  We study the 2D Navier-Stokes equations within the framework of a constraint
that ensures energy conservation throughout the solution. By employing the
Galerkin approximation method, we demonstrate the existence and uniqueness of a
global solution for the constrained Navier-Stokes equation on the torus
$\mathbb{T}^2$. Moreover, we investigate the linearized system associated with
the 2D-constrained Navier-Stokes equations, exploring its existence and
uniqueness. Subsequently, we establish the Lipschitz continuity and
Fr$\'{e}$chet differentiability properties of the solution mapping. Finally,
employing the formal Lagrange method, we prove the first-order necessary
optimality conditions.
