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.

[2]

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

[3]

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

[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.

[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.

[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.

[7]

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

[8]

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

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

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.

[2]

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

[3]

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

[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.

[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.

[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.

[7]

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

[8]

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

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

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