Uniqueness of the optimal control for a Lotka-Volterra control problem with a large crowding effect. (English) Zbl 0868.49004
An optimal control problem with steady-state equation being a partial differential equation of Lotka-Volterra type is studied. The unknown function \(u\) is understood as the construction of the biological species.
The authors developed several conditions to guarantee the uniqueness of the optimal control of the system and its approximation. More specifically, it is shown that, under suitable conditions, any optimal control of the system can be expressed in terms of a solution of a certain optimality system. Several conditions are provided to assure the uniqueness of the solution of the optimality system and so the uniqueness of the optimal control. At the end of the paper, an iterative scheme is given to approximate the unique optimal control. It is shown that this iterative scheme converges to the unique solution of the optimality system.
The authors developed several conditions to guarantee the uniqueness of the optimal control of the system and its approximation. More specifically, it is shown that, under suitable conditions, any optimal control of the system can be expressed in terms of a solution of a certain optimality system. Several conditions are provided to assure the uniqueness of the solution of the optimality system and so the uniqueness of the optimal control. At the end of the paper, an iterative scheme is given to approximate the unique optimal control. It is shown that this iterative scheme converges to the unique solution of the optimality system.
Reviewer: Yi Lin (Slippery Rock)
MSC:
| 49J20 | Existence theories for optimal control problems involving partial differential equations |
| 93C20 | Control/observation systems governed by partial differential equations |
| 92D25 | Population dynamics (general) |
Keywords:
admissible control; iterative approximation; optimal control; partial differential equation of Lotka-Volterra type; optimality systemReferences:
| [1] | H. Amann: Fixed points equations and nonlinear eigenvalue problems in ordered banach spaces, SIAM review, 18, 4, October 1976, 620-709. Zbl0345.47044 MR415432 · Zbl 0345.47044 · doi:10.1137/1018114 |
| [2] | H. Berestycki and P.L. Lions: Some applications of the method of super and subsolutions, Lect. Not. Math., 782, Springer-Verlag, 1980, 16-42. Zbl0433.35023 MR572249 · Zbl 0433.35023 |
| [3] | J. Blat and K.J. Brown: Bifurcation of steady-state solutions in predator-prey and competition systems, Proc. of the Roy. Soc. Edinburg, 97, 1984, 21-34. Zbl0554.92012 MR751174 · Zbl 0554.92012 · doi:10.1017/S0308210500031802 |
| [4] | A. Cañada, J.L. Gámez and J.A. Montero: An optimal control problem for a nonlinear elliptic equation arising from population dynamics, in Calculus of Variation, Applications and Computations, Longman, Pitman Research Notes in Mathematics Series, C. Bandle, J. Bemelmans, M. Chipot, J. Saint Jean Paulin and I. Shafrir, editors, 326, 1995, 35-40. Zbl0828.49006 MR1419332 · Zbl 0828.49006 |
| [5] | A. Cañada, J.L. Gámez and J.A. Montero: Study of a nonlinear optimal control problem for diffusive Volterra-Lotka equations, to appear. · Zbl 0916.49003 |
| [6] | D. Gilbarg and N.S. Trudinger: Elliptic Partial Differential Equations of Second Order, 2nd Edition, Springer-Verlag, Berlin, 1983. Zbl0562.35001 MR737190 · Zbl 0562.35001 |
| [7] | P. Hess: Periodic-parabolic boundary value problems and posuivity, Longman Group U.K. Limited, 1991. Zbl0731.35050 MR1100011 · Zbl 0731.35050 |
| [8] | F. He, A. Leung and S. Stojanovic: Periodic optimal control for parabolic Volterra-Lotka type equations, Math. Methods Appl. Sci., 18, 1995, 127-146. Zbl0818.49002 MR1311374 · Zbl 0818.49002 · doi:10.1002/mma.1670180204 |
| [9] | A. Leung: Systems of nonlinear partial differential equations, The Netherlands, Kluwer Acacemic Publishers, 1989. Zbl0691.35002 MR1621827 · Zbl 0691.35002 |
| [10] | A. Leung and S. Stojanovic: Direct methods for some distributed games, Diff. and Int. Eqns., 3, 1990, 1113-1125. Zbl0722.90093 MR1073061 · Zbl 0722.90093 |
| [11] | A. Leung and S. Stojanovic: Optimal control for elliptic Volterra-Lotka equations, J. Math. Anal. Appl., 173 1993, 603-619. Zbl0796.49005 MR1209343 · Zbl 0796.49005 · doi:10.1006/jmaa.1993.1091 |
| [12] | L. Li and R. Logan: Positive solutions to general elliptic competition models, Diff. and Int. Eqns., 4, 1991, 817-834. Zbl0751.35014 MR1108062 · Zbl 0751.35014 |
| [13] | S. Stojanovic: Optimal damping control and nonlinear elliptic Systems, SIAM J. Control Optim., 29, 1991, 594-608. Zbl0742.49017 MR1089146 · Zbl 0742.49017 · doi:10.1137/0329033 |
| [14] | J. Smoller: Shock waves and reaction-diffusion equations, NewYork, Springer, 1983. Zbl0508.35002 MR688146 · Zbl 0508.35002 |
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