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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.

Theorem

In the system of differential equations,

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

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

$\underset{t\to \mathrm{\infty }}{\lim }x(t)=0,\underset{t\to \mathrm{\infty }}{\lim }y(t)=c.$

References

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