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

We consider families of linear systems that are defined by matrix pairs $\big( A(\theta),B(\theta) \big)$ which depending on a parameter $\theta$ that is varying over a compact set in the plane. The focus of this paper is on the task of steering a family of initial states $x_0(\theta)$ in finite time arbitrarily close to a given family of desired terminal states $x^*(\theta)$ via a parameter-independent open-loop control input. In this case the pair $\big( A(\theta),B(\theta) \big)$ is called ensemble reachable. Using well-known characterizations of approximate controllability for systems in Banach spaces, ensemble reachability of $\big( A(\theta),B(\theta) \big)$ is equivalent to an infinite-dimensional extension of the Kalman rank condition. In this paper we investigate structural properties and prove a decomposition theorem according to the spectra of the matrices $A(\theta)$. Based on this results together with results from complex approximation and functional analysis we show necessary and sufficient conditions in terms of $\big( A(\theta),B(\theta) \big)$ for ensemble reachability for families of linear systems $\big( A(\theta),B(\theta) \big)$ defined on the Banach spaces of continuous functions and $L^q$-functions. The paper also presents results on output ensemble reachability for families $\big( A(\theta),B(\theta),C(\theta) \big)$ of parameter-dependent linear systems