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On an optimal control problem of time-fractional advection-diffusion equation
China university of Geosciences, Wuhan, Hubei province, China |
We consider an optimal control problem of an advection-diffusion equation with Caputo time-fractional derivative. By convex duality method we obtain as optimality condition a forward-backward coupled system. We then prove the existence of a solution to this coupled system using Schauder fixed point theorem. The uniqueness of the solution is also established under certain monotonicity condition on the cost functional.
References:
| [1] |
Y. Achdou, M. Bardi and M. Cirant,
Mean field games models of segregation, Math. Models Methods Appl. Sci., 27 (2017), 75-113.
doi: 10.1142/S0218202517400036. |
| [2] |
M. Allen, L. Caffarelli and A. Vasseur,
A parabolic problem with a fractional time derivative, Arch. Ration. Mech. Anal., 221 (2016), 603-630.
doi: 10.1007/s00205-016-0969-z. |
| [3] |
M. Annunziato, A. Borzì, M. Magdziarz and A. Weron,
A fractional Fokker-Planck control framework for subdiffusion processes, Optimal Control Appl. Methods, 37 (2016), 290-304.
doi: 10.1002/oca.2168. |
| [4] |
H. Antil, E. Otárola and A. J. Salgado,
A space-time fractional optimal control problem: Analysis and discretization, SIAM J. Control Optim., 54 (2016), 1295-1328.
doi: 10.1137/15M1014991. |
| [5] |
J.-D. Benamou and Y. Brenier,
A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393.
doi: 10.1007/s002110050002. |
| [6] |
J.-D. Benamou, G. Carlier and F. Santambrogio, Variational mean field games, in Active Particles, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 1 (2017), 141–171.
doi: 10.1007/978-3-319-49996-3_4. |
| [7] |
J.-D. Benamou, G. Carlier, S. D. Marino and L. Nenna,
An entropy minimization approach to second-order variational mean-field games, Math. Models Methods Appl. Sci., 29 (2019), 1553-1583.
doi: 10.1142/S0218202519500283. |
| [8] |
J.-P. Bouchaud and A. Georges,
Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep., 195 (1990), 127-293.
doi: 10.1016/0370-1573(90)90099-N. |
| [9] |
M. Burger, M. D. Francesco, P. A. Markowich and M.-T. Wolfram,
Mean field games with nonlinear mobilities in pedestrian dynamics, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1311-1333.
doi: 10.3934/dcdsb.2014.19.1311. |
| [10] |
F. Camilli, R. D. Maio and E. Iacomini,
A Hopf-Lax formula for Hamilton-Jacobi equations with Caputo time derivative, J. Math. Anal. Appl., 477 (2019), 1019-1032.
doi: 10.1016/j.jmaa.2019.04.069. |
| [11] |
F. Camilli and R. D. Maio,
A time-fractional mean field game, Adv. Differential Equations, 24 (2019), 531-554.
|
| [12] |
P. Cardaliaguet, Notes on Mean Field Games from P.L. Lions' lectures at Collége de France, 2010. |
| [13] |
P. Cardaliaguet, Weak solutions for first order mean field games with local coupling, in Analysis and Geometry in Control Theory and its Applications, Springer INdAM Ser., Spriner, Cham, 11 (2015), 111–158.
doi: 10.1007/978-3-319-06917-3_5. |
| [14] |
P. Cardaliaguet, P. J. Graber, A. Porretta and D. Tonon,
Second order mean field games with degenerate diffusion and local coupling, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1287-1317.
doi: 10.1007/s00030-015-0323-4. |
| [15] |
A. Cesaroni, M. Cirant, S. Dipierro, M. Novaga and E. Valdinoci,
On stationary fractional mean field games, J. Math. Pures Appl., 122 (2019), 1-22.
doi: 10.1016/j.matpur.2017.10.013. |
| [16] |
M. Cirant and A. Goffi,
On the existence and uniqueness of solutions to time-dependent fractional MFG, SIAM J. Math. Anal., 51 (2019), 913-954.
doi: 10.1137/18M1216420. |
| [17] |
D. A. Gomes and J. Saúde,
Mean field games models——A brief survey, Dyn. Games Appl., 4 (2014), 110-154.
doi: 10.1007/s13235-013-0099-2. |
| [18] |
Y. Giga and T. Namba,
Well-posedness of Hamilton-Jacobi equations with Caputo's time fractional derivative, Comm. Partial Differential Equations, 42 (2017), 1088-1120.
