  In natural resource management, or more generally in the study of
sustainability issues, often the objective is to maintain the state of a given
system within a desirable configuration, typically established in terms of
standards or thresholds. For instance, in fisheries management, the procedure
of designing policies may include keeping the spawning stock biomass over a
critical threshold and also ensuring minimal catches. Given a controlled
dynamical system in discrete-time, representing the evolution of some natural
resources under the action of controls and uncertainties, and an initial
endowment of the resources, the aim of this paper is to characterize the set of
robust sustainable thresholds, that is, the thresholds for which there exists
some control path, along with its corresponding state trajectory, satisfying
for all possible uncertainty scenarios, prescribed mixed constraints
parametrized by such thresholds. This set provides useful information to users
and decision-makers, illustrating the trade-offs between constraints and it is
strongly related to the robust viability, one of the key concepts in viability
theory, discipline that study the consistency between a controlled dynamical
system and given constraints. Specifically, we are concerned with
characterizing the weak and strong Pareto fronts of the set of robust
sustainable thresholds, providing a practical method for computing such objects
based on optimal control theory and a level-set approach. A numerical example,
relying on renewable resource management, is shown to demonstrate the proposed
method.
