  We explore incompressible homogeneous isotropic turbulence within the
(fourth-order model) formulation of optimal control and optimization, in
contrast to the classical works of Proudman and Reid (1954) and Tatsumi (1957),
with the intention to fix specially their defect of negative energy spectrum
values being developed and to examine generally the conventional closure
schemes. The isotropic forms for the general and spatially degenerated fourth
order correlations of fluctuating velocity are obtained and the corresponding
primary dynamical equations are derived. The degenerated fourth order
correlation contains four scalar functions $D_i$, $i=1,2,3,4$, whose
determination is the focus of closure. We discuss the constraints of equality
for these functions as required by the self-consistency of the definition of
the degenerated. Furthermore, we develop the constraints of inequality for the
scalar functions based on the application of the Cauchy-Schwarz inequality, the
non-negativity of the variance of products, and the non-negativity of the
turbulent energy spectrum. We intend to indicate the difficulty for a
conventional scheme to satisfy all these constraints. As an alternative, we
employ the turbulent energy per unit volume as the objective function to be
maximized, under the constraints and the dynamical equations, with the four
scalar functions as the control variables, which is a second-order cone
programming problem. We then treat the asymptotic state solutions at large time
and focus especially on the sub-model where the third order correlation is
taken as the control variable, considering the computing resources available.
