Control Problems and Invariant Subspaces for the Sabra Shell Model of Turbulence

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.