\(\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.

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