Consider system

𝐱˙=A𝐱\dot{\mathbf{x}} = A\mathbf{x}

Solution to system is

𝐱(t)=eAt𝐱(0)\mathbf{x}(t) = e^{At}\mathbf{x}(0)

If eigenvalues of AA all lie in open left half-plane, 𝐱(t)\mathbf{x}(t) asymptomatically approaches origin as time t→∞t \to \infty, i.e. asymptomatically stable.

If eigenvalues of AA lie in the closed left half-plane, (some eigenvalues possibly on imaginary axis),

References

  1. https://crrl.poly.edu/6253/lectures/lect4.pdf