Cost cumulant control has been gaining momentum in recent years and has proven to be competitive with other controllers when applied to various structural control problems. These control methods have generalized the well-known LQG control. Similarly, in this paper cost cumulants are used to generalize the $$H_2/H_{\infty}$$ control problem. Initially the development will be done for a class of nonlinear systems. The treatment will then consider the case when the system is linear and the costs are quadratic. The multi-cumulant, multiobjective control strategy, as well as the worst case disturbance, will be found through a Nash game approach. Uncertainties in the stiffness and damping coefficients will be taken into account and this control strategy will be applied to a four story building subject to a seismic disturbance. Also, a connection of the building problem to circuits will be given.