The Dubovitskii and Milyutin Methodology Applied to an Optimal Control Problem Originating in an Ecological System
Abstract
:1. Introduction
2. Assumptions and Statements of Main Results
2.1. Assumptions
2.2. Statements of Main Results
- (a)
- The sets
- (b)
- The application M is Gâteaux differentiable and the derivative of M in is defined by if and only if
- (c)
- The application M is strictly differentiable and is a surjective operator.
- (d)
- The following sets
3. Preliminaries
4. Proofs of Main Results
4.1. Proof of Theorem 1
- Step 1:
- Local solutions via a truncated problem. Let us consider the truncated Cauchy problem
- Step 2:
- The local solution is a global solution. To prove that the local solution on is a global solution it suffices to prove that is bounded on . Indeed, from (2c), the positivity of on deduced on Step 1 together with the positivity of g on given by Assumption 4, the strictly positivity of assumed in Assumption 2, and the fact that and is strictly positive on Ω (considered on Assumption 3), we deduce that for . Then, .
- Step 3:
4.2. Proof of Lemma 1
4.3. Proof of Theorem 2
5. Conclusions and Future Work
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Conflicts of Interest
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Coronel, A.; Huancas, F.; Lozada, E.; Rojas-Medar, M. The Dubovitskii and Milyutin Methodology Applied to an Optimal Control Problem Originating in an Ecological System. Mathematics 2021, 9, 479. https://doi.org/10.3390/math9050479
Coronel A, Huancas F, Lozada E, Rojas-Medar M. The Dubovitskii and Milyutin Methodology Applied to an Optimal Control Problem Originating in an Ecological System. Mathematics. 2021; 9(5):479. https://doi.org/10.3390/math9050479
Chicago/Turabian StyleCoronel, Aníbal, Fernando Huancas, Esperanza Lozada, and Marko Rojas-Medar. 2021. "The Dubovitskii and Milyutin Methodology Applied to an Optimal Control Problem Originating in an Ecological System" Mathematics 9, no. 5: 479. https://doi.org/10.3390/math9050479