September  2018, 8(3&4): 707-720. doi: 10.3934/mcrf.2018030

Null controllability of the Lotka-McKendrick system with spatial diffusion

1. 

Institut de Mathématiques de Bordeaux, Université de Bordeaux/Bordeaux INP/CNRS, 351 Cours de la Libération, 33 405 Talence, France

2. 

Institut de Mathématiques de Bordeaux UMR 5251, Université de Bordeaux/Bordeaux INP/CNRS, 351 Cours de la Libération, 33 405 Talence, France

* Corresponding author: Marius Tucsnak

Received  November 2017 Revised  May 2018 Published  September 2018

We consider the infinite dimensional linear control system described by the population dynamics model of Lotka-McKendrick with spatial diffusion. Considering control functions localized with respect to the spatial variable but active for all ages, we prove that the whole population can be steered to zero in any positive time. The main novelty we bring is that, unlike the existing results in the literature, we can also control the population of ages very close to 0. Another novelty brought in is the employed methodology: as far as we know, the present work is the first one remarking that the null controllability of the considered system can be obtained by using the Lebeau-Robbiano strategy, originally developed for the null-controllability of the heat equation.

Citation: Nicolas Hegoburu, Marius Tucsnak. Null controllability of the Lotka-McKendrick system with spatial diffusion. Mathematical Control & Related Fields, 2018, 8 (3&4) : 707-720. doi: 10.3934/mcrf.2018030
References:
[1]

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

[2]

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

[3]

B. Ainseba and M. Iannelli, Exact controllability of a nonlinear population-dynamics problem, Differential and Integral Equations, 16 (2003), 1369-1384.   Google Scholar

[4]

B. Ainseba and M. Langlais, On a population dynamics control problem with age dependence and spatial structure, J. Math. Anal. Appl., 248 (2000), 455-474.  doi: 10.1006/jmaa.2000.6921.  Google Scholar

[5]

S. Aniţa, Analysis and Control of Age-Dependent Population Dynamics, Theory and Applications, Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-94-015-9436-3.  Google Scholar

[6]

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

[7]

K. Beauchard, Null controllability of Kolmogorov-type equations, Math. Control Signals Systems, 26 (2014), 145-176.  doi: 10.1007/s00498-013-0110-x.  Google Scholar

[8]

K. Beauchard and K. Pravda-Starov, Null-controllability of hypoelliptic quadratic differential equations, J. Éc. polytech. Math., 5 (2018), 1-43.   Google Scholar

[9]

B. Z. Guo and W. L. Chan, On the semigroup for age dependent population dynamics with spatial diffusion, J. Math. Anal. Appl., 184 (1994), 190-199.  doi: 10.1006/jmaa.1994.1193.  Google Scholar

[10]

N. HegoburuP. Magal and M. Tucsnak, Controllability with positivity constraints of the Lotka-McKendrick system, SIAM J. Control Optim., 56 (2018), 723-750.  doi: 10.1137/16M1103087.  Google Scholar

[11]

W. Huyer, Semigroup formulation and approximation of a linear age-dependent population problem with spatial diffusion, Semigroup Forum, 49 (1994), 99-114.  doi: 10.1007/BF02573475.  Google Scholar

[12]

H. Inaba, Age-structured Population Dynamics in Demography and Epidemiology, Springer, Singapore, 2017. doi: 10.1007/978-981-10-0188-8.  Google Scholar

[13]

D. Jerison and G. Lebeau, Nodal sets of sums of eigenfunctions, in Harmonic Analysis and Partial Differential Equations (Chicago, IL, 1996), Chicago Lectures in Math., (1999), 223-239.  Google Scholar

[14]

F. Kappel and K. 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

[15]

O. Kavian and O. Traore, Approximate controllability by birth control for a nonlinear population dynamics model, ESAIM Control Optim. Calc. Var., 17 (2011), 1198-1213.  doi: 10.1051/cocv/2010043.  Google Scholar

[16]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356.  doi: 10.1080/03605309508821097.  Google Scholar

[17]

G. Lebeau and L. Robbiano, Stabilisation de l'équation des ondes par le bord, Duke Math. J., 86 (1997), 465-491.  doi: 10.1080/03605309508821097.  Google Scholar

[18]

Y. Netrusov and Y. Safarov, Weyl asymptotic formula for the Laplacian on domains with rough boundaries, Comm. Math. Phys., 253 (2005), 481-509.  doi: 10.1007/s00220-004-1158-8.  Google Scholar

[19]

T. I. Seidman, How violent are fast controls. Ⅲ, J. Math. Anal. Appl., 339 (2008), 461-468.  doi: 10.1016/j.jmaa.2007.07.008.  Google Scholar

[20]

J. SongJ. Y. YuX. Z. ZhangS. J. HuZ. X. ZhaoJ. Q. Liu and D. X. Feng, Spectral properties of population operator and asymptotic behaviour of population semigroup, Acta Math. Sci. (English Ed.), 2 (1982), 139-148.   Google Scholar

[21]

