Lyapunov’s direct method, cont’d
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.
Discrete time systems.
Bounded Input Bounded Output
GES implies BIBO; not vice versa.