Lyapunov Function

A Lyapunov function is a scalar function  defined on a region  that is continuous, positive definite,  for all ), and has continuous first-order partial derivatives at every point of . The derivative of  with respect to the system , written as  is defined as the dot product

(1)

The existence of a Lyapunov function for which  on some region  containing the origin, guarantees the stability of the zero solution of , while the existence of a Lyapunov function for which  is negative definite on some region  containing the origin guarantees the asymptotical stability of the zero solution of .

For example, given the system

			(2)
			(3)

and the Lyapunov function , we obtain

(4)

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

REFERENCES:

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.