SHARP TIME NULL CONTROLLABILITY OF A POPULATION DYNAMICS MODEL WITH AGE STRUCTURING AND DIFFUSION

Debayan Maity (1) , Marius Tucsnak (1) , Enrique Zuazua (2, 3)

1 IMB - Institut de Mathématiques de Bordeaux
2 Departamento de Matematicas [Madrid]
3 DEUSTO - Universidad de Deusto

In this article we study the null controllability of a linear system coming from a population dynamics model with age structuring and spatial diffusion (of Lotka-McKendrick type). The control is localized in the space variable as well as with respect to the ages. The first novelty we bring in is that the age interval in which the control needs to be active can be arbitrarily small and does not need to contain a neighborhood of 0. The second one is that we prove that the whole population can be steered into zero in an uniform time, without, as in the existing literature, excluding some interval of low ages. Finally, we improve the existing estimates of the controllability time and we show that our estimates are sharp, at least when the control is active for very low ages. The method of proof, combining final-state observability estimates with the use of characteristics, avoids the explicit use of parabolic Carleman estimates.

Mots clés: Population dynamics; Null controllability

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

Debayan Maity, Marius Tucsnak, Enrique Zuazua. SHARP TIME NULL CONTROLLABILITY OF A POPULATION DYNAMICS MODEL WITH AGE STRUCTURING AND DIFFUSION. 2018. ⟨hal-01764865v1⟩