These notes are courtesy of Daniel Hagen. They should give you a fairly good idea of what I write on board during lectures. Needless to say, Dan's handwriting beats mine by orders of magnitude - great job, Dan!
Index
Course mechanics; Topics; Introduction to nonlinear systems
Examples of nonlinear systems; Equilibrium points; Linearization
Range of nonlinear phenomena (multiple isolated equilibrium points, finite escape time, limit cycles, chaos); Fold bifurcations
Transcritical and pitchfork bifurcations; Phase portraits of 2nd order linear systems
Phase portraits of nonlinear systems near hyperbolic equilibria (Hartman-Grobman); Bendixson's theorem
Bendixson's theorem (examples); Positively invariant sets; Poincare-Bendixson theorem
Poincare-Bendixson theorem; Hopf bifurcations
Center manifold theory; Invariant manifolds
Existence and uniqueness of solutions; Lipschitz continuity
Continuous dependence on initial conditions and parameters; Sensitivity equations
Stability of equilibrium points
Lyapunov functions; Positive definiteness; Radial unboundedness
Lyapunov functions (examples); LaSalle's invariance principle
Lyapunov functions for LTI systems; Algebraic Lyapunov equation
LaSalle's invariance principle for LTI systems; Stability via linearization
Discussion of HW3; Region of attraction; Comparison functions
Comparison functions (continued); Uniform stability; Exponential stability
Lyapunov based stability analysis of time-varying systems; Stability using comparison functions
Exponential stability; Differential Lyapunov equation for linear time-varying system
“Generalization” of LaSalle's invariance principle to linear time-varying systems (much weaker result that for time-invariant systems); Uniform observability; Estimation of constant unknown parameters
Convergence of gradient update law for estimation of constant unknown parameters
Convergence of gradient update law for estimation of constant unknown parameters (examples)
Example of model reference adaptive control (MRAC)
Introduction to integrator backstepping
Integrator backstepping
Control Lyapunov functions; Relative degree and zero dynamics
Feedback linearization; Input-output linearization; Zero dynamics
Normal form
Input-to-State Stability
Notes on “Proximal Augmented Lagrangian Method for Non-Smooth Composite Optimization” slides