\(\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}}\)

#### Lyapunov’s direct method, cont’d

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.

#### Quadratic forms

Coordinate forms. Representation as a mapping \(V\to V^*\). Euclidean norm.

Coordinate system yields a quadratic form (the one where the unit vectors are orthonormal). Two bases produce the same quadratic form iff they are related by an orthogonal transformation.

Any quadratic form + Euclidean form can be turned into an endomorphism, and one can talk about eigenvectors.

Theorem: Each quadratic form has real eigenvalues, and an orthonormal basis of eigenvalues.

Signatures. Positive, etc. definite forms.

For complex spaces quadratic forms are not really suitable (if one wants just a real number as a result, the signatures are all \((n,n)\)

Sylvester criterion for positive definiteness.

(Bonus: Rayleigh characterization of eigenvalues.)

#### Interlude: complex eigenvalues and eigenvectors.

Jordan normal forms in real vector spaces.

#### Quadratic forms as Lyapunov functions

If the operator defines an asymptotically stable system, it is Hurwitz. For a Hurwitz operator, a quadratic form exists which is a (strict) Lyapunov function. A strict Lyapunov function implies AS.

Lyapunov equation.

BIBO

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