
-
Previous Article
Global well-posedness and large time behavior of classical solutions to the diffusion approximation model in radiation hydrodynamics
- DCDS Home
- This Issue
-
Next Article
A unified approach to weighted Hardy type inequalities on Carnot groups
Performance bounds for the mean-field limit of constrained dynamics
1. | Department of Mathematics, IGPM, RWTH Aachen University, Templergraben 55, Aachen, 52062, Germany |
2. | Department of Mathematics and Computer Science, University of Ferrara, Via N. Machiavelli 35, Ferrara, 44121, Italy |
In this work we are interested in the mean-field formulation of kinetic models under control actions where the control is formulated through a model predictive control strategy (MPC) with varying horizon. The relation between the (usually hard to compute) optimal control and the MPC approach is investigated theoretically in the mean-field limit. We establish a computable and provable bound on the difference in the cost functional for MPC controlled and optimal controlled system dynamics in the mean-field limit. The result of the present work extends previous findings for systems of ordinary differential equations. Numerical results in the mean-field setting are given.
References:
[1] |
G. Albi, M. Bongini, E. Cristiani and D. Kalise,
Invisible control of self-organizing agents leaving unknown environments, SIAM Journal of Applied Mathematics, 76 (2016), 1683-1710.
doi: 10.1137/15M1017016. |
[2] |
G. Albi, M. Herty and L. Pareschi,
Kinetic description of optimal control problems and applications to opinion consensus, Communications in Mathematical Sciences, 13 (2015), 1407-1429.
doi: 10.4310/CMS.2015.v13.n6.a3. |
[3] |
G. Albi, L. Pareschi and M. Zanella, Boltzmann type control of opinion consensus through leaders Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 372 (2014), 20140138, 18pp.
doi: 10.1098/rsta.2014.0138. |
[4] |
G. Albi, L. Pareschi and M. Zanella, Uncertainty quantification in control problems for flocking models Mathematical Problems in Engineering 2015 (2015), 14 pages.
doi: 10.1155/2015/850124. |
[5] |
D. Armbruster and C. Ringhofer,
Thermalized kinetic and fluid models for re-entrant supply chains, SIAM Journal on Multiscale Modeling & Simulation, 3 (2005), 782-800.
doi: 10.1137/030601636. |
[6] |
D. Balagué, J. A. Carrillo and Y. Yao,
Confinement for repulsive-attractive kernels, Discrete and Continuous Dynamical Systems -Series B, 19 (2014), 1227-1248.
doi: 10.3934/dcdsb.2014.19.1227. |
[7] |
N. Bellomo, G. A. Marsan and A. Tosin,
Complex Systems and Society. Modeling and Simulation SpringerBriefs in Mathematics, 2013.
doi: 10.1007/978-1-4614-7242-1. |
[8] |
N. Bellomo and J. Soler, On the mathematical theory of the dynamics of swarms viewed as complex systems Mathematical Models and Methods in Applied Sciences 22 (2012), 1140006, 29pp.
doi: 10.1142/S0218202511400069. |
[9] |
A. Bensoussan, J. Frehse and P. Yam,
Mean Field Games and Mean Field Type Control Theory SpringerBriefs in Mathematics, New York, 2013.
doi: 10.1007/978-1-4614-8508-7. |
[10] |
A. Blanchet and G. Carlier, From Nash to Cournot-Nash equilibria via the Monge–Kantorovich problem, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 372 (2014), 20130398, 11pp.
doi: 10.1098/rsta.2013.0398. |
[11] |
M. Bongini, M. Fornasier and D. Kalise,
(UN)conditional consensus emergence under perturbed and decentralized feedback controls, Discrete and Continuous Dynamical Systems -Series A, 35 (2015), 4071-4094.
doi: 10.3934/dcds.2015.35.4071. |
[12] |
E. F. Camacho and C. Bordons Alba, Model Predictive Control Springer-Verlag London, 2007. Google Scholar |
[13] |
M. Caponigro, M. Fornasier, B. Piccoli and E. Trélat,
Sparse stabilization and optimal control of the Cucker--Smale model, Mathematical Control and Related Fields, 3 (2013), 447-466.
doi: 10.3934/mcrf.2013.3.447. |
[14] |
M. Caponigro, M. Fornasier, B. Piccoli and E. Trélat,
Sparse stabilization and control of alignment models, Mathematical Models and Methods in Applied Sciences, 25 (2015), 521-564.
