A Velocity Functional for an Analysis of Stability in Delay-Differential Equations

Abstract

The author considers the linear delay-differential system (*)$\dot x(t) = A_0 x(t) + \sum\nolimits_1^p {A_i } x(t - h_i )$. It is shown that there is a velocity functional which along with its Lie derivative is analogous in the theory of delay-differential operators to the velocity Lyapunov function $V(x) = \langle {\langle {Ax,Ax} \rangle } \rangle $, which along with its Lie derivative $x^T (A + A^T )x$ is used in the analysis of the ordinary differential equation $\dot x(t) = Ax(t)$. The Lie derivative can be written as $2\langle {(A_0 + \sum\nolimits_1^p {\sigma _i } A_i )\phi ,\phi } \rangle $ in a suitable inner product $\langle , \rangle $ for $C^1 $ vector functions $\phi $ given over $[ - \eta ,0]$, where the $\sigma _i $ signify delay operators having length $h_i $ and $\eta = \max (h_1 , \ldots ,h_p )$. Next considering the nonlinear delay-differential equation (†)$\dot x(t) = f(x(t),x(t - h_1 ), \ldots ,x(t - h_p ))$, the author gives a velocity functional having Lie derivative which locally resembles that for the linear system (*). It is proven that if the linear operator associated with this Lie derivative everywhere satisfies a certain stability property, then the nonlinear system (†) will be globally contractive to a unique equilibrium.