  A variational formulation incorporating the full Navier-Stokes equations is
used to identify initial perturbations with finite kinetic energy E_{0} which
generate the largest gain in perturbation kinetic energy (across all possible
time intervals) for plane Couette flow. Two different representative flow
geometries are chosen corresponding to those used previously by Butler &
Farrell (1992) and Monokrousos et al. (2011). In the former (smaller geometry)
case as E_{0} increases from 0, we find an optimal which is a smooth nonlinear
continuation of the well-known linear result at $E_{0} = 0$. At $E_{0} =
E_{c}$, however, completely unrelated states are uncovered which trigger
turbulence and our algorithm consequently fails to converge. As $E_{0}
\rightarrow E^{+}_{c}, we find good evidence that the turbulence triggering
initial conditions approach a 'minimal seed' which corresponds to the state of
lowest energy on the laminar-turbulent basin boundary or 'edge'. This situation
is repeated in the Monokrousos et al. (2011) (larger) geometry albeit with one
notable new feature - the appearance of a nonlinear optimal (as found recently
in pipe flow by Pringle & Kerswell (2010) and boundary layer flow by Cherubini
et al. (2010)) at finite $E_{0} < E_{c}$ which has a very different structure
to the linear optimal. Again the minimal seed at $E_{0} = E_{c}$ does not
resemble the linear or now the nonlinear optimal. Our results support the first
of two conjectures recently posed by Pringle et al. (2011) but contradict the
second. Importantly, their prediction that the form of the functional optimised
is not important for identifying E_{c} providing heightened values are produced
by turbulent flows is confirmed: we find the what looks to be the same E_{c}
and minimal seed using energy gain as opposed to total dissipation in the
Monokrousos et al. (2011) geometry.
