Runge-Kutta methods for affinely controlled nonlinear systems

Rooted tree analysis is adapted from stochastic differential equations to derive systematically general Runge-Kutta methods for deterministic affinely controlled nonlinear systems. Order conditions are found and some specific coefficients for second- and third-order methods are determined, which are then used for simulations compared with the Taylor methods for affinely controlled nonlinear systems derived by Grüne and Kloeden.