  We propose the {\alpha}-suboptimal covering number to characterize multi-task
control problems where the set of dynamical systems and/or cost functions is
infinite, analogous to the cardinality of finite task sets. This notion may
help quantify the function class expressiveness needed to represent a good
multi-task policy, which is important for learning-based control methods that
use parameterized function approximation. We study suboptimal covering numbers
for linear dynamical systems with quadratic cost (LQR problems) and construct a
class of multi-task LQR problems amenable to analysis. For the scalar case, we
show logarithmic dependence on the "breadth" of the space. For the matrix case,
we present experiments 1) measuring the efficiency of a particular constructive
cover, and 2) visualizing the behavior of two candidate systems for the lower
bound.
