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Shell models of turbulence are representation of turbulence equations inFourier domain. Various shell models along with numerical simulations have beenstudied earlier. One of the most suitable shell model of turbulence is socalled sabra shell model. The existence, uniqueness and regularity property ofthis model are extensively studied in \cite{PBT}. In this paper we haveaddressed stabilization problems related to sabra shell model of turbulence. Wehave studied internal stabilization via finite dimensional controller. Moreoverwe have also studied optimal robust control problem by solving an infinite timehorizon max-min control problem. We first prove the $H^ \infty$ stabilizationof the linearized system and charatarize it in terms of a feedback operator bysolving an algebric ricatti equation. Finally we show that the control willasymptotically stabilize the nonlinear system.

Interior and H$^\infty$ feedback stabilization for sabra Shell model of turbulence