Hybrid optimal control for ecological-economic problems with regime shifts

• Applicant: Privatdozent Dr. Thorsten Upmann
• Subject Area: Economic Theory ; Mathematics
• Term: since 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 448981368

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.

• DFG Programme: Research Grants
• International Connection: Russia
• Partner Organisation: Russian Foundation for Basic Research, until 3/2022
• Cooperation Partner: Dmitry Gromov