  The problem of finding bounds of time-averaged characteristics of dynamical
systems, such as the bound on the mean energy dissipation rate in a turbulent
flow governed by incompressible Navier-Stokes equations, is considered. It is
shown that the direct method described by Seis (J. Fluid Mech., 777:591-603,
2015) and the auxiliary functional method by Chernyshenko et al. (Phil. Trans.
R. Soc. B., 372:20130350, 2014) are related and can lead to the same bound. The
well-known background flow method of Doering and Constantin is equivalent to
the auxiliary functional method with a quadratic auxiliary functional. The
known implementations of the direct method apparently also correspond to
quadratic auxiliary functionals. The findings are illustrated by the analysis
of the plane Couette flow. Three routes of further progress using non-quadratic
auxiliary functionals and at the same time allowing to utilise the experience
accumulated with the background flow method are proposed: making the balance
parameter dependent on the energy, making the background flow time-dependent in
a specific way, and adding helicity to the auxiliary functional.