doi: 10.1142/S0218202515400059. |
[15] |
M. Caponigro, A. C. Lai and B. Piccoli,
A nonlinear model of opinion formation on the sphere, Discrete and Continuous Dynamical Systems -Series A, 35 (2015), 4241-4268.
doi: 10.3934/dcds.2015.35.4241. |
[16] |
P. Cardaliaguet, Notes on Mean Field Games P. -L. Lions' lectures at Collège de France, 2010. Google Scholar |
[17] |
R. Carmona, J.-P. Fouque and L.-H. Sun,
Mean field games and systemic risk, Communications in Mathematical Sciences, 13 (2015), 911-933.
doi: 10.4310/CMS.2015.v13.n4.a4. |
[18] |
J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic
models of swarming, Mathematical modeling of collective behavior in socio-economic and life
sciences, Model. Simul. Sci. Eng. Technol. , Birkh¨auser Boston, Inc. , Boston, MA, (2010),
297–336.
doi: 10.1007/978-0-8176-4946-3_12. |
[19] |
R. M. Colombo, M. Herty and M. Mercier,
Control of the continuity equation with a non local flow, ESAIM: Control, Optimization and Calculus of Variations, 17 (2011), 353-379.
doi: 10.1051/cocv/2010007. |
[20] |
R. M. Colombo and N. Pogodaev,
Confinement strategies in a model for the interaction between individuals and a continuum, SIAM Journal on Applied Dynamical Systems, 11 (2012), 741-770.
doi: 10.1137/110854321. |
[21] |
S. Cordier, L. Pareschi and G. Toscani,
On a kinetic model for a simple market economy, Journal of Statistical Physics, 120 (2005), 253-277.
doi: 10.1007/s10955-005-5456-0. |
[22] |
I. Couzin, J. Krause, N. Franks and S. Levin,
Effective leadership and decision-making in animal groups on the move, Nature, 433 (2005), 513-516.
doi: 10.1038/nature03236. |
[23] |
E. Cristiani, B. Piccoli and A. Tosin,
Multiscale Modeling of Pedestrian Dynamics MS & A: Modeling, Simulation and Applications, 12. Springer, Cham, 2014.
doi: 10.1007/978-3-319-06620-2. |
[24] |
F. Cucker and S. Smale,
Emergent behavior in flocks, IEEE Transactions on Automatic Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
[25] |
P. Degond, M. Herty and J. -G. Liu, Meanfield games and model predictive control, Preprint, arXiv(2014). Google Scholar |
[26] |
P. Degond, J.-G. Liu, S. Motsch and V. Panferov,
Hydrodynamic models of self-organized dynamics: Derivation and existence theory, Methods and Applications of Analysis, 20 (2013), 89-114.
doi: 10.4310/MAA.2013.v20.n2.a1. |
[27] |
P. Degond, J. -G. Liu and C. Ringhofer, Evolution of wealth in a nonconservative economy driven by local Nash equilibria Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 372 (2014), 20130394, 15pp.
doi: 10.1098/rsta.2013.0394. |
[28] |
M. Di Francesco and M. D. Rosini,
Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit, Archive for Rational Mechanics and Analysis, 217 (2015), 831-871.
doi: 10.1007/s00205-015-0843-4. |
[29] |
B. Düring, P. Markowich, J.-F. Pietschmann and M.-T. Wolfram,
Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 465 (2009), 3687-3708.
doi: 10.1098/rspa.2009.0239. |
[30] |
M. Fornasier, B. Piccoli and F. Rossi, Mean-field sparse optimal control Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 372 (2014), 20130400, 21pp.
doi: 10.1098/rsta.2013.0400. |
[31] |
M. Fornasier and F. Solombrino,
Mean-field optimal control, ESAIM: Control, Optimization, and Calculus of Variations, 20 (2014), 1123-1152.
doi: 10.1051/cocv/2014009. |
[32] |
S. Galam, Y. Gefen and Y. Shapir, Sociophysics: A new approach of sociological collective behavior, Journal of Mathematical Sociology, 9 (1982), 1-13. Google Scholar |
[33] |
G. Grimm, M. J. Messina, S. E. Tuna and A. R. Teel,
Model predictive control: For want of a local control Lyapunov function, all is not lost, IEEE Transactions on Automatic Control, 50 (2005), 546-558.
