Abstract
Shell models of turbulence are representation of turbulence equations in
Fourier domain. Various shell models are studied for their mathematical
relevance and the numerical simulations which exhibit at most
resemblance with turbulent flows. One of the mathematically well studied
shell model of turbulence is called sabra shell model. This work
concerns with two important issues related to shell model namely
feedback stabilization and robust stabilization. We first address
stabilization problem related to sabra shell model of turbulence and
prove that the system can be stabilized via finite dimensional
controller. Thus only finitely many modes of the shell model would
suffice to stabilize the system. Later we study robust stabilization in
the presence of the unknown disturbance and corresponding control
problem by solving an infinite time horizon max-min control problem. We
first prove the $H^ \infty$ stabilization of the
associated linearized system and characterize the optimal control in
terms of a feedback operator by solving an algebraic riccati equation.
Using the same riccati operator we establish asymptotic stability of the
nonlinear system.