Backstepping design for a semi-discretized Ginzburg-Landau model of vortex shedding

We design a state feedback controller that achieves global asymptotic stabilization of a nonlinear Ginzburg-Landau model of vortex shedding from bluff bodies. Stabilization is obtained in two steps. First, the upstream and downstream parts of the system are shown to exhibit the ISS property with respect to certain boundary input terms governed by the core flow in the vicinity of the bluff body. Second, a finite difference approximation of arbitrary order of the core flow is stabilized using the backstepping method. Consequently, all the states in the core flow are driven to zero, including the boundary input terms of the upstream and downstream subsystems. The control design is valid for any Reynolds number. Numerical simulations are provided in order to demonstrate the performance of the controller, along with a potential of using low order discretizations for the control design, and thereby reducing the number of sensors needed for implementation.