doi: 10.1109/TAC.2005.847055. |
[34] |
L. Grüne,
Analysis and design of unconstrained nonlinear MPC schemes for finite and infinite dimensional systems, SIAM Journal on Control and Optimization, 48 (2009), 1206-1228.
doi: 10.1137/070707853. |
[35] |
L. Grüne, J. Pannek, M. Seehafer and K. Worthmann,
Analysis of unconstrained nonlinear MPC schemes with time varying control horizon, SIAM Journal on Control and Optimization, 48 (2010), 4938-4962.
doi: 10.1137/090758696. |
[36] |
M. Herty and C. Ringhofer,
Feedback controls for continuous priority models in supply chain management, Computational Methods in Applied Mathematics, 11 (2011), 206-213.
doi: 10.2478/cmam-2011-0011. |
[37] |
M. Herty, S. Steffensen and L. Pareschi,
Mean-field control and Riccati equations, Networks and Heterogeneous Media, 10 (2015), 699-715.
doi: 10.3934/nhm.2015.10.699. |
[38] |
Y. Huang and A. Bertozzi,
Asymptotics of blowup solutions for the aggregation equation, Discrete and Continuous Dynamical Systems -Series B, 17 (2012), 1309-1331.
doi: 10.3934/dcdsb.2012.17.1309. |
[39] |
A. Jadbabaie and J. Hauser,
On the stability of receding horizon control with a general terminal cost, IEEE Transactions on Automatic Control, 50 (2005), 674-678.
doi: 10.1109/TAC.2005.846597. |
[40] |
M. Krstic, I. Kanellakopoulos and P. Kokotovic, Nonlinear and Adaptive Control Design John Wiley & Sons Inc. , New York, 1995. Google Scholar |
[41] |
J.-M. Lasry and P.-L. Lions,
Mean field games, Japanese Journal of Mathematics, 2 (2007), 229-260.
doi: 10.1007/s11537-007-0657-8. |
[42] |
D. Q. Mayne, J. B. Rawlings, C. V. Rao and P. O. M. Scokaert,
Constrained model predictive control: Stability and optimality, Automatica, 36 (2000), 789-814.
doi: 10.1016/S0005-1098(99)00214-9. |
[43] |
H. Michalska and D. Q. Mayne,
Robust receding horizon control of constrained nonlinear systems, IEEE Transactions on Automatic Control, 38 (1993), 1623-1633.
doi: 10.1109/9.262032. |
[44] |
H. Michalska and D. Q. Mayne,
Moving horizon observers and observer-based control, IEEE Transactions on Automatic Control, 40 (1995), 995-1006.
doi: 10.1109/9.388677. |
[45] |
S. Motsch and E. Tadmor,
A new model for self-organized dynamics and its flocking behavior, Journal of Statistical Physics, 144 (2011), 923-947.
doi: 10.1007/s10955-011-0285-9. |
[46] |
S. Motsch and E. Tadmor,
Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.
doi: 10.1137/120901866. |
[47] |
L. Pareschi and G. Toscani,
Self-similarity and power-like tails in nonconservative kinetic models, Journal of Statistical Physics, 124 (2006), 747-779.
doi: 10.1007/s10955-006-9025-y. |
[48] |
L. Pareschi and G. Toscani, Interacting Multi-Agent Systems. Kinetic Equations & Monte Carlo Methods Oxford University Press, 2013. Google Scholar |
[49] |
E. D. Sontag,
Mathematical Control Theory: Deterministic Finite Dimensional Systems Second edition. Texts in Applied Mathematics, 6. Springer-Verlag, New York, 1998.
doi: 10.1007/978-1-4612-0577-7. |
[50] |
G. Tadmor,
Receding horizon revisited: An easy way to robustly stabilize an LTV system, Systems Control Letters, 18 (1992), 285-294.
doi: 10.1016/0167-6911(92)90058-Z. |
[51] |
G. Toscani,
Kinetic models of opinion formation, Communications in Mathematical Sciences, 4 (2006), 481-496.
doi: 10.4310/CMS.2006.v4.n3.a1. |
show all references
References:
[1] |
G. Albi, M. Bongini, E. Cristiani and D. Kalise,
Invisible control of self-organizing agents leaving unknown environments, SIAM Journal of Applied Mathematics, 76 (2016), 1683-1710.
