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

TL;DR

This paper addresses stabilization of the sabra shell model of turbulence using finite-dimensional controllers and $H^ Infty$ control techniques, demonstrating asymptotic stabilization of the nonlinear system.

Contribution

It introduces an $H^ Infty$ stabilization approach for the sabra shell model, including controller design via Riccati equations and analysis of nonlinear system stabilization.

Findings

Proved $H^ Infty$ stabilization of the linearized system.

Characterized the feedback operator through algebraic Riccati equation.

Showed the control asymptotically stabilizes the nonlinear system.

Abstract

Shell models of turbulence are representation of turbulence equations in Fourier domain. Various shell models 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 \cite{PBT}. In this paper we have addressed stabilization problems related to sabra shell model of turbulence. We have studied internal stabilization via finite dimensional controller. Moreover we have also studied optimal robust control problem by solving an infinite time horizon max-min control problem. We first prove the $H^ \infty$ stabilization of the linearized system and charatarize it in terms of a feedback operator by solving an algebric ricatti equation. Finally we show that the control will asymptotically stabilize the nonlinear system.