Polystability for a class of nonlinear systems with delay

This paper treats the polystability, which means the zero solution of a system has one kind of stability with respect to partial variables while the other variables have other kind of stability, problem for a class of nonlinear system with delay. Based on Lyapunov stability theory, cauchy matrix and some inequality techniques, one sufficient condition is proposed to realize the polystability for the zero solution of the delayed system. Under some assumptions upon the nonlinear terms of the nonlinear system with delay, the main result that the zero solution of the nonlinear system with delay is uniformly stable and exponentially stable with respect to partial variables is obtained. Finally, one numerical example is given to show the effectiveness of the method.