Explicit integral operator feedback for local stabilization of nonlinear thermal convection loop PDEs

A state feedback boundary control law that stabilizes fluid flow in a 2D thermal convection loop is presented. The fluid is enclosed between two cylinders, heated from above and cooled from below, which makes its motion unstable for a large enough Rayleigh number. The actuation is at the boundary through rotation (direct velocity actuation) and heat flux (heating or cooling) of the outer boundary. The design is a new approach for this kind of a coupled PDE problem, based on a combination of singular perturbation theory and the backstepping method for infinite dimensional linear systems. Stability is proved by Lyapunov method. Though only a linearized version of the plant is considered in the design, an extensive closed loop simulation study of the nonlinear PDE model shows that the result holds for reasonably large initial conditions. A highly accurate approximation to the control law is found in closed form.