doi: 10.1080/03605302.2017.1324880. |
| [19] |
O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications, in Paris-Princeton Lectures on Mathematical Finance 2010, Lecture Notes in Math., 2003, Springer, Berlin, 2011,205–266.
doi: 10.1007/978-3-642-14660-2_3. |
| [20] |
M. Hahn, K. Kobayashi and S. Umarov,
SDEs driven by a time-changed Lévy process and their associated time-fractional order pseudo-differential equations, J. Theoret. Probab., 25 (2012), 262-279.
doi: 10.1007/s10959-010-0289-4. |
| [21] |
G. H. Hardy and J. E. Littlewood,
Some properties of fractional integrals. I., Math. Z., 27 (1928), 565-606.
doi: 10.1007/BF01171116. |
| [22] |
M. E. Hernández-Hernández and V. N. Kolokoltsov,
Probabilistic solutions to nonlinear fractional differential equations of generalized Caputo and Riemann–Liouville type, Stochastics, 90 (2018), 224-255.
doi: 10.1080/17442508.2017.1334059. |
| [23] |
M. Huang, P. E. Caines and R. P. Malhame,
Large-population cost-coupled LQG problems with non uniform agents: Individual-mass behaviour and decentralized ε-Nash equilibria, IEEE Trans. Automat. Control, 52 (2007), 1560-1571.
doi: 10.1109/TAC.2007.904450. |
| [24] |
J. Kemppainen and R. Zacher,
Long-time behaviour of non-local in time Fokker-Planck equations via the entropy method, Math. Models Methods Appl. Sci., 29 (2019), 209-235.
doi: 10.1142/S0218202519500076. |
| [25] |
V. Kolokoltsov and M. Veretennikova, A fractional Hamilton-Jacobi Bellman equation for scaled limits of controlled continuous time random walks, Commun. Appl. Ind. Math., 6 (2014), e-484, 18 pp.
doi: 10.1685/journal.caim.484. |
| [26] |
V. Kolokoltsov and M. Veretennikova,
Well-posedness and regularity of the Cauchy problem for nonlinear fractional in time and space equations, Fract. Differ. Calc., 4 (2014), 1-30.
doi: 10.7153/fdc-04-01. |
| [27] |
A. Kubica and M. Yamamoto,
Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients, Fract. Calc. Appl. Anal., 21 (2018), 276-311.
doi: 10.1515/fca-2018-0018. |
| [28] |
J.-M. Lasry and P.-L. Lions,
Jeux à champ moyen. Ⅱ – Horizon fini et contrôle optimal, C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684.
doi: 10.1016/j.crma.2006.09.018. |
| [29] |
J.-M. Lasry and P.-L. Lions,
Mean field games, Jpn. J. Math., 2 (2007), 229-260.
doi: 10.1007/s11537-007-0657-8. |
| [30] |
L. Li and J. Liu,
Some compactness criteria for weak solutions of time fractional PDEs, SIAM J. Math. Anal., 50 (2018), 3963-3995.
doi: 10.1137/17M1145549. |
| [31] |
L. Li, J.-G. Liu and L. Wang,
Cauchy problems for Keller–Segel type time–space fractional diffusion equation, J. Differential Equations, 265 (2018), 1044-1096.
doi: 10.1016/j.jde.2018.03.025. |
| [32] |
A. R. Mészáros and F. J. Silva,
A variational approach to second order mean field games with density constraints: The stationary case, J. Math. Pures Appl., 104 (2015), 1135-1159.
doi: 10.1016/j.matpur.2015.07.008. |
| [33] |
R. Metzler and J. Klafter,
The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.
doi: 10.1016/S0370-1573(00)00070-3. |
| [34] |
S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. |
| [35] |
Q. Tang and F. Camilli, Variational time-fractional mean field games, Dynamic Games and Applications, Springer US, 2019, 1–16.
doi: 10.1007/s13235-019-00330-2. |
| [36] |
E. Topp and M. Yangari,
Existence and uniqueness for parabolic problems with Caputo time derivative, J. Differential Equations, 262 (2017), 6018-6046.
doi: 10.1016/j.jde.2017.02.024. |
| [37] |
R. Zacher,
Boundedness of weak solutions to evolutionary partial integro-differential equations with discontinuous coefficients, J. Math. Anal. Appl., 348 (2008), 137-149.
