The article investigates a fisheries management model described by a phase-constrained optimal control problem. Two cases are considered: the problem has a singular regime and the problem is without a singular regime. Optimal harvest strategies are derived for each case in analytical form. The article also presents numerical results for certain parameter values that are relevant in applications.

SUMMARYWe derive a closed‐form solution for a well‐known fisheries harvesting model with an additional state constraint. The problem is linear in the control and previous solutions appearing in the literature have been numerical in nature. The so‐called direct adjoining approach is used in our derivation and the optimal solutions turn out to be a mixture of bang‐bang and boundary arcs. Copyright © 2013 John Wiley & Sons, Ltd.

We consider the time optimization problem for a biological model describing the process of the growth of bacterial cells, more precisely, the problem of transition to balanced in minimum time. By a change of variables, a three-dimensional problem is reduced to a two-dimensional one for which we construct an optimal synthesis and present a complete proof of the optimality. In particular, we show that the optimal control has at most one switching point and construct the switching line of the optimal control. We represent a formula for the computation of the optimal time of transition into the terminal state from an arbitrary initial point.

This work examines a biological model that describes the growth of microbial cells. The problem of optimum control is to maximize the amount of structural biomass at the final moment in time. This work studies two control modes. It is shown that starting from a certain point in time, one mode is better than the other in terms of maximizing the structural biomass of the cells.

A special model of resource allocation over an infinite interval of time is studied. Using the Pontryagin maximum principle, an extreme solution is constructed whose optimality is proven with the help of a theorem on sufficient conditions, in the form of constructions of Pontryagin’s maximum principle. A concrete example in which the classical maximum principle is inapplicable is considered.

The minimal convex hull of a subset of finite-dimensional space is constructed in a discrete fashion: at each step, we construct a new set that includes the inherited set. The procedure for finding the chain of sets involves geometrically evident constructions. This chain becomes stationary; i.e., from some number onward, all the sets of the chain being constructed coincide with the minimal convex hull. Our approach uses the principal result (Caratheodory’s theorem) underlying the conventional approach to constructing a minimal convex hull.