Birkhoff系统的离散最优控制理论及其应用

本项目主要基于Birkhoff系统的保结构算法研究Birkhoff系统的离散最优控制理论，进而实现更加精确有效地求解最优控制问题。通常，由于非线性，最优控制问题往往依赖于数值求解而非解析求解。直接法作为数值求解方法之一，主要是通过利用传统差分格式直接离散目标泛函和受控运动方程，将最优控制问题转化为一有限维的非线性规划问题进行求解。借鉴直接法的求解思想，本项目拟采用更加精确有效的Birkhoff系统的保结构算法近似最优控制问题中的受控运动方程，通过诱导更加真实的非线性规划问题来获得更加精确的离散最优控制。除了构建上述Birkhoff系统的离散最优控制框架，本项目进一步致力于突破现有数值求解最优控制问题方法的局限性，即局部最优性，通过将所发展的离散最优控制方法与路径规划技巧相结合，有效解决求解非线性规划问题时所涉及的初值选取问题，进而确定出一个数值求解最优控制问题全局最优解的可行方案。

The project mainly studies the discrete optimal control theory for Birkhoffian systems to solve optimal control problems more accurately and efficiently, based on structure preserving algorithms. Usually, optimal control problems rely on numerical rather than analytical solving methods, due to their nonlinearity. The direct method, one of the numerical solving methods, is mainly to transform the optimal control problem into a nonlinear optimization problem with finite dimensions, via directly discretizing the objective functional and the forced dynamical equations according to traditional difference schemes. Drawing on the idea of the direct method, this project employs structure preserving algorithms for Birkhoffian systems, which are more accurate and efficient than classical difference schemes, to approximate the forced dynamical equations. This way can give more faithful optimization problems and consequently yield more accurate discrete optimal control. Besides constructing the discrete optimal control framework for Birkhoffian systems, the project ulteriorly attempts to overcome the major limitation of existing numerical solving methods for optimal control problems, namely the restriction to local optimality. By utilizing the presented discrete optimal control method and combining it with the motion planning technique, the project seeks to efficiently solve the initial guess problem for nonlinear optimization, and give a feasible strategy for finding the global optimal solution of optimal control problems.