  A fundamental concept in control theory is that of controllability, where any
system state can be reached through an appropriate choice of control inputs.
Indeed, a large body of classical and modern approaches are designed for
controllable linear dynamical systems. However, in practice, we often encounter
systems in which a large set of state variables evolve exogenously and
independently of the control inputs; such systems are only partially
controllable. The focus of this work is on a large class of partially
controllable linear dynamical systems, specified by an underlying sparsity
pattern. Our main results establish structural conditions and finite-sample
guarantees for learning to control such systems. In particular, our structural
results characterize those state variables which are irrelevant for optimal
control, an analysis which departs from classical control techniques. Our
algorithmic results adapt techniques from high-dimensional statistics --
specifically soft-thresholding and semiparametric least-squares -- to exploit
the underlying sparsity pattern in order to obtain finite-sample guarantees
that significantly improve over those based on certainty-equivalence. We also
corroborate these theoretical improvements over certainty-equivalent control
through a simulation study.