G. Tenenbaum and M. Tucsnak, On the null-controllability of diffusion equations, ESAIM Control Optim. Calc. Var., 17 (2011), 1088-1100.  doi: 10.1051/cocv/2010035.  Google Scholar

[22]

O. Traore, Null controllability of a nonlinear population dynamics problem Int. J. Math. Math. Sci., (2006), Art. ID 49279, 20 pp. doi: 10.1155/IJMMS/2006/49279.  Google Scholar

[23]

J. Zabczyk, Remarks on the algebraic Riccati equation in Hilbert space, Appl. Math. Optim., 2 (1975/76), 251-258.  doi: 10.1007/BF01464270.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

B. Ainseba and M. Iannelli, Exact controllability of a nonlinear population-dynamics problem, Differential and Integral Equations, 16 (2003), 1369-1384.   Google Scholar

[4]

B. Ainseba and M. Langlais, On a population dynamics control problem with age dependence and spatial structure, J. Math. Anal. Appl., 248 (2000), 455-474.  doi: 10.1006/jmaa.2000.6921.  Google Scholar

[5]

S. Aniţa, Analysis and Control of Age-Dependent Population Dynamics, Theory and Applications, Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-94-015-9436-3.  Google Scholar

[6]

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

[7]

K. Beauchard, Null controllability of Kolmogorov-type equations, Math. Control Signals Systems, 26 (2014), 145-176.  doi: 10.1007/s00498-013-0110-x.  Google Scholar

[8]

K. Beauchard and K. Pravda-Starov, Null-controllability of hypoelliptic quadratic differential equations, J. Éc. polytech. Math., 5 (2018), 1-43.   Google Scholar

[9]

B. Z. Guo and W. L. Chan, On the semigroup for age dependent population dynamics with spatial diffusion, J. Math. Anal. Appl., 184 (1994), 190-199.  doi: 10.1006/jmaa.1994.1193.  Google Scholar

[10]

N. HegoburuP. Magal and M. Tucsnak, Controllability with positivity constraints of the Lotka-McKendrick system, SIAM J. Control Optim., 56 (2018), 723-750.  doi: 10.1137/16M1103087.  Google Scholar

[11]

W. Huyer, Semigroup formulation and approximation of a linear age-dependent population problem with spatial diffusion, Semigroup Forum, 49 (1994), 99-114.  doi: 10.1007/BF02573475.  Google Scholar

[12]

H. Inaba, Age-structured Population Dynamics in Demography and Epidemiology, Springer, Singapore, 2017. doi: 10.1007/978-981-10-0188-8.  Google Scholar

[13]

D. Jerison and G. Lebeau, Nodal sets of sums of eigenfunctions, in Harmonic Analysis and Partial Differential Equations (Chicago, IL, 1996), Chicago Lectures in Math., (1999), 223-239.  Google Scholar

[14]

F. Kappel and K. 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

[15]

O. Kavian and O. Traore, Approximate controllability by birth control for a nonlinear population dynamics model, ESAIM Control Optim. Calc. Var., 17 (2011), 1198-1213.  doi: 10.1051/cocv/2010043.  Google Scholar

[16]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356.  doi: 10.1080/03605309508821097.  Google Scholar

[17]

G. Lebeau and L. Robbiano, Stabilisation de l'équation des ondes par le bord, Duke Math. J., 86 (1997), 465-491.  doi: 10.1080/03605309508821097.  Google Scholar

[18]

Y. Netrusov and Y. Safarov, Weyl asymptotic formula for the Laplacian on domains with rough boundaries, Comm. Math. Phys., 253 (2005), 481-509.  doi: 10.1007/s00220-004-1158-8.  Google Scholar

[19]

T. I. Seidman, How violent are fast controls. Ⅲ, J. Math. Anal. Appl., 339 (2008), 461-468.  doi: 10.1016/j.jmaa.2007.07.008.  Google Scholar

[20]

J. SongJ. Y. YuX. Z. ZhangS. J. HuZ. X. ZhaoJ. Q. Liu and D. X. Feng, Spectral properties of population operator and asymptotic behaviour of population semigroup, Acta Math. Sci. (English Ed.), 2 (1982), 139-148.   Google Scholar

[21]

G. Tenenbaum and M. Tucsnak, On the null-controllability of diffusion equations, ESAIM Control Optim. Calc. Var., 17 (2011), 1088-1100.  doi: 10.1051/cocv/2010035.  Google Scholar

[22]

O. Traore, Null controllability of a nonlinear population dynamics problem Int. J. Math. Math. Sci., (2006), Art. ID 49279, 20 pp. doi: 10.1155/IJMMS/2006/49279.  Google Scholar

[23]

J. Zabczyk, Remarks on the algebraic Riccati equation in Hilbert space, Appl. Math. Optim., 2 (1975/76), 251-258.  doi: 10.1007/BF01464270.  Google Scholar

Figure 1.  The spectrum of the free diffusion operator $A_0$ (green crosses) and of $-\Delta$ (red circles)
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