分数阶Birkhoff系统的变分积分子和最优控制

Birkhoff系统是Hamilton系统的一种自然推广，能够包含更多的动力学系统，应用到更广泛的领域。首先，本项目拟由离散分数阶Pfaff–Birkhoff变分原理，导出离散型分数阶Birkhoff方程和广义Birkhoff方程，进而构建基于变分积分子的分数阶Birkhoff系统的数值算法，并研究其稳定性、收敛性、保结构等问题。. 其次，提出分数阶Birkhoff模型的最优控制问题的一般表示，由分数阶变分原理导出相应的分数阶Birkhoff形式的控制方程，基于变分积分子构建求解该最优控制问题的数值方法，讨论其稳定性和收敛性，研究新模型的对称性与守恒量等问题。

The Birkhoffian system is a kind of natural generalization of Hamiltonian system. It can contain more dynamical systems and be applied to more extensive fields. Firstly, from the discrete fractional Pfaff–Birkhoff variational principle, the discrete fractional Birkhoffian equations and generalized Birkhoffian equations will be derived in this program. Further, based on the variational integrators, the numerical schemes will be constructed for the fractional Birkhoffian system. The stabilities, convergence, structure-preserving, etc., of the numerical methods will be discussed.. Secondly, the general formulation of optimal control problems for fractional Birkhoffian models will be developed. The fractional Birkhoff-type control equations will be derived from the fractional variational principle. The numerical schemes, which will be used to solve the optimal control program, will be constructed based on the variational integrators. Moreover, the stabilities and convergence of the numerical methods will be considered and the symmetries and conserved quantities of the new models will be investigated.