Supplementary material from "Hamiltonian control of socio-ecological multiplexes under Fokker–Planck dynamics"

Le résumé fourni par la source

Socio-ecological systems (SESs) couple human activities and ecological processes through networks of economic, social and environmental interactions. These systems are often treated as if their connectivity were given, yet the resources required to preserve network structure under uncertainty are rarely quantified. We develop a stochastic optimal control framework for connectivity-constrained SES on a multiplex network whose edge weights are Mahalanobis distances between node-level feature vectors. The law of these distances evolves under a controlled Fokker–Planck partial differential equation, and we formulate a Hamiltonian via an infinite-dimensional minimum principle. A Feynman–Kac representation links the adjoint (co-state) field to discounted shadow prices for marginal reductions in edge distances. Numerically, we instantiate the framework on a stylized U.S. corn-ethanol corridor modeled as a tri-layer socio-ecological multiplex and compute an open-loop stabilization policy. The optimal control is front-loaded: early interventions compress economic, social and environmental distances and keep the supra-Laplacian within a resilience band, after which the policy relaxes to a low-intensity maintenance regime. To improve numerical performance, we augment the base control with a recurrent neural-network residual that learns network-level corrections. The results show how these controllers can stabilize multiplex SES against shocks and reveal how policies respond to volatility parameters.

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