Lyapunov Function

A Lyapunov function is a scalar function V(y) defined on a region D that is continuous, positive definite, V(y)>0 for all y!=0), and has continuous first-order partial derivatives at every point of D. The derivative of V with respect to the system y^'=f(y), written as V^*(y) is defined as the dot product

 V^*(y)=del V(y)·f(y).

The existence of a Lyapunov function for which V^*(y)<=0 on some region D containing the origin, guarantees the stability of the zero solution of y^'=f(y), while the existence of a Lyapunov function for which V^*(y) is negative definite on some region D containing the origin guarantees the asymptotical stability of the zero solution of y^'=f(y).

For example, given the system


and the Lyapunov function V(y,z)=(y^2+z^2)/2, we obtain


which is nonincreasing on every region containing the origin, and thus the zero solution is stable.

See also

Linear Stability, Nonlinear Stability

This entry contributed by Martin Keller-Ressel

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Boyce, W. E. and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 5th ed. New York: Wiley, pp. 502-512, 1992.Brauer, F. and Nohel, J. A. The Qualitative Theory of Ordinary Differential Equations: An Introduction. New York: Dover, 1989.Hahn, W. Theory and Application of Liapunov's Direct Method. Englewood Cliffs, NJ: Prentice-Hall, 1963.Jordan, D. W. and Smith, P. Nonlinear Ordinary Differential Equations. Oxford, England: Clarendon Press, p. 283, 1977.Kalman, R. E. and Bertram, J. E. "Control System Analysis and Design Via the 'Second Method' of Liapunov, I. Continuous-Time Systems." J. Basic Energ. Trans. ASME 82, 371-393, 1960.Oguztöreli, M. N.; Lakshmikantham, V.; and Leela, S. "An Algorithm for the Construction of Liapunov Functions." Nonlinear Anal. 5, 1195-1212, 1981.Zwillinger, D. "Liapunov Functions." §120 in Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, pp. 429-432, 1997.

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Lyapunov Function

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Keller-Ressel, Martin. "Lyapunov Function." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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