Editing Lyapunov stability (section)

==Stability for linear state space models==
A linear [[State space (controls)|state space]] model

:<math>\dot{\textbf{x}} = A\textbf{x}</math>,

where <math> A</math> is a finite matrix, is asymptotically stable (in fact, [[Exponential stability|exponentially stable]]) if all real parts of the [[eigenvalue]]s of <math> A</math> are negative.  This condition is equivalent to the following one:<ref>{{cite journal |last1=Goh |first1=B. S. |title=Global stability in many-species systems |journal=The American Naturalist |date=1977 |volume=111 |issue=977 |pages=135–143 |doi=10.1086/283144 |bibcode=1977ANat..111..135G |s2cid=84826590 }}</ref>

:<math>A^\textsf{T}M + MA</math>

is negative definite for some [[Positive-definite matrix|positive definite]] matrix <math>M = M^\textsf{T}</math>. (The relevant Lyapunov function is <math>V(x) = x^\textsf{T}Mx</math>.)

Correspondingly, a time-discrete linear [[State space (controls)|state space]] model

:<math>\textbf{x}_{t+1} = A\textbf{x}_t</math>

is asymptotically stable (in fact, exponentially stable) if all the eigenvalues of <math> A</math> have a [[Absolute value|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 <math>\{A_1, \dots, A_m\}</math>)

:<math>{\textbf{x}_{t+1}} = A_{i_t}\textbf{x}_t,\quad A_{i_t} \in \{A_1, \dots, A_m\}</math>

is asymptotically stable (in fact, exponentially stable) if the [[joint spectral radius]] of the set <math>\{A_1, \dots, A_m\}</math> is smaller than one.