  Feedback heating from AGN in massive galaxies and galaxy clusters can be
thought of as a naturally occurring control system which plays a significant
role in regulating both star formation rates and the X-ray luminosity of the
surrounding hot gas. In the simplest case, negative feedback can be viewed as a
system response that is `optimised' to minimise deviations from equilibrium,
such that the system rapidly evolves towards a steady state. However, a general
solution of this form appears to be incompatible with radio observations which
indicate intermittent AGN outbursts. Here, we explore an energetically
favourable scenario in which feedback is required to both balance X-ray gas
cooling, and minimise the sum of the energy radiated by the gas and the energy
injected by the AGN. This specification is equivalent to ensuring that AGN
heating balances the X-ray gas cooling with minimum black hole growth. It is
shown that minimum energy heating occurs in discrete events, and not at a
continuous, constant level. Furthermore, systems with stronger feedback
experience proportionally more powerful heating events, but correspondingly
smaller duty cycles. Interpreting observations from this perspective would
imply that stronger feedback occurs in less massive objects - elliptical
galaxies, rather than galaxy clusters. One direct consequence of this effect
would be that AGN heating events are sufficiently powerful to expel hot gas
from the gravitational potential of a galaxy, but not a galaxy cluster, which
is consistent with theoretical explanations for the steepening of the L_X-T
relation at temperatures below 1-2 keV.
