Mathematical Sciences: Optimality Conditions and Algorithm Covergence Behavior for Optimal Control Problems

最优控制问题通常被自然地表述为无限维函数空间中特殊结构的数学程序。在这种情况下，研究极限无限维程序的算法可以预测在近似有限维空间中实际实现的标准计算方案的行为，并且还可以提出在有限维数学程序的标准方法中没有对应的新算法。特别是，最近的研究表明，标准非线性规划算法最适合于控制输入向量中哈密顿量为一致凸的最优控制问题，而其他最优控制问题可能需要基于固有无限维庞特里亚金必要条件的强变分方法。提出的研究有以下直接目标：(i)建立牛顿投影方法在控制函数的Chebychev范数邻域和根均方邻域的收敛性，这些邻域满足最近发展的非凸非二次约束输入调节器问题的局部最优性充分条件；（ii）将(i)中的分析扩展到混合算法，该算法采用牛顿投影和拉格朗日增广技术来解决具有控制变量和状态变量约束的调节器问题；（iii）对于控制输入向量中具有非凸哈密顿量的最优控制问题，在满足尖锐局部最优性的根均方范数的根均方域内，建立了强变分方法的收敛性；（iv）证实(i)- (iii)中无限维收敛分析的预测，用于固定网格和嵌套网格的近似有限维计算。最优控制问题出现在飞机和航天器轨道计算、核和化学反应堆控制、结构和气动设计、生态系统管理以及许多其他应用中。在一般的最优控制问题中，在不同的时间和空间点上选择大量的控制变量（力、电压、温度等），使某些物理系统进入期望的状态，并以尽可能低的成本（操纵时间、燃料或功率消耗等）完成这一任务。控制和状态变量值通常也受到硬件或安全考虑的约束，状态转换通常由复杂的常微分方程或偏微分方程系统控制。因此，相关的优化任务是苛刻的，并且要求算法在计算其连续逼近时有效地利用控制问题结构，并且对所讨论的问题表现出良好的全局和局部收敛特性。从广义上讲，所提出的研究的目标是了解在最优控制的背景下，哪种算法类型对给定的问题结构最有效。＊＊＊

9500908 Dunn Optimal control problems are often naturally formulated as specially structured mathematical programs in infinite-dimensional function spaces. In such cases, a study of algorithms for the limiting infinite-dimensional programs can predict the behavior of standard computational schemes that are actually implemented in approximating finite-dimensional spaces, and can also suggest new algorithms that have no counterparts in the standard methodology for finite-dimensional mathematical programs. In particular, recent studies have shown that standard nonlinear programming algorithms are best suited to optimal control problems with Hamiltonians that are uniformly convex in the control input vector, while other optimal control problems may require strong variation methods based on the inherently infinite-dimensional necessary condition of Pontryagin. The proposed investigation has the following immediate objectives: (i) to establish the convergence properties of Newtonian projection methods in Chebychev norm neighborhoods and root-mean-square neighborhoods of control functions that satisfy recently developed local optimality sufficient conditions for nonconvex nonquadratic constrained input regulator problems; (ii) to extend the analysis in (i) to hybrid algorithms that employ Newtonian projection and Lagrangian augmentation techniques for regulator problems with control variable and state variable constraints; (iii) to establish the convergence properties of strong variation methods in root-mean-square neighborhoods of control functions satisfying sharp local optimality sufficient conditions in the root-mean-square norm for optimal control problems with Hamiltonians that are not convex in the control input vector; (iv) to corroborate the predictions of the infinite dimensional convergence analyses in (i)--(iii) for approximate finite dimensional computations on fixed and nested grids. Optimal control problems arise in aircraft and spacecraft trajectory calcula tions, nuclear and chemical reactor control, structural and aerodynamic design, management of ecological systems, and many other applications. In the generic optimal control problem, a large number of control variables (forces, voltages, temperatures, etc.) are chosen at various points in time and space to bring some physical system into a desired state, and to accomplish this at the lowest possible cost (manuever duration, fuel or power consumption, etc.). Control and state variable values are also typically constrained by hardware or safety considerations, and state transitions are often governed by complicated systems of ordinary or partial differential equations. The associated optimization task is therefore demanding, and requires algorithms that effectively exploit control problem structure in the computation of their successive approximations, and also exhibit good global and local convergence characteristics for the problems in question. In broad terms, the goal of the proposed investigation is to understand which algorithm types are most effective for a given problem structure in the context of optimal control. ***