This course is a mathematical introduction to nonlinear control theory, a subject with roots in dynamical systems theory, mechanics, and differential geometry. The focus of this course is on the dynamical systems perspective. The material covered in this course finds application in fields as diverse as orbital mechanics and aerospace engineering, circuit theory, power systems, robotics, and mathematical biology, to name a few. The course is organized in four chapters, as follows:

1) Vector Fields and Dynamical Systems: Finite dimensional dynamical systems, vector fields, and their equivalence. Existence and uniqueness of solutions of ODEs. 2) Foundations of Dynamical Systems Theory: Invariant sets and their characterization by the Nagumo theorem. Limit sets as a tool to characterize the asymptotic behaviour of bounded orbits. Limit sets of two-dimensional systems: the Poincaré-Bendixson theorem. Poincaré theory of stability of closed orbits. Linearization of vector fields about equilibria. Linearization of vector fields about closed orbits. 3) Foundations of Stability Theory: Equilibrium stability and its characterization by means of Lyapunov’s theorem. Domain of attraction of an equilibrium. The Krasovskii-LaSalle invariance principle. Stability of LTI systems, and exponential stability of equilibria. Converse stability theorems. 4) Introduction to Nonlinear Stabilization: Control-Lyapunov functions. Parametrization of equilibrium stabilizers by CLFs (Artstein-Sontag theorem). Passive systems and passivity-based equilibrium stabilization. Passivity of mechanical control systems. Port-Hamiltonian systems.