Solving viability problems in dynamic games using individual strategies derived from guaranteed viability kernels: Application to an agricultural cooperative model

In this paper we address decentralized control problems in the context of sustainable management of discrete-time deterministic dynamical systems. We consider $N$ players, each with their own controlled dynamical system, which are linked through the dynamics of a global variable. We consider sustainable constraints as constraints in the state space. From this general problem defined in both viability theory and game theory, we define $N$ viability sub-problems. For each of these, we consider the guaranteed viability kernel, subset of the constraint set where it is possible to maintain the subsystem whatever the strategy of other players. When these kernels are not empty, we propose a general algorithm to compute individual strategies that ensure the sustainability of all. Numerical experiments on a model of agricultural cooperative show the feasibility of our approach and suggest that it can be the basis for the definition of new operating rules when the sustainability constraints cannot be ensured.