  Shell models of turbulence are representation of turbulence equations in
Fourier domain. Various shell models and their existence theory along with
numerical simulations have been studied earlier. In this work we study control
problems related to sabra shell model of turbulence. We associate two cost
functionals: one ensures minimizing turbulence in the system and the other
addresses the need of taking the flow near a priori known state. We derive
optimal controls in terms of the solution of adjoint equations for
corresponding linearized problems. In this work, we also establish feedback
controllers which would preserve prescribed physical constraints. Since fluid
equations have certain fundamental invariants, we would like to preserve these
quantities via a control in the feedback form. We utilize the theory of
nonlinear semi groups and represent the feedback control as a multi-valued
feedback term which lies in the normal cone of the convex constraint space
under consideration.
