巴拿赫格上的状态约束控制问题及其应用
State constrained control problems in Banach lattices and applications
摘要：
本文研究无穷维空间中一类确定性最优控制问题。这些问题的特点是存在一个积极的状态约束，这往往会出现在经济应用中。为了处理这样的约束，我们将问题建立在具有Riesz空间结构的巴拿赫空间中（即，一个典型的例子是紧集上的连续函数空间。在这种情况下，这似乎是新的，在这种情况下，我们能够找到显式的解决方案的Hamilton-Jacobi-Bellman（HJB）方程相关联的一个合适的辅助问题，并编写相应的最佳反馈控制。由于一类无穷维的Perron-Frobenius定理，我们使用这些结果来获得有关原问题的最优路径的信息。这是不可能的，在无限维的设置中使用的早期作品在这个问题上，其中的状态空间是一个$\mathrm L^2$空间。
英文摘要：
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.
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