doi: 10.1137/15M1017016. |
[2] |
G. Albi, M. Herty and L. Pareschi,
Kinetic description of optimal control problems and applications to opinion consensus, Communications in Mathematical Sciences, 13 (2015), 1407-1429.
doi: 10.4310/CMS.2015.v13.n6.a3. |
[3] |
G. Albi, L. Pareschi and M. Zanella, Boltzmann type control of opinion consensus through leaders Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 372 (2014), 20140138, 18pp.
doi: 10.1098/rsta.2014.0138. |
[4] |
G. Albi, L. Pareschi and M. Zanella, Uncertainty quantification in control problems for flocking models Mathematical Problems in Engineering 2015 (2015), 14 pages.
doi: 10.1155/2015/850124. |
[5] |
D. Armbruster and C. Ringhofer,
Thermalized kinetic and fluid models for re-entrant supply chains, SIAM Journal on Multiscale Modeling & Simulation, 3 (2005), 782-800.
doi: 10.1137/030601636. |
[6] |
D. Balagué, J. A. Carrillo and Y. Yao,
Confinement for repulsive-attractive kernels, Discrete and Continuous Dynamical Systems -Series B, 19 (2014), 1227-1248.
doi: 10.3934/dcdsb.2014.19.1227. |
[7] |
N. Bellomo, G. A. Marsan and A. Tosin,
Complex Systems and Society. Modeling and Simulation SpringerBriefs in Mathematics, 2013.
doi: 10.1007/978-1-4614-7242-1. |
[8] |
N. Bellomo and J. Soler, On the mathematical theory of the dynamics of swarms viewed as complex systems Mathematical Models and Methods in Applied Sciences 22 (2012), 1140006, 29pp.
doi: 10.1142/S0218202511400069. |
[9] |
A. Bensoussan, J. Frehse and P. Yam,
Mean Field Games and Mean Field Type Control Theory SpringerBriefs in Mathematics, New York, 2013.
doi: 10.1007/978-1-4614-8508-7. |
[10] |
A. Blanchet and G. Carlier, From Nash to Cournot-Nash equilibria via the Monge–Kantorovich problem, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 372 (2014), 20130398, 11pp.
doi: 10.1098/rsta.2013.0398. |
[11] |
M. Bongini, M. Fornasier and D. Kalise,
(UN)conditional consensus emergence under perturbed and decentralized feedback controls, Discrete and Continuous Dynamical Systems -Series A, 35 (2015), 4071-4094.
doi: 10.3934/dcds.2015.35.4071. |
[12] |
E. F. Camacho and C. Bordons Alba, Model Predictive Control Springer-Verlag London, 2007. Google Scholar |
[13] |
M. Caponigro, M. Fornasier, B. Piccoli and E. Trélat,
Sparse stabilization and optimal control of the Cucker--Smale model, Mathematical Control and Related Fields, 3 (2013), 447-466.
doi: 10.3934/mcrf.2013.3.447. |
[14] |
M. Caponigro, M. Fornasier, B. Piccoli and E. Trélat,
Sparse stabilization and control of alignment models, Mathematical Models and Methods in Applied Sciences, 25 (2015), 521-564.
doi: 10.1142/S0218202515400059. |
[15] |
M. Caponigro, A. C. Lai and B. Piccoli,
A nonlinear model of opinion formation on the sphere, Discrete and Continuous Dynamical Systems -Series A, 35 (2015), 4241-4268.
doi: 10.3934/dcds.2015.35.4241. |
[16] |
P. Cardaliaguet, Notes on Mean Field Games P. -L. Lions' lectures at Collège de France, 2010. Google Scholar |
[17] |
R. Carmona, J.-P. Fouque and L.-H. Sun,
Mean field games and systemic risk, Communications in Mathematical Sciences, 13 (2015), 911-933.
doi: 10.4310/CMS.2015.v13.n4.a4. |
[18] |
J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic
models of swarming, Mathematical modeling of collective behavior in socio-economic and life
sciences, Model. Simul. Sci. Eng. Technol. , Birkh¨auser Boston, Inc. , Boston, MA, (2010),
297–336.
