December  2019, 9(4): 697-718. doi: 10.3934/mcrf.2019047

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

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

Received  October 2018 Revised  February 2019 Published  November 2019

This work is devoted to establish a bang-bang principle of time optimal controls for a controlled age-structured population evolving in a bounded domain of $ \mathbb{R}^n $. Here, the bang-bang principle is deduced by an $ L^\infty $ null-controllability result for the Lotka-McKendrick equation with spatial diffusion. This $ L^\infty $ null-controllability result is obtained by combining a methodology employed by Hegoburu and Tucsnak - originally devoted to study the null-controllability of the Lotka-McKendrick equation with spatial diffusion in the more classical $ L^2 $ setting - with a strategy developed by Wang, originally intended to study the time optimal internal controls for the heat equation.

Citation: Nicolas Hegoburu. Time optimal internal controls for the Lotka-McKendrick equation with spatial diffusion. Mathematical Control & Related Fields, 2019, 9 (4) : 697-718. doi: 10.3934/mcrf.2019047
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. (electronic).  Google Scholar

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B. Ainseba and M. Iannelli, Exact controllability of a nonlinear population-dynamics problem, Differential and Integral Equations. An International Journal for Theory & Applications, 16 (2003), 1369-1384.   Google Scholar

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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

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S. Aniţa, Optimal harvesting for a nonlinear age-dependent population dynamics, J. Math. Anal. Appl., 226 (1998), 6-22.  doi: 10.1006/jmaa.1998.6064.  Google Scholar

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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

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S. Aniţa and N. Hegoburu, Null controllability via comparison results for nonlinear age-structured population dynamics, Mathematics of Control, Signals, and Systems, 31 (2019), Art. 2, 38pp. doi: 10.1007/s00498-019-0232-x.  Google Scholar

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S. AniţaM. IannelliM.-Y. Kim and E.-J. Park, Optimal harvesting for periodic age-dependent population dynamics, SIAM J. Appl. Math., 58 (1998), 1648-1666.  doi: 10.1137/S0036139996301180.  Google Scholar

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J. Apraiz and L. Escauriaza, Null-control and measurable sets, ESAIM Control Optim. Calc. Var.), 19 (2013), 239-254.  doi: 10.1051/cocv/2012005.  Google Scholar

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J. ApraizL. EscauriazaG. Wang and C. Zhang, Observability inequalities and measurable sets, J. Eur. Math. Soc. (JEMS), 16 (2014), 2433-2475.  doi: 10.4171/JEMS/490.  Google Scholar

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V. Barbu and M. Iannelli, Optimal control of population dynamics, J. Optim. Theory Appl., 102 (1999), 1-14.  doi: 10.1023/A:1021865709529.  Google Scholar

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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

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M. Brokate, Pontryagin's principle for control problems in age-dependent population dynamics, J. Math. Biol., 23 (1985), 75-101.  doi: 10.1007/BF00276559.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

N. Hritonenko and Y. Yatsenko, The structure of optimal time- and age-dependent harvesting in the Lotka-McKendrik population model, Math. Biosci., 208 (2007), 48-62.  doi: 10.1016/j.mbs.2006.09.008.  Google Scholar

[18]

N. Hritonenko, Y. Yatsenko, R.-U. Goetz and A. Xabadia, A bang–bang regime in optimal harvesting of size-structured populations, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), e2331–e2336. Google Scholar

[19]

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

[20]

M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Giardini Editori e Stampatori in Pisa, 1995. Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

J.-L. Lions, Contrôle Optimal de Systèmes Gouvernés par des Équations aux Dérivées Partielles, Dunod, Paris; Gauthier-Villars, Paris, 1968.  Google Scholar

[27]

Q. Lü, Bang-bang principle of time optimal controls and null controllability of fractional order parabolic equations, Acta Math. Sin. (Engl. Ser.), 26 (2010), 2377-2386.  doi: 10.1007/s10114-010-9051-1.  Google Scholar

[28]

Q. Lü, A lower bound on local energy of partial sum of eigenfunctions for Laplace-Beltrami operators, ESAIM Control Optim. Calc. Var., 19 (2013), 255-273.  doi: 10.1051/cocv/2012008.  Google Scholar

[29]

D. Maity, On the null controllability of the Lotka-Mckendrick system, working paper, 2018. Google Scholar

[30]

D. MaityM. 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.  doi: 10.1016/j.matpur.2018.12.006.  Google Scholar

[31]

N. Medhin, Optimal harvesting in age-structured populations, J. Optim. Theory Appl., 74 (1992), 413-423.  doi: 10.1007/BF00940318.  Google Scholar

[32]

S. MicuI. Roventa and M. Tucsnak, Time optimal boundary controls for the heat equation, Journal of Functional Analysis, 263 (2012), 25-49.  doi: 10.1016/j.jfa.2012.04.009.  Google Scholar

[33]

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

[34]

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.  doi: 10.1016/S0252-9602(18)30629-5.  Google Scholar

[35]

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

[36]

