  This paper introduces a framework for quantitative characterization of the
controllability of time-varying linear systems (or networks) in terms of input
novelty. The motivation for such an approach comes from the study of
biophysical sensory networks in the brain, wherein responsiveness to both
energy and salience (or novelty) are presumably critical for mediating behavior
and function. Here, we use an inner product to define the angular separation of
the current input with respect to the past input history. Then, by constraining
input energy, we define a non-convex optimal control problem to obtain the
minimally novel input that effects a given state transfer. We provide
analytical conditions for existence and uniqueness in continuous-time, as well
as an explicit closed-form expression for the solution. In discrete time, we
show that a relaxed convex optimization formulation provides the global optimal
solution of the original non-convex problem. Finally, we show how the minimum
novelty control can be used as a metric to study control properties of large
scale recurrent neuronal networks and other complex linear systems. In
particular, we highlight unique aspects of a system's controllability that are
captured through the novelty-based metric. The result suggests that a
multifaceted approach, combining energy-based analysis with other specific
metrics, may be useful for obtaining more complete controllability
characterizations.
