A linear state space model

where formula_48 is a finite matrix, is asymptotically stable (in fact, exponentially stable) if all real parts of the eigenvalues of formula_48 are negative. This condition is equivalent to the following one:

is negative definite for some positive definite matrix formula_51. (The relevant Lyapunov function is formula_52.)

Correspondingly, a time-discrete linear state space model

is asymptotically stable (in fact, exponentially stable) if all the eigenvalues of formula_48 have a modulus smaller than one.

This latter condition has been generalized to switched systems: a linear switched discrete time system (ruled by a set of matrices
formula_55)

is asymptotically stable (in fact, exponentially stable) if the joint spectral radius of the set formula_55 is smaller than one.

Stability for linear state space models
Stability for systems with inputs
ଇତିହାସ
Definition for discrete-time systems