Geometric Control Theory and Applications 

Introductory material manifolds, vector fields, tangent spaces; orbits of vector fields and Frobenius Theorem; controllablity and Chow Theorem; drift versus driftless systems, accessibility versus controllability

Bilinear control systems bilinear systems and matrix transition Lie groups; structure of matrix Lie groups (homogeneous spaces, transitivity, exponential map and canonical coordinates)  Lie algebras (Levi decomposition, semisimpicity, solvability, nilpotency, Cartan criteria); controllability properties for bilinear control systems on matrix Lie groups; series expansions of control systems; nonholonomic behavior in robotic systems: examples systems with of first order nonholonomic constraints (trailer systems, chained form); examples of control systems on Lie groups (rigid bodies on SO(3) and SE(3); system on a sphere)

Open-loop control methods for bilinear systems piecewise-constant controls feedback linearization system inversion averaging and oscillatory control

Application of bilinear systems: finite dimensional Schrodinger equation matrix transition Lie group of a Schordinger equation: SU(N); controllability on semisimple Lie algebras; control by piecewise constant inputs; functional series expansions in the controls: Magnus expansion and product of exponentials expansions;  application to elementary unitary gates synthesis for "quantum circuits"; example two-level system: qbit.