Lyapunov–Malkin theorem

The Lyapunov–Malkin theorem (named for Aleksandr Lyapunov and Ioel 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

↑ Zenkov, D. V.; Bloch, A. M.; Marsden, J. E. (2002). "Lyapunov–Malkin Theorem and Stabilization of the Unicycle Rider". Systems and Control Letters. 45 (4): 293–302. doi:10.1016/S0167-6911(01)00187-6.

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[Zenkov-1] Zenkov, D. V.; Bloch, A. M.; Marsden, J. E. (2002). "Lyapunov–Malkin Theorem and Stabilization of the Unicycle Rider". Systems and Control Letters. 45 (4): 293–302. doi:10.1016/S0167-6911(01)00187-6.