# Lyapunov–Malkin theorem

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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,

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

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

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

## References

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