  Learning, say through direct policy updates, often requires assumptions such
as knowing a priori that the initial policy (gain) is stabilizing, or
persistently exciting (PE) input-output data, is available. In this paper, we
examine online regulation of (possibly unstable) partially unknown linear
systems with no prior access to an initial stabilizing controller nor PE
input-output data; we instead leverage the knowledge of the input matrix for
online regulation. First, we introduce and characterize the notion of
"regularizability" for linear systems that gauges the extent by which a system
can be regulated in finite-time in contrast to its asymptotic behavior
(commonly characterized by stabilizability/controllability). Next, having
access only to the input matrix, we propose the Data-Guided Regulation (DGR)
synthesis procedure that -- as its name suggests -- regulates the underlying
state while also generating informative data that can subsequently be used for
data-driven stabilization or system identification. We further improve the
computational performance of DGR via a rank-one update and demonstrate its
utility in online regulation of the X-29 aircraft.
