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

TL;DR

This paper investigates control strategies for the Sabra shell model of turbulence, focusing on minimizing turbulence and maintaining physical invariants through optimal and feedback controls derived from adjoint equations and nonlinear semigroup theory.

Contribution

It introduces new control methods for the Sabra shell model, including optimal controls and feedback controllers that preserve physical invariants of the system.

Findings

Derived optimal controls using adjoint equations.

Established feedback controllers preserving invariants.

Represented feedback control as a multi-valued term in the normal cone.

Abstract

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.