A Criterion for the Existence of Nondestructive Controls in the Problem of Optimal Exploitation of a Binary-Structured System

Earlier the authors proved the equivalence of a sustainable exploitation problem for a system of renewable resources and a certain mathematical program. In this paper we study the properties of a map describing the dependence of the state vector of the system on the control. In the particular case of a structured population described by the binary Leslie model, conditions for the objective function are characterized under which there are optimal controls preserving all structural divisions of the system. In this case, we use the notion of local irreducibility, which generalizes the classical notion of map irreducibility.