Summary: In this paper, we consider families of linear systems (linear ensembles) defined by matrix pairs \(\bigl(A(\theta), B(\theta)\bigr)\) depending on a parameter \(\theta \in \mathbf{P}\) that is varying over a compact subset \(\mathbf{P}\) of the complex plane. In particular, we investigate the existence of open-loop controls which are independent of the parameter \(\theta \in \mathbf{P}\) and steer a given family of initial states \(x_0 (\theta)\) arbitrarily close to a desired family of terminal states \(f(\theta)\) in finite time. Here, the maps \(\theta \mapsto x_0 (\theta)\) and \(\theta \mapsto f(\theta)\) are assumed to lie in a common appropriately chosen Banach space \(X_n (\mathbf{P})\) of \(\mathbb{C}^n\)-valued functions. If this task is solvable for all initial and terminal states, the pair \((A(\theta), B(\theta))\) is called (completely) ensemble controllable with respect to \(X_n(\mathbf{P})\).
Using a well-known infinite-dimensional version of the Kalman rank condition for systems on Banach spaces, we derive sufficient conditions for cascade and parallel connections of linear ensembles. Moreover, we prove an abstract decomposition theorem which results from a spectral splitting of the matrix family \(A(\theta)\). Based on these findings as well as on cyclicity conditions for multiplication operators and approximation theory, we obtain necessary and sufficient conditions for ensemble controllability (reachability) with respect to the space of continuous functions and the space of \(L^q\)-functions. In the last section, results on averaged controllability (reachability) for linear families \((A(\theta), B(\theta), C(\theta))\) are presented.