G. Wang, $L^\infty$-null controllability for the heat equation and its consequences for the time optimal control problem, SIAM J. Control Optim., 47 (2008), 1701-1720.  doi: 10.1137/060678191.  Google Scholar

[37]

G. Wang and C. Zhang, Observability inequalities from measurable sets for some abstract evolution equations, SIAM J. Control Optim., 55 (2017), 1862-1886.  doi: 10.1137/15M1051907.  Google Scholar

[38]

G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcel Dekker, Inc., New York, Dordrecht, 1985. Google Scholar

[39]

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. (electronic).  Google Scholar

[3]

B. Ainseba and M. Iannelli, Exact controllability of a nonlinear population-dynamics problem, Differential and Integral Equations. An International Journal for Theory & Applications, 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, Optimal harvesting for a nonlinear age-dependent population dynamics, J. Math. Anal. Appl., 226 (1998), 6-22.  doi: 10.1006/jmaa.1998.6064.  Google Scholar

[6]

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

[7]

S. Aniţa and N. Hegoburu, Null controllability via comparison results for nonlinear age-structured population dynamics, Mathematics of Control, Signals, and Systems, 31 (2019), Art. 2, 38pp. doi: 10.1007/s00498-019-0232-x.  Google Scholar

[8]

S. AniţaM. IannelliM.-Y. Kim and E.-J. Park, Optimal harvesting for periodic age-dependent population dynamics, SIAM J. Appl. Math., 58 (1998), 1648-1666.  doi: 10.1137/S0036139996301180.  Google Scholar

[9]

J. Apraiz and L. Escauriaza, Null-control and measurable sets, ESAIM Control Optim. Calc. Var.), 19 (2013), 239-254.  doi: 10.1051/cocv/2012005.  Google Scholar

[10]

J. ApraizL. EscauriazaG. Wang and C. Zhang, Observability inequalities and measurable sets, J. Eur. Math. Soc. (JEMS), 16 (2014), 2433-2475.  doi: 10.4171/JEMS/490.  Google Scholar

[11]

V. Barbu and M. Iannelli, Optimal control of population dynamics, J. Optim. Theory Appl., 102 (1999), 1-14.  doi: 10.1023/A:1021865709529.  Google Scholar

[12]

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

[13]

M. Brokate, Pontryagin's principle for control problems in age-dependent population dynamics, J. Math. Biol., 23 (1985), 75-101.  doi: 10.1007/BF00276559.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

N. Hritonenko and Y. Yatsenko, The structure of optimal time- and age-dependent harvesting in the Lotka-McKendrik population model, Math. Biosci., 208 (2007), 48-62.  doi: 10.1016/j.mbs.2006.09.008.  Google Scholar

[18]

N. Hritonenko, Y. Yatsenko, R.-U. Goetz and A. Xabadia, A bang–bang regime in optimal harvesting of size-structured populations, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), e2331–e2336. Google Scholar

[19]

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

[20]

M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Giardini Editori e Stampatori in Pisa, 1995. Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

J.-L. Lions, Contrôle Optimal de Systèmes Gouvernés par des Équations aux Dérivées Partielles, Dunod, Paris; Gauthier-Villars, Paris, 1968.  Google Scholar

[27]

Q. Lü, Bang-bang principle of time optimal controls and null controllability of fractional order parabolic equations, Acta Math. Sin. (Engl. Ser.), 26 (2010), 2377-2386.  doi: 10.1007/s10114-010-9051-1.  Google Scholar

[28]

Q. Lü, A lower bound on local energy of partial sum of eigenfunctions for Laplace-Beltrami operators, ESAIM Control Optim. Calc. Var., 19 (2013), 255-273.  doi: 10.1051/cocv/2012008.  Google Scholar

[29]

D. Maity, On the null controllability of the Lotka-Mckendrick system, working paper, 2018. Google Scholar

[30]

D. MaityM. 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.  doi: 10.1016/j.matpur.2018.12.006.  Google Scholar

[31]

N. Medhin, Optimal harvesting in age-structured populations, J. Optim. Theory Appl., 74 (1992), 413-423.  doi: 10.1007/BF00940318.  Google Scholar

[32]

S. MicuI. Roventa and M. Tucsnak, Time optimal boundary controls for the heat equation, Journal of Functional Analysis, 263 (2012), 25-49.  doi: 10.1016/j.jfa.2012.04.009.  Google Scholar

[33]

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

[34]

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.  doi: 10.1016/S0252-9602(18)30629-5.  Google Scholar

[35]

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

[36]

G. Wang, $L^\infty$-null controllability for the heat equation and its consequences for the time optimal control problem, SIAM J. Control Optim., 47 (2008), 1701-1720.  doi: 10.1137/060678191.  Google Scholar

[37]

G. Wang and C. Zhang, Observability inequalities from measurable sets for some abstract evolution equations, SIAM J. Control Optim., 55 (2017), 1862-1886.  doi: 10.1137/15M1051907.  Google Scholar

[38]

G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcel Dekker, Inc., New York, Dordrecht, 1985. Google Scholar

[39]

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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