Explore the application of the Koopman operator in simulating and controlling nonlinear dynamical systems using linear dynamical systems on observable subspaces. Delve into a 32-minute lecture by Steve Brunton, based on the paper "Koopman observable subspaces and finite linear representations of nonlinear dynamical systems for control." Gain insights into the Koopman operator overview, practical examples including the logistic map, and optimal control techniques. Understand the significance of this approach in simplifying complex nonlinear systems for more effective analysis and control. Access additional resources and related papers for a deeper understanding of the subject matter.