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Overtaking optimal control problem of age-dependent populations with infinite horizon. (English) Zbl 0707.92017

The authors consider optimal birth control of an age-dependent population model with unbounded time interval. The model involves a bilinear boundary control of a distributed system described by a first order linear partial differential equation. Since no assumptions on the convergence of the improper integral representing a cost functional are made, the optimal control problem is solved in the weaker sense known as overtaking optimality [D. A. Carlson, A. Haurie and A. Jabrane, SIAM J. Control Optimization 25, No.6, 1517-1541 (1987; Zbl 0658.49003)].
The main idea is the use of Pontryagin’s maximum principle for an associated finite horizon optimal control problem and afterwards formulation of conditions which ensure the turnpike property. This property says that an optimal trajectory on any finite horizon stays most of the time in the vicinity of an extremal steady state and ultimately converges to it if the time interval becomes unbounded.
Nevertheless the paper has one failure: there are too many mistakes. For example, in the introduction the authors refer to five sections of the paper while in fact it consists of only three sections.
Reviewer: A.Šwierniak

MSC:

92D25 Population dynamics (general)
49K20 Optimality conditions for problems involving partial differential equations
49K99 Optimality conditions

Citations:

Zbl 0658.49003
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References:

[1] Chan, W. L.; Zhu, Guo Bao, Optimal birth control of population dynamics, J. Math. Anal. Appl., 144, 532-552 (1989) · Zbl 0708.92017
[2] W. L. Chan and Guo Bao ZhuJ. Math. Anal. Appl.; W. L. Chan and Guo Bao ZhuJ. Math. Anal. Appl. · Zbl 0736.92016
[3] Brokate, M., Pontryagin’s principle for control problems in age-dependent population dynamics, J. Math. Biol., 23, 75-102 (1985) · Zbl 0599.92017
[4] Carlson, D. A.; Haurie, A.; Jabrane, A., Existence of overtaking solutions to infinite dimensional control problems on unbounded time intervals, SIAM J. Control Optim., 25, 1517-1541 (1987) · Zbl 0658.49003
[5] Song, J.; Yu, J.-Y, Population System Control (1987), Springer-Verlag: Springer-Verlag New York
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