Discrete gradient integration method for generalized Hamilton systems

For generalized Hamilton system and generalized Hamilton control system, by Hamilton function based on energy and the discrete gradient method, the numerical solution method that can preserve Hamilton function properties of Hamilton system is given. Firstly, the generalized Hamilton system models and the energy change rates based on energy concept of the dynamic systems are established, then based on the definition of discrete gradient, the discrete gradient integration method is proposed and the proof of the integration algorithm precision is given. It is proved that the integration method can effectively preserve the rate of Hamilton function with respect to time. The numerical example shows that the method is effective. The paper will provide a new way for the further studies of Hamilton dynamic system.