Transfer operator method for control in fluid flows

In this paper, we present a systematic approach inspired from the ergodic theory of dynamical system for the optimal control of complex fluid flows. The infinite dimensional Navier Stokes equation describing the complex fluid flow is first reduced to a finite set of coupled ordinary differential equations. We utilize Proper Orthogonal Decomposition techniques to obtain a reduced order model. The linear transfer, Perron Frobenius (P-F), operator-based Lyapunov measure framework is used for the nonlinear stability analysis and optimal control design of the reduced order system. Efficient numerical schemes that leverage the linearity of the framework have been developed for analysis and control design. The framework is utilized to analyze a benchmark problem in fluid dynamics: the control of recirculation in a two-dimensional channel flow with a backward facing step.