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Optimal control of a population dynamics model with missing birth rate. (English) Zbl 1453.49004

The authors establish the existence of a localized distributed control for a model describing the dynamics of a population with age dependence and spatial structure, with an unknown birth rate. Having in view that the population is thought to represent an invasive species, of concern is bringing the distribution of individuals to a desired distribution, and this is done by acting only on a part of the domain.
The authors are then lead to solving the so-called no-regret control problem [J.-L. Lions, C. R. Acad. Sci., Paris, Sér. I 315, No. 12, 1253–1257 (1992; Zbl 0766.93033)]. They do so by establishing first several preliminary regularity and convergence results. By regularizing the no-regret control problem, they obtain a low-regret control problem and, by using the Legendre-Fenchel transform, prove that the latter is equivalent to a classical optimal control problem. They then introduce an appropriate Hilbert space and apply the Aubin-Lions lemma to obtain the convergence of a family of low-regret controls towards the no-regret control, finding also a singular optimality system that characterizes the no-regret control.

MSC:

49J20 Existence theories for optimal control problems involving partial differential equations
49N30 Problems with incomplete information (optimization)
92D25 Population dynamics (general)
93C41 Control/observation systems with incomplete information

Citations:

Zbl 0766.93033
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References:

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