Figure 1.
The spectrum of the free diffusion operator $ A_0 $ (green crosses) and of $ -\Delta $ (red circles)
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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 |
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.
Keywords:
Population dynamics,
controllability,
diffusion,
semigroup,
spectral decomposition,
bang-bang principle,
time optimal control.
Mathematics Subject Classification:
93B03, 93B05, 92D25, 34K35, 35Q93.
Citation:
Nicolas Hegoburu. Time optimal internal controls for the Lotka-McKendrick equation with spatial diffusion. Mathematical Control and Related Fields,
2019, 9
(4)
: 697-718.
doi: 10.3934/mcrf.2019047
![]()
References:
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Exact and approximate controllability of the age and space population dynamics structured model, Journal of Mathematical Analysis and Applications, 275 (2002), 562-574.
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Exact controllability of a nonlinear population-dynamics problem, Differential and Integral Equations. An International Journal for Theory & Applications, 16 (2003), 1369-1384.
|
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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. |
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S. Aniţa,
Optimal harvesting for a nonlinear age-dependent population dynamics, J. Math. Anal. Appl., 226 (1998), 6-22.
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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. |
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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.
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S. Aniţa, M. Iannelli, M.-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. |
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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. |
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J. Apraiz, L. Escauriaza, G. Wang and C. Zhang,
Observability inequalities and measurable sets, J. Eur. Math. Soc. (JEMS), 16 (2014), 2433-2475.
doi: 10.4171/JEMS/490. |
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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. |
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V. Barbu, M. 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. |
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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. |
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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. |
| [15] |
N. Hegoburu, P. 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. |
| [16] |
N. Hegoburu and M. Tucsnak,
Null controllability of the Lotka-McKendrick system with spatial diffusion, Math. Control Relat. Fields, 8 (2018), 707-720.
|
| [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. |
| [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. |
| [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. |
| [20] |
M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Giardini Editori e Stampatori in Pisa, 1995. |
| [21] |
H. Inaba, Age-structured Population Dynamics in Demography and Epidemiology, Springer, Singapore, 2017.
doi: 10.1007/978-981-10-0188-8. |
| [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. |
| [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. |
| [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. |
| [25] |
G. Lebeau and L. Robbiano,
Stabilisation de l'équation des ondes par le bord, Duke Math. J., 86 (1997), 465-491.
|
| [26] |
J.-L. Lions, Contrôle Optimal de Systèmes Gouvernés par des Équations aux Dérivées Partielles, Dunod, Paris; Gauthier-Villars, Paris, 1968. |
| [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. |
| [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. |
| [29] |
D. Maity, On the null controllability of the Lotka-Mckendrick system, working paper, 2018. |
| [30] |
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.
doi: 10.1016/j.matpur.2018.12.006. |
| [31] |
N. Medhin,
Optimal harvesting in age-structured populations, J. Optim. Theory Appl., 74 (1992), 413-423.
doi: 10.1007/BF00940318. |
| [32] |
S. Micu, I. 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. |
| [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. |
| [34] |
J. Song, J. Y. Yu, X. Z. Zhang, S. J. Hu, Z. X. Zhao, J. 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. |
| [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. |
| [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. |
| [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. |
| [38] |
G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcel Dekker, Inc., New York, Dordrecht, 1985. |
| [39] |
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. (electronic). |
| [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.
|
| [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,
Optimal harvesting for a nonlinear age-dependent population dynamics, J. Math. Anal. Appl., 226 (1998), 6-22.
doi: 10.1006/jmaa.1998.6064. |
| [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. |
| [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. |
| [8] |
S. Aniţa, M. Iannelli, M.-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. |
| [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. |
| [10] |
J. Apraiz, L. Escauriaza, G. Wang and C. Zhang,
Observability inequalities and measurable sets, J. Eur. Math. Soc. (JEMS), 16 (2014), 2433-2475.
doi: 10.4171/JEMS/490. |
| [11] |
V. Barbu and M. Iannelli,
Optimal control of population dynamics, J. Optim. Theory Appl., 102 (1999), 1-14.
doi: 10.1023/A:1021865709529. |
| [12] |
V. Barbu, M. 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. |
| [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. |
| [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. |
| [15] |
N. Hegoburu, P. 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. |
| [16] |
N. Hegoburu and M. Tucsnak,
Null controllability of the Lotka-McKendrick system with spatial diffusion, Math. Control Relat. Fields, 8 (2018), 707-720.
|
| [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. |
| [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. |
| [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. |
| [20] |
M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Giardini Editori e Stampatori in Pisa, 1995. |
| [21] |
H. Inaba, Age-structured Population Dynamics in Demography and Epidemiology, Springer, Singapore, 2017.
doi: 10.1007/978-981-10-0188-8. |
| [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. |
| [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. |
| [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. |
| [25] |
G. Lebeau and L. Robbiano,
Stabilisation de l'équation des ondes par le bord, Duke Math. J., 86 (1997), 465-491.
|
| [26] |
J.-L. Lions, Contrôle Optimal de Systèmes Gouvernés par des Équations aux Dérivées Partielles, Dunod, Paris; Gauthier-Villars, Paris, 1968. |
| [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. |
| [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. |
| [29] |
D. Maity, On the null controllability of the Lotka-Mckendrick system, working paper, 2018. |
| [30] |
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.
doi: 10.1016/j.matpur.2018.12.006. |
| [31] |
N. Medhin,
Optimal harvesting in age-structured populations, J. Optim. Theory Appl., 74 (1992), 413-423.
doi: 10.1007/BF00940318. |
| [32] |
S. Micu, I. 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. |
| [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. |
| [34] |
J. Song, J. Y. Yu, X. Z. Zhang, S. J. Hu, Z. X. Zhao, J. 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. |
| [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. |
| [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. |
| [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. |
| [38] |
G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcel Dekker, Inc., New York, Dordrecht, 1985. |
| [39] |
J. Zabczyk,
Remarks on the algebraic Riccati equation in Hilbert space, Appl. Math. Optim., 2 (1975/76), 251-258.
doi: 10.1007/BF01464270. |
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