doi: 10.1007/978-0-8176-4946-3_12. |
[19] |
R. M. Colombo, M. Herty and M. Mercier,
Control of the continuity equation with a non local flow, ESAIM: Control, Optimization and Calculus of Variations, 17 (2011), 353-379.
doi: 10.1051/cocv/2010007. |
[20] |
R. M. Colombo and N. Pogodaev,
Confinement strategies in a model for the interaction between individuals and a continuum, SIAM Journal on Applied Dynamical Systems, 11 (2012), 741-770.
doi: 10.1137/110854321. |
[21] |
S. Cordier, L. Pareschi and G. Toscani,
On a kinetic model for a simple market economy, Journal of Statistical Physics, 120 (2005), 253-277.
doi: 10.1007/s10955-005-5456-0. |
[22] |
I. Couzin, J. Krause, N. Franks and S. Levin,
Effective leadership and decision-making in animal groups on the move, Nature, 433 (2005), 513-516.
doi: 10.1038/nature03236. |
[23] |
E. Cristiani, B. Piccoli and A. Tosin,
Multiscale Modeling of Pedestrian Dynamics MS & A: Modeling, Simulation and Applications, 12. Springer, Cham, 2014.
doi: 10.1007/978-3-319-06620-2. |
[24] |
F. Cucker and S. Smale,
Emergent behavior in flocks, IEEE Transactions on Automatic Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
[25] |
P. Degond, M. Herty and J. -G. Liu, Meanfield games and model predictive control, Preprint, arXiv(2014). Google Scholar |
[26] |
P. Degond, J.-G. Liu, S. Motsch and V. Panferov,
Hydrodynamic models of self-organized dynamics: Derivation and existence theory, Methods and Applications of Analysis, 20 (2013), 89-114.
doi: 10.4310/MAA.2013.v20.n2.a1. |
[27] |
P. Degond, J. -G. Liu and C. Ringhofer, Evolution of wealth in a nonconservative economy driven by local Nash equilibria Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 372 (2014), 20130394, 15pp.
doi: 10.1098/rsta.2013.0394. |
[28] |
M. Di Francesco and M. D. Rosini,
Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit, Archive for Rational Mechanics and Analysis, 217 (2015), 831-871.
doi: 10.1007/s00205-015-0843-4. |
[29] |
B. Düring, P. Markowich, J.-F. Pietschmann and M.-T. Wolfram,
Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 465 (2009), 3687-3708.
doi: 10.1098/rspa.2009.0239. |
[30] |
M. Fornasier, B. Piccoli and F. Rossi, Mean-field sparse optimal control Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 372 (2014), 20130400, 21pp.
doi: 10.1098/rsta.2013.0400. |
[31] |
M. Fornasier and F. Solombrino,
Mean-field optimal control, ESAIM: Control, Optimization, and Calculus of Variations, 20 (2014), 1123-1152.
doi: 10.1051/cocv/2014009. |
[32] |
S. Galam, Y. Gefen and Y. Shapir, Sociophysics: A new approach of sociological collective behavior, Journal of Mathematical Sociology, 9 (1982), 1-13. Google Scholar |
[33] |
G. Grimm, M. J. Messina, S. E. Tuna and A. R. Teel,
Model predictive control: For want of a local control Lyapunov function, all is not lost, IEEE Transactions on Automatic Control, 50 (2005), 546-558.
doi: 10.1109/TAC.2005.847055. |
[34] |
L. Grüne,
Analysis and design of unconstrained nonlinear MPC schemes for finite and infinite dimensional systems, SIAM Journal on Control and Optimization, 48 (2009), 1206-1228.
doi: 10.1137/070707853. |
[35] |
L. Grüne, J. Pannek, M. Seehafer and K. Worthmann,
Analysis of unconstrained nonlinear MPC schemes with time varying control horizon, SIAM Journal on Control and Optimization, 48 (2010), 4938-4962.
doi: 10.1137/090758696. |
[36] |
M. Herty and C. Ringhofer,
Feedback controls for continuous priority models in supply chain management, Computational Methods in Applied Mathematics, 11 (2011), 206-213.
doi: 10.2478/cmam-2011-0011. |
[37] |
M. Herty, S. Steffensen and L. Pareschi,
Mean-field control and Riccati equations, Networks and Heterogeneous Media, 10 (2015), 699-715.
