An optimal control problem for a pest, predator, and plant system. (English) Zbl 1239.49049
Summary: We study an optimal control problem for a system of three nonlinear parabolic equations from population dynamics. The equations model a trophic chain consisting of a predator, a pest and a plant species. The existence and uniqueness of the positive solution for the system are proved. The control variable is connected with the action of a pesticide. Our goal is to minimize the density of the pest and to maximize the plant density. The existence of the optimal solution is proved. The first and second order optimality conditions are established.
MSC:
| 49N90 | Applications of optimal control and differential games |
| 92D25 | Population dynamics (general) |
| 37N25 | Dynamical systems in biology |
Keywords:
dual system; first order optimality conditions; reaction; diffusion equations; second order optimality conditions; trophic chainReferences:
| [1] | Apreutesei, N., Necessary optimality conditions for a Lotka-Volterra three species system, Math. Model. Nat. Phenom., 1, 123-135 (2006) · Zbl 1201.92058 |
| [2] | Haimovici, A., A control problem for a Volterra three species system, Math. Rev. Anal. Numér. Théor. Approximation, Math., 23(46), 1, 35-41 (1981) · Zbl 0483.92015 |
| [3] | Apreutesei, G., Some convergences of convex functions and its \(\varepsilon \)-subdifferentials, An. Stiint. Univ. Al. I. Cuza Iaşi, 43, 1, 139-149 (1996) · Zbl 0984.46048 |
| [4] | Casas, E., Pontryagin’s principle for state-constrained boundary control problems of semilinear parabolic equations, SIAM J. Control Optim., 35, 4, 1297-1327 (1997) · Zbl 0893.49017 |
| [5] | Casas, E.; Mateos, M.; Raymond, J. P., Pontryagin’s principle for the control of parabolic equations with gradient state constraints, Nonlinear Anal. TMA, 46, 7, 933-956 (2001) · Zbl 0998.49014 |
| [6] | Casas, E.; De Los Reyes, J. C.; Tröltzsch, F., Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints, SIAM J. Optim., 19, 2, 616-643 (2008) · Zbl 1161.49019 |
| [7] | Raymond, J. P.; Tröltzsch, F., Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints, Discrete Contin. Dyn. Syst., 6, 2, 431-450 (2000) · Zbl 1010.49015 |
| [8] | Rösch, A.; Tröltzsch, F., Sufficient second-order optimality conditions for a parabolic optimal control problem with pointwise control-state constraints, SIAM J. Control Optim., 42, 1, 138-154 (2003) · Zbl 1038.49028 |
| [9] | Griesse, R.; Volkwein, S., A primal-dual active set strategy for optimal boundary control of a nonlinear reaction-diffusion system, SIAM J. Control Optim., 44, 2, 467-494 (2005) · Zbl 1090.49024 |
| [10] | Yunusov, M., Optimal control of an ecosystem of three trophic levels, Dokl. Akad. Nauk Tadzhik. SSR, 30, 5, 277-281 (1987), (in Russian) · Zbl 0641.92017 |
| [11] | Georgescu, P.; Hsieh, Y. H., Global dynamics for a predator-prey model with stage structure for predator, SIAM J. Appl. Math., 67, 1379-1395 (2007) · Zbl 1120.92045 |
| [12] | Garvie, M.; Trenchea, C., Optimal control of a nutrient-phytoplankton-zooplankton-fish system, SIAM J. Control Optim., 46, 3, 775-791 (2007) · Zbl 1357.49090 |
| [13] | Feichtinger, G.; Veliov, V., On a distributed control problem arising in dynamic optimization of a fixed-size population, SIAM J. Optim., 18, 3, 980-1003 (2007) · Zbl 1262.49025 |
| [14] | Jia, C. H.; Feng, D. X., An optimal control problem of a coupled nonlinear parabolic population system, Acta Math. Appl. Sin. Engl. Ser., 23, 3, 377-388 (2007) · Zbl 1135.49003 |
| [15] | Barbu, V., Mathematical Methods in Optimization of Differential Systems (1994), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0819.49002 |
| [16] | Pazy, A., (Semigroups of Linear Operators and Applications to Partial Differential Equations. Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44 (1983), Springer-Verlag: Springer-Verlag New York), etc. · Zbl 0516.47023 |
| [17] | Vrabie, I., \((C_0\)-Semigroups and Applications. \(C_0\)-Semigroups and Applications, Math. Stud., vol. 191 (2003), North-Holland) · Zbl 1119.47044 |
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