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[edit]

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(xy), Y(xy) represent higher order nonlinear terms. If all eigenvalues of the matrix 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

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

References[edit]

  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.