  In this paper, we consider families of linear systems (linear ensembles)
defined by matrix pairs $\big( A(\theta),B(\theta) \big)$ depending on a
parameter $\theta \in \p$ that is varying over a compact subset $\p$ of the
complex plane. In particular, we investigate the following control task: Find
an open-loop control which is {\it independent} of the parameter $\theta \in
\p$ and steers 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 $X_n(\p)$ of $\C^n$-valued
functions}. If this task is solvable for all initial and terminal states, the
pair $\big( A(\theta),B(\theta) \big)$ is called {(completely)} ensemble
controllable with respect to $X_n(\p)$.
  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 linear ensembles. Moreover, we prove an abstract
decomposition theorem which results from a spectral splitting of the matrix
family $A(\theta)$. Based on thses findings as well as approximation theory and
cyclicity conditions of multiplications operators, we obtain necessary and
sufficient conditions for ensemble controllability (reachability) with respect
to the Banach spaces of continuous functions and $L^q$-functions. In the last
section, results on {averaged} controllability (reachability) for linear
families $\big( A(\theta),B(\theta),C(\theta) \big)$ are presented.
