December  2019, 9(4): 719-728. doi: 10.3934/mcrf.2019048

On the null controllability of the Lotka-Mckendrick system

Institut de Mathématiques, Université de Bordeaux, Bordeaux INP, CNRS F-33400 Talence, France

Received  January 2019 Revised  May 2019 Published  November 2019

Fund Project: Debayan Maity acknowledges the support of the Agence Nationale de la Recherche - Deutsche Forschungsgemeinschaft (ANR - DFG), project INFIDHEM, ID ANR-16-CE92-0028.

In this work, we study null-controllability of the Lotka-McKendrick system of population dynamics. The control is acting on the individuals in a given age range. The main novelty we bring in this work is that the age interval in which the control is active does not necessarily contain a neighbourhood of $ 0. $ The main result asserts the whole population can be steered into zero in large time. The proof is based on final-state observability estimates of the adjoint system with the use of characteristics.

Citation: Debayan Maity. On the null controllability of the Lotka-Mckendrick system. Mathematical Control & Related Fields, 2019, 9 (4) : 719-728. doi: 10.3934/mcrf.2019048
References:
[1]

B. Ainseba, Exact and approximate controllability of the age and space population dynamics structured model, J. Math. Anal. Appl., 275 (2002), 562-574.  doi: 10.1016/S0022-247X(02)00238-X.  Google Scholar

[2]

B. Ainseba and S. Aniţa, Local exact controllability of the age-dependent population dynamics with diffusion, Abstr. Appl. Anal., 6 (2001), 357-368.  doi: 10.1155/S108533750100063X.  Google Scholar

[3]

B. Ainseba and S. Aniţa, Internal exact controllability of the linear population dynamics with diffusion, Electron. J. Differential Equations, 2004 (2004), 11pp.  Google Scholar

[4]

V. BarbuM. Iannelli and M. Martcheva, On the controllability of the Lotka-McKendrick model of population dynamics, J. Math. Anal. Appl., 253 (2001), 142-165.  doi: 10.1006/jmaa.2000.7075.  Google Scholar

[5]

N. HegoburuP. Magal and M. Tucsnak, Controllability with positivity constraints of the lotka–mckendrick system, SIAM Journal on Control and Optimization, 56 (2018), 723-750.  doi: 10.1137/16M1103087.  Google Scholar

[6]

N. Hegoburu and M. Tucsnak, Null controllability of the Lotka-McKendrick system with spatial diffusion, Math. Control Relat. Fields, 8 (2018), 707-720.  doi: 10.3934/mcrf.2018030.  Google Scholar

[7]

F. Kappel and K. P. Zhang, Approximation of linear age-structured population models using Legendre polynomials, J. Math. Anal. Appl., 180 (1993), 518-549.  doi: 10.1006/jmaa.1993.1414.  Google Scholar

[8]

D. Maity, M. Tucsnak and E. Zuazua, Controllability and positivity constraints in population dynamics with age structuring and diffusion, Journal de Mathématiques Pures et Appliquées, 129 (2019), 153–179, URL http://www.sciencedirect.com/science/article/pii/S0021782418301740. doi: 10.1016/j.matpur.2018.12.006.  Google Scholar

[9]

J. SongJ. Y. YuX. Z. WangS. J. HuZ. X. ZhaoJ. Q. LiuD. X. Feng and G. T. Zhu, Spectral properties of population operator and asymptotic behaviour of population semigroup, Acta Math. Sci. (English Ed.), 2 (1982), 139-148.  doi: 10.1016/S0252-9602(18)30629-5.  Google Scholar

[10]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

[11]

G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, vol. 89 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1985.  Google Scholar

show all references

References:
[1]

B. Ainseba, Exact and approximate controllability of the age and space population dynamics structured model, J. Math. Anal. Appl., 275 (2002), 562-574.  doi: 10.1016/S0022-247X(02)00238-X.  Google Scholar

[2]

B. Ainseba and S. Aniţa, Local exact controllability of the age-dependent population dynamics with diffusion, Abstr. Appl. Anal., 6 (2001), 357-368.  doi: 10.1155/S108533750100063X.  Google Scholar

[3]

B. Ainseba and S. Aniţa, Internal exact controllability of the linear population dynamics with diffusion, Electron. J. Differential Equations, 2004 (2004), 11pp.  Google Scholar

[4]

V. BarbuM. Iannelli and M. Martcheva, On the controllability of the Lotka-McKendrick model of population dynamics, J. Math. Anal. Appl., 253 (2001), 142-165.  doi: 10.1006/jmaa.2000.7075.  Google Scholar

[5]

N. HegoburuP. Magal and M. Tucsnak, Controllability with positivity constraints of the lotka–mckendrick system, SIAM Journal on Control and Optimization, 56 (2018), 723-750.  doi: 10.1137/16M1103087.  Google Scholar

[6]

N. Hegoburu and M. Tucsnak, Null controllability of the Lotka-McKendrick system with spatial diffusion, Math. Control Relat. Fields, 8 (2018), 707-720.  doi: 10.3934/mcrf.2018030.  Google Scholar

[7]

F. Kappel and K. P. Zhang, Approximation of linear age-structured population models using Legendre polynomials, J. Math. Anal. Appl., 180 (1993), 518-549.  doi: 10.1006/jmaa.1993.1414.  Google Scholar

[8]

D. Maity, M. Tucsnak and E. Zuazua, Controllability and positivity constraints in population dynamics with age structuring and diffusion, Journal de Mathématiques Pures et Appliquées, 129 (2019), 153–179, URL http://www.sciencedirect.com/science/article/pii/S0021782418301740. doi: 10.1016/j.matpur.2018.12.006.  Google Scholar

[9]

J. SongJ. Y. YuX. Z. WangS. J. HuZ. X. ZhaoJ. Q. LiuD. X. Feng and G. T. Zhu, Spectral properties of population operator and asymptotic behaviour of population semigroup, Acta Math. Sci. (English Ed.), 2 (1982), 139-148.  doi: 10.1016/S0252-9602(18)30629-5.  Google Scholar

[10]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

[11]

G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, vol. 89 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1985.  Google Scholar

Figure 1.  Minimal time required for observability inequality to hold for the transport equation. For both cases, $ \widetilde q(T, \cdot) = 0 $ on the purple region
Figure 2.  An illustration of estimate of $ q(t, 0) $ with $ a_{2} = a_{b}. $
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