Explore the stabilization of nonlinear systems using oscillating controls in this comprehensive lecture by Alexander ZUYEV from MPI, Magdeburg, Germany. Delve into the challenges of asymptotic stability and controllability, and discover why trigonometric polynomials are crucial in this context. Examine bracket generating systems, Lie bracket extensions, and control design schemes through examples like Brockett's problem and unicycle systems. Investigate systems with drift, including rotating rigid bodies and hydrodynamical models, and learn about Lie brackets and energy cascades. Conclude with an analysis of stabilization in Galerkin systems, gaining valuable insights into nonholonomic and fluid dynamics applications.