Comparison theorem for viability kernels via conic preorders

Abstract

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.