Controllability and positivity constraints in population dynamics with age structuring and diffusion

Abstract

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 age. 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 neighbourhood of 0. The second one is that we prove that the whole population can be steered into zero in a uniform time, without, as in the existing literature, excluding some interval of low ages. Moreover, 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. Finally, we show that the system can be steered between two positive steady states by controls preserving the positivity of the state trajectory. The method of proof, combining final-state observability estimates with the use of characteristics and with L ∞ estimates of the associated semigroup, avoids the explicit use of parabolic Carleman estimates.

On the null controllability of the Lotka-Mckendrick system

“…The invertibility of the operator (λI − A) follows from Song et. al [9, Theorem 1(i)].Since η ∈ D(A 0 ), we get ψ ∈ D(A 0 ) and ψ solves(8). This completes the proof of the proposition.…”

“…In their case, the control is localized in space variable but active for all ages. Recently, Maity, Tucsnak and Zuazua showed [8] that the same result can be achieved by means of control localized in space variable as well as with respect to age, not necessarily containing a neighbourhood of zero.…”

Time optimal internal controls for the Lotka-McKendrick equation with spatial diffusion

“…The case when the control acts in a spatial subdomain ω and also only for small age classes was investigated by B. Ainseba and S. Aniţa [2], for initial data p 0 in a neighborhood of the target p. As already mentioned, Hegoburu and Tucsnak proved the null controllability of system (1), using an adaptation of the Lebeau Robbiano strategy originally developed for the null-controllability of the heat equation. This result has been recently improved by Maity, Tucsnak and Zuazua [30], assuming that the young individuals are not able to reproduce before some age a b > 0 , where the control function u in system (1) has support in some interval of ages [a 1 , a 2 ], where 0 a 1 < a 2 a † . In [30] the authors proved the null controllability result with this additional age restriction, provided that the control time τ is large enough, and the age a 1 is smaller than a b .…”

“…This result has been recently improved by Maity, Tucsnak and Zuazua [30], assuming that the young individuals are not able to reproduce before some age a b > 0 , where the control function u in system (1) has support in some interval of ages [a 1 , a 2 ], where 0 a 1 < a 2 a † . In [30] the authors proved the null controllability result with this additional age restriction, provided that the control time τ is large enough, and the age a 1 is smaller than a b . Related approximate and exact controllability issues have also been studied in Ainseba and Langlais [4], Ainseba and Iannelli [3], Traore [35], Kavian and Traore [23].…”

“…, so that the mapping K is a bounded linear extension of the mapping K on L 1 (ω). Notice that, using (30), for every z ∈ L 1 (ω) we have…”