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

9500908邓恩最优控制问题通常被自然地表示为无限维函数空间中的特殊结构的数学规划。在这种情况下，对极限无限维程序的算法的研究可以预测在逼近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，标准的非线性规划算法最适合于控制输入向量一致凸的哈密顿量的最优控制问题，而其他最优控制问题可能需要基于Pontryagin内在无限维必要条件的强变分方法。所提出的研究有以下直接目的：(I)建立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的控制函数的Chebychev范数邻域和均方根邻域上的牛顿投影方法的收敛性质；(Ii)将文[1]中的分析推广到使用牛顿投影和拉格朗日增广技术的混合算法，以求解具有控制变量和状态变量约束的调节器问题；(3)建立了控制函数均方根邻域内强变分方法的收敛性质；(4)证实了(1)-(3)中关于固定网格和嵌套网格上的近似有限维计算的无穷维收敛分析的预言。最优控制问题出现在飞机和航天器轨迹计算、核反应堆和化学反应堆控制、结构和空气动力学设计、生态系统管理和许多其他应用中。在一般的最优控制问题中，存在大量的控制变量(力、电压、温度等)。在不同的时间点和空间点选择，以使某些物理系统进入所需状态，并以尽可能低的成本(人工持续时间、燃料或功率消耗等)实现这一点。控制和状态变量值通常还受到硬件或安全考虑的约束，并且状态转换通常由复杂的常微分方程组或偏微分方程组控制。因此，相关的优化任务要求很高，并要求算法在计算其逐次逼近时有效地利用控制问题的结构，并且对所讨论的问题表现出良好的全局和局部收敛特性。总而言之，拟议调查的目标是了解在最优控制的背景下，对于给定的问题结构，哪些算法类型是最有效的。***

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. ***