  It is well known that highly volatile control laws, while theoretically
optimal for certain systems, are undesirable from an engineering perspective,
being generally deleterious to the controlled system. In this article we are
concerned with the temporal volatility of the control process of the regulator
in discrete time Linear Quadratic Regulators (LQRs). Our investigation in this
paper unearths a surprising connection between the cost functional which an LQR
is tasked with minimizing and the temporal variations of its control laws.
  We first show that optimally controlling the system always implies high
levels of control volatility, i.e., it is impossible to reduce volatility in
the optimal control process without sacrificing cost. We also show that, akin
to communication systems, every LQR has a $Capacity~Region$ associated with it,
that dictates and quantifies how much cost is achievable at a given level of
control volatility. This additionally establishes the fact that no admissible
control policy can simultaneously achieve low volatility and low cost. We then
employ this analysis to explain the phenomenon of temporal price volatility
frequently observed in deregulated electricity markets.
