The purpose of Optimal Control is to influence the behavior of a dynamical system in order to achieve a desired goal. Optimal control has a large variety of applications where the dynamics can be controlled optimally, such as aerospace, aeronautics, chemical plants, mechanical systems, finance and economics, but also to solve inverse problems where the goal is to determine input data in an equation from its solution values. An important application we will study in several settings is to determine the " data " in differential equations models using optimally controlled reconstructions of measured " solution " values. Inverse problems are typically harder to solve numerically than forward problems since they are often ill-posed (in contrast to forward problems), where ill-posed is the opposite of well-posed and a problem is defined to be well-posed if the following three properties holds (1) there is a solution, (2) the solution is unique, and (3) the solution depends continuously on the data. It is clear that a solution that does not depend continuously on its data is difficult to approximate accurately, since a tiny perturbation of the data (either as measurement error and/or as numerical approximation error) may 99

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最优控制和逆问题

最优控制的目的是影响动态系统的行为以实现预期目标。最优控制具有多种应用，其中可以对动力学进行最优控制，例如航空航天、航空、化工厂、机械系统、金融和经济，但也用于解决逆问题，其目标是确定方程中的输入数据其解值。我们将在几种设置中研究的一个重要应用是使用测量的“解”值的最佳控制重建来确定微分方程模型中的“数据”。逆问题通常比正向问题更难用数值解决，因为它们通常是不适定的（与正向问题相反），其中不适定与适定相反，如果以下三个性质成立，则问题被定义为适定问题：（1）存在解，（2）解是唯一的，以及（3）解取决于不断在数据上。很明显，不连续依赖其数据的解决方案很难准确近似，因为数据的微小扰动（作为测量误差和/或数值近似误差）可能 99