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.
Good homogeneous functions: 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.