论文标题
巴纳克晶格和应用中的状态受限控制问题
State constrained control problems in Banach lattices and applications
论文作者
论文摘要
本文旨在研究无限尺寸空间中确定性最佳控制问题的家族。此类问题的特殊特征是存在阳性状态限制，这通常是在经济应用中引起的。为了处理这种约束，我们在带有Riesz空间结构（即Banach晶格）的Banach空间中设置了问题，不一定是反思的：一个典型的例子是紧凑型集合上连续功能的空间。在这种情况下，在这种情况下似乎是新的，我们能够找到与合适的辅助问题相关的Hamilton-Jacobi-Bellman（HJB）方程的明确解决方案，并编写相应的最佳反馈控制。得益于一种无限的尺寸perron-frobenius定理，我们使用这些结果来获取有关原始问题的最佳路径的信息。在此主题的早期作品中使用的无限尺寸设置中，这是不可能的，在该主题上，状态空间为$ \ mathrm l^2 $空间。
This paper aims to study a family of deterministic optimal control problems in infinite dimensional spaces. The peculiar feature of such problems is the presence of a positivity state constraint, which often arises in economic applications. To deal with such constraints, we set up the problem in a Banach space with a Riesz space structure (i.e., a Banach lattice) not necessarily reflexive: a typical example is the space of continuous functions on a compact set. In this setting, which seems to be new in this context, we are able to find explicit solutions to the Hamilton-Jacobi-Bellman (HJB) equation associated to a suitable auxiliary problem and to write the corresponding optimal feedback control. Thanks to a type of infinite dimensional Perron-Frobenius Theorem, we use these results to get information about the optimal paths of the original problem. This was not possible in the infinite dimensional setting used in earlier works on this subject, where the state space was an $\mathrm L^2$ space.
