New Liapunov Function for Nonlinear Time-Varying Systems

A new method of generating Liapunov functions is described that is useful for time-varying nonlinear differential equations. Constant coefficient quadratic forms are often used as Liapunov functions for linear constant coefficient differential equations. However, if the differential equation coefficients are time-varying and nonlinear, better results are usually obtained by using variable quadratic forms as Liapunov functions. This variation in V often introduces undesirable terms in V˙, which are cancelled by modifying V by subtracting integrals of certain partial derivatives of V with respect to the dependent variables. With few restrictions, V is proved to remain positive definite after the modification. The method often directly extends a Liapunov function useful for constant coefficient differential equations to cover the case when the coefficients are time-varying and nonlinear. Two examples are presented, including the incremental circuit for a time-varying nonlinear transmission line with hysteresis and the equations for an N-body collision avoidance problem.