We consider an optimal control problem with dynamics that switch between several subsystems of nonlinear differential equations. Each subsystem is assumed to satisfy a linear growth condition. Furthermore, each subsystem switch is accompanied by an instantaneous change in the state. These instantaneous changes-called "state jumps"-are influenced by a set of control parameters that, together with the subsystem switching times, are decision variables to be selected optimally. We show that an approximate solution for this optimal control problem can be computed by solving a sequence of conventional dynamic optimization problems. Existing optimization techniques can be used to solve each problem in this sequence. A convergence result is also given to justify this approach.