Proportional control of oscillatory thermocapillary convection in a toy model

Simple model equations were formulated to examine the influence of linear feedback control on oscillations in thermocapillary convection. Limiting the solutions to have a few wavenumbers and roughly assuming the other spatial profiles, the system of equations is reduced to a set of ordinary differential equations. This toy model is able to recreate basic features of the uncontrolled system such as standing/traveling wave structures and bifurcation characteristics. The control was realized by locally heating and cooling the free surface by linearly feeding back the temperature signals measured in local positions. Implementing the proportional feedback control in this toy model, we could capture some of the essential qualitative features of the influence of the control observed in the previous experiments, including the limitation of the control. The formulation of the toy model can be used to gain physical insight in the control problem.