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September  2018, 7(3): 417-445. doi: 10.3934/eect.2018021

## Control problems and invariant subspaces for sabra shell model of turbulence

 School of Mathematics, IISER Thiruvananthapuram, Maruthamala PO, Vithura, Thiruvananthapuram, Kerala, 695 551, India

* Corresponding author

Received  November 2017 Revised  March 2018 Published  July 2018

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. One of the most suitable shell model of turbulence is so called sabra shell model. The existence, uniqueness and regularity property of this model are extensively studied in [11]. We follow the same functional setup given in [11] and study control problems related to it. 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. Another interesting problem studied in this work is to 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 certain assumptions. Moreover, one of the most interesting result of this work is that we can design a feedback control with only finitely many modes, which is able to preserve the flow in the neighborhood of the constraint set.

Citation: Tania Biswas, Sheetal Dharmatti. Control problems and invariant subspaces for sabra shell model of turbulence. Evolution Equations & Control Theory, 2018, 7 (3) : 417-445. doi: 10.3934/eect.2018021
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