Stabilization of two-dimensional Rayleigh–Bénard convection by means of optimal feedback control

We consider the problem of suppressing the Rayleigh–Bénard convection in a finite domain by adjusting the heat flux profile at the bottom of the system while keeping the heat input the same. The appropriate profile of heat flux at the bottom is determined by the optimal feedback control. When most of the convection modes are taken into consideration in the construction of the feedback controller, the suppressed state is found to be stable for all range of Rayleigh number investigated. With the feedback controller constructed by employing only the dominant convection modes, however, there exists a threshold Rayleigh number beyond which a new convective state emerges due to the hydrodynamic instability. The threshold Rayleigh number is found to increase with the number of modes taken in the feedback controller.