# september 23

$$\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}}$$

#### Global asymptotic stability for LTI.

Equivalent to local stability.

Hurwitz operators. Multiple eigenvalues and Jordan normal forms.

#### Lyapunov’s direct method

Positive definite functions.

Lyapunov functions, definition.

Lyapunov’s theorem.

Convenient Lyapunov functions: homogeneous ones.

Strict homogeneous Lyapunov function for a linear system remains one after a perturbation of the system.

Examples.

Good homogeneous functions: quadratic forms.