  In this paper, we examine the fundamental performance limitations in the
control of stochastic dynamical systems; more specifically, we derive generic
$\mathcal{L}_p$ bounds that hold for any causal (stabilizing) controllers and
any stochastic disturbances, by an information-theoretic analysis. We first
consider the scenario where the plant (i.e., the dynamical system to be
controlled) is linear time-invariant, and it is seen in general that the lower
bounds are characterized by the unstable poles (or nonminimum-phase zeros) of
the plant as well as the conditional entropy of the disturbance. We then
analyze the setting where the plant is assumed to be (strictly) causal, for
which case the lower bounds are determined by the conditional entropy of the
disturbance. We also discuss the special cases of $p = 2$ and $p = \infty$,
which correspond to minimum-variance control and controlling the maximum
deviations, respectively. In addition, we investigate the power-spectral
characterization of the lower bounds as well as its relation to the
Kolmogorov-Szeg\"o formula.