doi: 10.1016/j.jmaa.2008.06.054. |
| [38] |
R. Zacher,
A De Giorgi–Nash type theorem for time fractional diffusion equations, Math. Ann., 356 (2013), 99-146.
doi: 10.1007/s00208-012-0834-9. |
show all references
References:
| [1] |
Y. Achdou, M. Bardi and M. Cirant,
Mean field games models of segregation, Math. Models Methods Appl. Sci., 27 (2017), 75-113.
doi: 10.1142/S0218202517400036. |
| [2] |
M. Allen, L. Caffarelli and A. Vasseur,
A parabolic problem with a fractional time derivative, Arch. Ration. Mech. Anal., 221 (2016), 603-630.
doi: 10.1007/s00205-016-0969-z. |
| [3] |
M. Annunziato, A. Borzì, M. Magdziarz and A. Weron,
A fractional Fokker-Planck control framework for subdiffusion processes, Optimal Control Appl. Methods, 37 (2016), 290-304.
doi: 10.1002/oca.2168. |
| [4] |
H. Antil, E. Otárola and A. J. Salgado,
A space-time fractional optimal control problem: Analysis and discretization, SIAM J. Control Optim., 54 (2016), 1295-1328.
doi: 10.1137/15M1014991. |
| [5] |
J.-D. Benamou and Y. Brenier,
A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393.
doi: 10.1007/s002110050002. |
| [6] |
J.-D. Benamou, G. Carlier and F. Santambrogio, Variational mean field games, in Active Particles, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 1 (2017), 141–171.
doi: 10.1007/978-3-319-49996-3_4. |
| [7] |
J.-D. Benamou, G. Carlier, S. D. Marino and L. Nenna,
An entropy minimization approach to second-order variational mean-field games, Math. Models Methods Appl. Sci., 29 (2019), 1553-1583.
doi: 10.1142/S0218202519500283. |
| [8] |
J.-P. Bouchaud and A. Georges,
Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep., 195 (1990), 127-293.
doi: 10.1016/0370-1573(90)90099-N. |
| [9] |
M. Burger, M. D. Francesco, P. A. Markowich and M.-T. Wolfram,
Mean field games with nonlinear mobilities in pedestrian dynamics, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1311-1333.
doi: 10.3934/dcdsb.2014.19.1311. |
| [10] |
F. Camilli, R. D. Maio and E. Iacomini,
A Hopf-Lax formula for Hamilton-Jacobi equations with Caputo time derivative, J. Math. Anal. Appl., 477 (2019), 1019-1032.
doi: 10.1016/j.jmaa.2019.04.069. |
| [11] |
F. Camilli and R. D. Maio,
A time-fractional mean field game, Adv. Differential Equations, 24 (2019), 531-554.
|
| [12] |
P. Cardaliaguet, Notes on Mean Field Games from P.L. Lions' lectures at Collége de France, 2010. |
| [13] |
P. Cardaliaguet, Weak solutions for first order mean field games with local coupling, in Analysis and Geometry in Control Theory and its Applications, Springer INdAM Ser., Spriner, Cham, 11 (2015), 111–158.
doi: 10.1007/978-3-319-06917-3_5. |
| [14] |
P. Cardaliaguet, P. J. Graber, A. Porretta and D. Tonon,
Second order mean field games with degenerate diffusion and local coupling, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1287-1317.
doi: 10.1007/s00030-015-0323-4. |
| [15] |
A. Cesaroni, M. Cirant, S. Dipierro, M. Novaga and E. Valdinoci,
On stationary fractional mean field games, J. Math. Pures Appl., 122 (2019), 1-22.
doi: 10.1016/j.matpur.2017.10.013. |
| [16] |
M. Cirant and A. Goffi,
On the existence and uniqueness of solutions to time-dependent fractional MFG, SIAM J. Math. Anal., 51 (2019), 913-954.
doi: 10.1137/18M1216420. |
| [17] |
D. A. Gomes and J. Saúde,
Mean field games models——A brief survey, Dyn. Games Appl., 4 (2014), 110-154.
doi: 10.1007/s13235-013-0099-2. |
| [18] |
Y. Giga and T. Namba,
Well-posedness of Hamilton-Jacobi equations with Caputo's time fractional derivative, Comm. Partial Differential Equations, 42 (2017), 1088-1120.
doi: 10.1080/03605302.2017.1324880. |
| [19] |
O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications, in Paris-Princeton Lectures on Mathematical Finance 2010, Lecture Notes in Math., 2003, Springer, Berlin, 2011,205–266.
