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| | ==Theorem== |
| | In the system of [[differential equation]]s, |
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| | :<math>\dot x = Ax + X(x,y),\quad\dot y = Y(x,y)\ </math> |
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| | where, <math>x \in \mathbb{R}^m</math>, <math>y \in \mathbb{R}^n</math>, <math>A</math> in an ''m'' × ''m'' [[Matrix (mathematics)|matrix]], and ''X''(''x'', ''y''), ''Y''(''x'', ''y'') represent higher order nonlinear terms. If all [[eigenvalue]]s of the matrix <math>A</math> 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 |
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| | :<math>\lim_{t \to \infty}x(t) = 0,\quad \lim_{t \to \infty}y(t) = c.\ </math> |
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| | ==References== |
| | {{reflist}} |
| | |
| | {{DEFAULTSORT:Lyapunov-Malkin theorem}} |
| | [[Category:Theorems in dynamical systems]] |
| | [[Category:Stability theory]] |
Latest revision as of 09:39, 6 January 2014
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,
where, 
, 
, 
in an m × m matrix, and X(x, y), Y(x, y) represent higher order nonlinear terms. If all eigenvalues of the matrix 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
References
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- ↑ 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.