Discrete Approximations in Variational Problems

9704912这一建议涉及到无限维变分问题的数值分析，例如涉及常微分方程组、微分包含和最优控制问题的变分不等式。利用最近的稳定性结果，将得到状态约束最优控制问题的离散近似的误差估计。对于具有控制和状态约束的线性/二次问题，将开发有效的算法来计算解。此外，稳定性理论将被用来对计算解中的误差产生先验和后验估计。对于微分包含，离散化的发展将具有足够的一般性，以涵盖具有不连续右侧和状态约束的问题。本提案中将开发的算法和理论适用于工业中的各种问题，如与机器人的最佳性能、航空航天的飞行控制、具有所需性能的薄膜设计或化学过程的优化工程相关的问题。当这些问题中的任何一个在计算机上被解决时，连续的过程被分解成微小的时间步长。这项建议研究了以这种方式离散这一连续过程时所造成的误差。

9704912 Hager This proposal concerns the numerical analysis of infinite dimensional variational problems such as variational inequalities involving ordinary differential equations, differential inclusions, and optimal control problems. Using recent stability results, errors estimates will be derived for discrete approximations of state constrained optimal control problems. For linear/quadratic problems with control and state constraints, efficient algorithms will be developed for computing a solution. In addition, the stability theory will be applied to generate both a priori and a posteriori estimates for the errors in the computed solutions. For differential inclusions, discretizations will be developed with sufficient generality to encompass problems with discontinuous right sides and state constraints. The algorithms and theory that will be developed in this proposal are applicable to a variety of problems in industry such as those associated with the optimal performance of robots, flight control in aerospace, the design of thin films with desired properties, or the optimal engineering of chemical processes. When any of these problems is solved on a computer, the continuous process is decomposed into tiny time steps. This proposal studies the errors that are caused when this continuous process is discretized in this way.