Exponential input-to-state stability of Runge-Kutta methods for neutral delay control systems

The aim of this paper is to find conditions under which the Runge-Kutta (RK) method reproduces the exponential input-to-state stability (exp-ISS) behavior of the nonlinear neutral delay control systems (NDCSs) without involving control Lyapunov function. A spectial continuous RK method is introduced which is equivalent to the descrete RK method for the exp-ISS. Under global Lipschitz condition, boundedness and an appropriate strong convergence are gotten. Under this strong convergent condition, it is shown that, for sufficiently small step-sizes, the exp-ISS of a NDCS holds if and only if that of the RK method is preserved.