Z$, $B:U{\rightarrow} Z$, and $C:X{\rightarrow} Y$ are bounded, where $Z$ is another Hilbert space. The operator $A$ is assumed to be closed and defined on some dense subspace $D(A)\subset X$. Mappings are defined that induce the notions of controllability and observability. The controllable states and the unobservable states are characterized by invariant subspaces. Based on that, Kalman decompositions of infinite-dimensional descriptor systems are presented.