Hybrid optimal control for ecological-economic problems with regime shifts

生态系统为自然和人类提供的生态服务可能容易受到突然和重大变化的影响。 为了防止或减轻这些服务的损害，需要确定和实施适当的生态保护和养护政策。 在规划这些保护和经营政策时，人们通常会诉诸于特定标准的最佳策略. 然而，对于这样的动态系统，跨时最优政策措施的选择是相当复杂的。 关于这些问题的文献包括一些结果，旨在为特定的，往往是相当stylisedproblems设计最优控制策略。 然而，在解决更复杂但实际上相关的问题方面，几乎没有做过什么工作。 具体而言，在多个潜在状态转移的情况下，生态环境系统的最优控制的研究很少;对于空间分布的、迁移（或扩散）的物种，面临着潜在的生存条件恶化，即状态转移，也是如此。与现有的结果相比，本文讨论了多个状态转移的可能性，如外生与~内生风险或可逆风险与~不可逆转的制度变迁-预计将大大影响最佳政策，从而可能导致精心设计的治理的新景象。 然而，在存在多个可能的政权转移的情况下解决控制或治理问题需要更强大的分析工具;对于空间最优控制问题也是如此，即使只有一个潜在的政权转移。 幸运的是，混合（最优）控制理论提供了一个自然的框架来处理动态发生突变的系统;该理论专门考虑由于内生或外生事件的发生而动态变化的系统。 然而，混合控制理论的研究范围大多局限于相当简单的“玩具”问题-但还没有认真的尝试将这一理论应用于解决生态经济问题，因此，本工作的目标是将混合控制理论的方法应用和扩展到一类广泛的生态经济问题与政权转移。 通过这种方式，我们的分析将有助于解决跨期的生态和环境问题的突变（变化，甚至灾难）的存在，影响物种的丰度或其动态的增长。 此外，我们还探讨了“紧缩”政策的可能性和条件，即，将系统保持在两种制度之间的边界上的政策以及类似于机器学习的动态可能会作为最佳解决方案出现，而不是作为建模假设本身。

Ecological systems are potentially vulnerable to abrupt and significant changes in theirecological services that they provide to nature and mankind. In order to prevent ormitigate impairments of these services suitable policies for ecological protection andconservation need to be identified and implemented. When planning those protective andoperational policies one typically resorts to strategies optimal with respect to specificcriteria. However, the selection of intertemporally optimal policy measures for suchdynamic systems is quite involved.Ecological systems that are subject to abrupt changes may be modelled as dynamical systems with regime shifts. The literature on those problems consists of a number of results aimed at designing optimal control strategies for specific and often rather stylisedproblems. Little work, though, has been done on solving more complex, but actuallyrelevant, problems. Specifically, there is only few work on the optimal control of ecological and environmental systems in the presence of multiple potential regime shifts; and similar is true for spatially distributed, migrating (or dispersing) species facing a potential deterioration in their living conditions, viz a regime shift.As compared with the existing results, the possibility of \emph{multiple} regimeshifts---such as exogenous vs.~endogenous risk or reversible vs.~irreversible regimeshifts---is expected to significantly affect optimal policies, and may thus lead to newinsights for well-designed governance. Yet, solving control or governance problems in thepresence of multiple possible regime shifts requires more powerful mathematicalinstruments; and the same is true for spatial optimal control problems even with a singlepotential regime shift. Fortunately, the theory of hybrid (optimal) control offers anatural framework to deal with systems whose dynamics undergo abrupt changes; this theory specifically considers systems whose dynamics change due to the occurrence of endogenous or exogenous events. However, the scope of hybrid control theory has been mostly restricted to rather simple "toy'' problems---but no serious attempt has yet been made to employ this theory to solve ecological-economic problems.The goal of this work is therefore to apply and extend the methods of hybrid controltheory to a wide class of ecological--economic problems with regime shifts. In this way,our analysis will help solve intertemporal ecological and environmental problems in thepresence of abrupt shifts (changes or even catastrophes) that affect either the abundances of the species or their dynamics of growth. Also, we explore the possibilities of and the conditions under which "tightrope'' policies---i.e., policies that keep the system on the boundary between two regimes---as well as hysteresis-like dynamics may emerge as an optimal solution rather than as a modelling assumption per se.