Lyapunov–Malkin theorem

The Lyapunov–Malkin theorem (named for Aleksandr Lyapunov and Ioel Gilevich Malkin) is a mathematical theorem detailing nonlinear stability of systems.^[1]

Theorem

In the system of differential equations,

$\dot x = Ax + X(x,y),\quad\dot y = Y(x,y)\$

where, $x \in \mathbb{R}^m$, $y \in \mathbb{R}^n$, $A$ in an m × m matrix, and X(x, y), Y(x, y) represent higher order nonlinear terms. If all eigenvalues of the matrix $A$ have negative real parts, and X(x, y), Y(x, y) vanish when x = 0, then the solution x = 0, y = 0 of this system is stable with respect to (x, y) and asymptotically stable with respect to  x. If a solution (x(t), y(t)) is close enough to the solution x = 0, y = 0, then

$\lim_{t \to \infty}x(t) = 0,\quad \lim_{t \to \infty}y(t) = c.\$

References

1. Zenkov, D.V., Bloch, A.M., & Marsden, J.E. (1999). "Lyapunov–Malkin Theorem and Stabilization of the Unicycle Rider." . Retrieved on 2009-10-18.