Abstract:An research has been made of the Noether theorem for fractional order optimal control problems of generalized Birkhoff systems. Firstly, based on Caputo fractional derivative, fractional optimal control problems have been proposed for generalized Birkhoff systems, thus obtaining the extremum conditions for fractional optimal control problems of generalized Birkhoff systems by using fractional calculus and fractional variations. Then, an investigation is made of the invariance of functionals under the action of infinitesimal transformation groups by adopting fractional calculus and variational methods, thus providing the Noether theorem for fractional optimal control problems under time invariant and time variation conditions, respectively. Finally, application examples are given to verify the effectiveness of the proposed method.