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 and 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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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}. $
[1]

Yacouba Simporé, Oumar Traoré. Null controllability of a nonlinear age, space and two-sex structured population dynamics model. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021052

[2]

Farid Ammar Khodja, Cherif Bouzidi, Cédric Dupaix, Lahcen Maniar. Null controllability of retarded parabolic equations. Mathematical Control and Related Fields, 2014, 4 (1) : 1-15. doi: 10.3934/mcrf.2014.4.1

[3]

Lianwen Wang. Approximate controllability and approximate null controllability of semilinear systems. Communications on Pure and Applied Analysis, 2006, 5 (4) : 953-962. doi: 10.3934/cpaa.2006.5.953

[4]

Lingyang Liu, Xu Liu. Controllability and observability of some coupled stochastic parabolic systems. Mathematical Control and Related Fields, 2018, 8 (3&4) : 829-854. doi: 10.3934/mcrf.2018037

[5]

Chun Zong, Gen Qi Xu. Observability and controllability analysis of blood flow network. Mathematical Control and Related Fields, 2014, 4 (4) : 521-554. doi: 10.3934/mcrf.2014.4.521

[6]

Ovidiu Cârjă, Alina Lazu. On the minimal time null controllability of the heat equation. Conference Publications, 2009, 2009 (Special) : 143-150. doi: 10.3934/proc.2009.2009.143

[7]

Piermarco Cannarsa, Genni Fragnelli, Dario Rocchetti. Null controllability of degenerate parabolic operators with drift. Networks and Heterogeneous Media, 2007, 2 (4) : 695-715. doi: 10.3934/nhm.2007.2.695

[8]

El Mustapha Ait Ben Hassi, Farid Ammar khodja, Abdelkarim Hajjaj, Lahcen Maniar. Carleman Estimates and null controllability of coupled degenerate systems. Evolution Equations and Control Theory, 2013, 2 (3) : 441-459. doi: 10.3934/eect.2013.2.441

[9]

Lahcen Maniar, Martin Meyries, Roland Schnaubelt. Null controllability for parabolic equations with dynamic boundary conditions. Evolution Equations and Control Theory, 2017, 6 (3) : 381-407. doi: 10.3934/eect.2017020

[10]

Lydia Ouaili. Minimal time of null controllability of two parabolic equations. Mathematical Control and Related Fields, 2020, 10 (1) : 89-112. doi: 10.3934/mcrf.2019031

[11]

Farid Ammar Khodja, Franz Chouly, Michel Duprez. Partial null controllability of parabolic linear systems. Mathematical Control and Related Fields, 2016, 6 (2) : 185-216. doi: 10.3934/mcrf.2016001

[12]

Abdelmouhcene Sengouga. Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints. Evolution Equations and Control Theory, 2020, 9 (1) : 1-25. doi: 10.3934/eect.2020014

[13]

Zhaoqiang Ge. Controllability and observability of stochastic implicit systems and stochastic GE-evolution operator. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 339-351. doi: 10.3934/naco.2021009

[14]

Ali Wehbe, Marwa Koumaiha, Layla Toufaily. Boundary observability and exact controllability of strongly coupled wave equations. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1269-1305. doi: 10.3934/dcdss.2021091

[15]

Chibueze Christian Okeke, Abdulmalik Usman Bello, Lateef Olakunle Jolaoso, Kingsley Chimuanya Ukandu. Inertial method for split null point problems with pseudomonotone variational inequality problems. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021037

[16]

Abdelaziz Khoutaibi, Lahcen Maniar. Null controllability for a heat equation with dynamic boundary conditions and drift terms. Evolution Equations and Control Theory, 2020, 9 (2) : 535-559. doi: 10.3934/eect.2020023

[17]

J. Carmelo Flores, Luz De Teresa. Null controllability of one dimensional degenerate parabolic equations with first order terms. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 3963-3981. doi: 10.3934/dcdsb.2020136

[18]

Irina F. Sivergina, Michael P. Polis. About global null controllability of a quasi-static thermoelastic contact system. Conference Publications, 2005, 2005 (Special) : 816-823. doi: 10.3934/proc.2005.2005.816

[19]

Umberto Biccari, Mahamadi Warma. Null-controllability properties of a fractional wave equation with a memory term. Evolution Equations and Control Theory, 2020, 9 (2) : 399-430. doi: 10.3934/eect.2020011

[20]

Nicolas Hegoburu, Marius Tucsnak. Null controllability of the Lotka-McKendrick system with spatial diffusion. Mathematical Control and Related Fields, 2018, 8 (3&4) : 707-720. doi: 10.3934/mcrf.2018030

2020 Impact Factor: 1.284

Metrics

  • PDF downloads (256)
  • HTML views (134)
  • Cited by (0)

Other articles
by authors

[Back to Top]