doi: 10.1007/978-3-642-14660-2_3. |
| [20] |
M. Hahn, K. Kobayashi and S. Umarov,
SDEs driven by a time-changed Lévy process and their associated time-fractional order pseudo-differential equations, J. Theoret. Probab., 25 (2012), 262-279.
doi: 10.1007/s10959-010-0289-4. |
| [21] |
G. H. Hardy and J. E. Littlewood,
Some properties of fractional integrals. I., Math. Z., 27 (1928), 565-606.
doi: 10.1007/BF01171116. |
| [22] |
M. E. Hernández-Hernández and V. N. Kolokoltsov,
Probabilistic solutions to nonlinear fractional differential equations of generalized Caputo and Riemann–Liouville type, Stochastics, 90 (2018), 224-255.
doi: 10.1080/17442508.2017.1334059. |
| [23] |
M. Huang, P. E. Caines and R. P. Malhame,
Large-population cost-coupled LQG problems with non uniform agents: Individual-mass behaviour and decentralized ε-Nash equilibria, IEEE Trans. Automat. Control, 52 (2007), 1560-1571.
doi: 10.1109/TAC.2007.904450. |
| [24] |
J. Kemppainen and R. Zacher,
Long-time behaviour of non-local in time Fokker-Planck equations via the entropy method, Math. Models Methods Appl. Sci., 29 (2019), 209-235.
doi: 10.1142/S0218202519500076. |
| [25] |
V. Kolokoltsov and M. Veretennikova, A fractional Hamilton-Jacobi Bellman equation for scaled limits of controlled continuous time random walks, Commun. Appl. Ind. Math., 6 (2014), e-484, 18 pp.
doi: 10.1685/journal.caim.484. |
| [26] |
V. Kolokoltsov and M. Veretennikova,
Well-posedness and regularity of the Cauchy problem for nonlinear fractional in time and space equations, Fract. Differ. Calc., 4 (2014), 1-30.
doi: 10.7153/fdc-04-01. |
| [27] |
A. Kubica and M. Yamamoto,
Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients, Fract. Calc. Appl. Anal., 21 (2018), 276-311.
doi: 10.1515/fca-2018-0018. |
| [28] |
J.-M. Lasry and P.-L. Lions,
Jeux à champ moyen. Ⅱ – Horizon fini et contrôle optimal, C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684.
doi: 10.1016/j.crma.2006.09.018. |
| [29] |
J.-M. Lasry and P.-L. Lions,
Mean field games, Jpn. J. Math., 2 (2007), 229-260.
doi: 10.1007/s11537-007-0657-8. |
| [30] |
L. Li and J. Liu,
Some compactness criteria for weak solutions of time fractional PDEs, SIAM J. Math. Anal., 50 (2018), 3963-3995.
doi: 10.1137/17M1145549. |
| [31] |
L. Li, J.-G. Liu and L. Wang,
Cauchy problems for Keller–Segel type time–space fractional diffusion equation, J. Differential Equations, 265 (2018), 1044-1096.
doi: 10.1016/j.jde.2018.03.025. |
| [32] |
A. R. Mészáros and F. J. Silva,
A variational approach to second order mean field games with density constraints: The stationary case, J. Math. Pures Appl., 104 (2015), 1135-1159.
doi: 10.1016/j.matpur.2015.07.008. |
| [33] |
R. Metzler and J. Klafter,
The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.
doi: 10.1016/S0370-1573(00)00070-3. |
| [34] |
S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. |
| [35] |
Q. Tang and F. Camilli, Variational time-fractional mean field games, Dynamic Games and Applications, Springer US, 2019, 1–16.
doi: 10.1007/s13235-019-00330-2. |
| [36] |
E. Topp and M. Yangari,
Existence and uniqueness for parabolic problems with Caputo time derivative, J. Differential Equations, 262 (2017), 6018-6046.
doi: 10.1016/j.jde.2017.02.024. |
| [37] |
R. Zacher,
Boundedness of weak solutions to evolutionary partial integro-differential equations with discontinuous coefficients, J. Math. Anal. Appl., 348 (2008), 137-149.
doi: 10.1016/j.jmaa.2008.06.054. |
| [38] |
R. Zacher,
A De Giorgi–Nash type theorem for time fractional diffusion equations, Math. Ann., 356 (2013), 99-146.
doi: 10.1007/s00208-012-0834-9. |
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