Uniform and $L^q$-Ensemble Reachability of Parameter-dependent Linear Systems

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.