Optimal Control for the Degenerate Elliptic Logistic Equation

Abstract

We consider the optimal control of the harvesting of the diffusive degenerate elliptic logistic equation. Under certain assumptions, we prove the existence and uniqueness of an optimal control. Moreover, the optimality system and a characterization of the optimal control are also derived. Sub-supersolution method, singular eigenvalue problem and differentiability with respect to the positive cone are the techniques used to get our results.

Study of the Optimal Harvesting Control and the Optimality System for an Elliptic Problem

“…The following result (whose proof can be found in [7]) shows that (2.4) possesses a unique solution in C 1 0 (Ω), it provides us of an useful estimate and properties of the solution.…”

“…In the following result, we collect these results and some properties of the principal eigenvalue, see [7]. …”

“…Observe that by the strong maximum principle for f ∈ C, any nontrivial and nonnegative solution u of (3.1) is strictly positive, this means that u ∈ int(P + ). The uniqueness of positive solution follows as in Theorem 1 of [9] and the continuity of the map f → u f as in Theorem 3.3 of [7].…”

“…In [7], the authors analyzed the case 0 < α < 1 ≤ β, b strictly positive and the cost functional (1.4) under more restrictive monotony assumptions on functions h, k. There, the controls are restricted to the set…”

Measure-Valued Solutions for a Differential Game Related to Fish Harvesting

“…Lebesgue measure on Ω. We stress that the study of optimal strategies within a space of Radon measures is a major difference between the present paper and earlier literature on the subject [CGM,DMS1,DMS2,LM,LW,N].…”