Comparison Theorem for Viability Kernels via Conic Preorders
Résumé
In natural resource management, decision-makers often aim at maintaining the
state of the system within a desirable set for all times.
For instance, fisheries management procedures include keeping the
spawning stock biomass over a critical threshold.
Another example is given by the peak control of an epidemic outbreak
that encompasses maintaining the
number of infected individuals below medical treatment capacities.
In mathematical terms, one controls a dynamical system.
Then, keeping the state of the system within a desirable set for all times is
possible when the initial state belongs to the so-called viability
kernel. We introduce the notion of conic quasimonotonicity reducibility.
With this property, we provide a comparison theorem by inclusion between two viability
kernels, corresponding to two control systems in the infinite horizon case. We
also derive conditions for equality. We illustrate the method with a model for the
biocontrol of a vector-transmitted epidemic.
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