Abstract:

This paper develops the mean-square exponential input-to-state stability (exp-ISS) of the Euler-Maruyama (EM) method for stochastic delay control systems (SDCSs). The definition of mean-square exp-ISS of numerical methods is established. The conditions of the exact and EM method for an SDCS with the property of mean-square exp-ISS are obtained without involving control Lyapunov functions or functional. Under the global Lipschitz coefficients and mean-square continuous measurable inputs, it is proved that the mean-square exp-ISS of an SDCS holds if and only if that of the EM method is preserved for a sufficiently small step size. The proposed results are evaluated by using numerical experiments to show their effectiveness.