| Course
outline |
|
- Introduction
to nonlinear systems: an overview of peculiar behaviors of
nonlinear systems vs. linear systems: multiple isolated equilibrium
points, finite escape times, limit cycles.
|
- Mathematical
preliminaries: Quick review of normed vector spaces. Induced norms.
Gronwall-Bellman inequality. Lipschitz condition.
|
- Fundamental
properties: Local and global existence and uniqueness of
solutions. Continuity with respect to initial conditions. Finite escape
times.
|
- Phase
plane analysis and local behavior of solutions near equilibria: Behavior
of two-dimensional autonomous systems.
Classification of equilibrium points. The Hartman-Grossman theorem.
Periodic orbits. Bendixon's criterion. Index theory.
|
- Geometric
properties: The flow of a vector field. Invariant sets.
Asymptotic behavior of solutions: limit sets and the Poincare-Bendixon
theorem.
|
- Stability
theory for autonomous systems: Stability of an equilibrium
solution. Attractivity and uniform attractivity. Lyapunov stability.
Global asymptotic stability: Barbashin-Krasovskii theorem. La Salle's
invariance principle.
Stability of linear systems: Lyapunov matrix equation. Stability by
linearization.
|
- Stability
theory of non autonomous systems: Comparison functions. Uniform
stability. Lyapunov theorems. Stability of linear time varying systems.
Stability by linearization.
|
- Introduction
to systems with inputs: Input-to-state stability. BIBO
stability. Dissipative systems. Passive
systems. Stabilization of passive systems.
|
- Feedback
linearization: Local relative degree. Lie derivatives. Lie bracket
of vector fields. Invariant and involutive distributions.
Zero-dynamics. Linearization by feedback. Input/output linearization. Sufficient
conditions for feedback
stabilization.
|
