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February  2020, 25(2): 761-779. doi: 10.3934/dcdsb.2019266

On an optimal control problem of time-fractional advection-diffusion equation

Qing Tang ,

China university of Geosciences, Wuhan, Hubei province, China

* Corresponding author: Qing Tang

Received  January 2019 Revised  April 2019 Published  November 2019

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.

Keywords: Optimal control, Caputo derivative, fractional Fokker-Planck equation, fractional Hamilton-Jacobi equation, variational Mean Field Games.
Mathematics Subject Classification: Primary: 35R11, 60H05, 26A33, 40L20.
Citation: Qing Tang. On an optimal control problem of time-fractional advection-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 761-779. doi: 10.3934/dcdsb.2019266
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[11]

F. Camilli and R. D. Maio, A time-fractional mean field game, Adv. Differential Equations, 24 (2019), 531-554.   Google Scholar

[12]

P. Cardaliaguet, Notes on Mean Field Games from P.L. Lions' lectures at Collége de France, 2010. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. I., Math. Z., 27 (1928), 565-606.  doi: 10.1007/BF01171116.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

F. Camilli and R. D. Maio, A time-fractional mean field game, Adv. Differential Equations, 24 (2019), 531-554.   Google Scholar

[12]

P. Cardaliaguet, Notes on Mean Field Games from P.L. Lions' lectures at Collége de France, 2010. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. I., Math. Z., 27 (1928), 565-606.  doi: 10.1007/BF01171116.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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