doi: 10.3934/nhm.2015.10.699. |
[38] |
Y. Huang and A. Bertozzi,
Asymptotics of blowup solutions for the aggregation equation, Discrete and Continuous Dynamical Systems -Series B, 17 (2012), 1309-1331.
doi: 10.3934/dcdsb.2012.17.1309. |
[39] |
A. Jadbabaie and J. Hauser,
On the stability of receding horizon control with a general terminal cost, IEEE Transactions on Automatic Control, 50 (2005), 674-678.
doi: 10.1109/TAC.2005.846597. |
[40] |
M. Krstic, I. Kanellakopoulos and P. Kokotovic, Nonlinear and Adaptive Control Design John Wiley & Sons Inc. , New York, 1995. Google Scholar |
[41] |
J.-M. Lasry and P.-L. Lions,
Mean field games, Japanese Journal of Mathematics, 2 (2007), 229-260.
doi: 10.1007/s11537-007-0657-8. |
[42] |
D. Q. Mayne, J. B. Rawlings, C. V. Rao and P. O. M. Scokaert,
Constrained model predictive control: Stability and optimality, Automatica, 36 (2000), 789-814.
doi: 10.1016/S0005-1098(99)00214-9. |
[43] |
H. Michalska and D. Q. Mayne,
Robust receding horizon control of constrained nonlinear systems, IEEE Transactions on Automatic Control, 38 (1993), 1623-1633.
doi: 10.1109/9.262032. |
[44] |
H. Michalska and D. Q. Mayne,
Moving horizon observers and observer-based control, IEEE Transactions on Automatic Control, 40 (1995), 995-1006.
doi: 10.1109/9.388677. |
[45] |
S. Motsch and E. Tadmor,
A new model for self-organized dynamics and its flocking behavior, Journal of Statistical Physics, 144 (2011), 923-947.
doi: 10.1007/s10955-011-0285-9. |
[46] |
S. Motsch and E. Tadmor,
Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.
doi: 10.1137/120901866. |
[47] |
L. Pareschi and G. Toscani,
Self-similarity and power-like tails in nonconservative kinetic models, Journal of Statistical Physics, 124 (2006), 747-779.
doi: 10.1007/s10955-006-9025-y. |
[48] |
L. Pareschi and G. Toscani, Interacting Multi-Agent Systems. Kinetic Equations & Monte Carlo Methods Oxford University Press, 2013. Google Scholar |
[49] |
E. D. Sontag,
Mathematical Control Theory: Deterministic Finite Dimensional Systems Second edition. Texts in Applied Mathematics, 6. Springer-Verlag, New York, 1998.
doi: 10.1007/978-1-4612-0577-7. |
[50] |
G. Tadmor,
Receding horizon revisited: An easy way to robustly stabilize an LTV system, Systems Control Letters, 18 (1992), 285-294.
doi: 10.1016/0167-6911(92)90058-Z. |
[51] |
G. Toscani,
Kinetic models of opinion formation, Communications in Mathematical Sciences, 4 (2006), 481-496.
doi: 10.4310/CMS.2006.v4.n3.a1. |



[1] |
Patrick Gerard, Christophe Pallard. A mean-field toy model for resonant transport. Kinetic & Related Models, 2010, 3 (2) : 299-309. doi: 10.3934/krm.2010.3.299 |
[2] |
Franco Flandoli, Enrico Priola, Giovanni Zanco. A mean-field model with discontinuous coefficients for neurons with spatial interaction. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3037-3067. doi: 10.3934/dcds.2019126 |
[3] |
Seung-Yeal Ha, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. Uniform stability and mean-field limit for the augmented Kuramoto model. Networks & Heterogeneous Media, 2018, 13 (2) : 297-322. doi: 10.3934/nhm.2018013 |
[4] |
Salah Eddine Choutri, Boualem Djehiche, Hamidou Tembine. Optimal control and zero-sum games for Markov chains of mean-field type. Mathematical Control & Related Fields, 2019, 9 (3) : 571-605. doi: 10.3934/mcrf.2019026 |
[5] |
Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control & Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97 |
[6] |
Xun Li, Jingrui Sun, Jiongmin Yong. Mean-field stochastic linear quadratic optimal control problems: closed-loop solvability. Probability, Uncertainty and Quantitative Risk, 2016, 1 (0) : 2-. doi: 10.1186/s41546-016-0002-3 |
[7] |
Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026 |
[8] |
Hancheng Guo, Jie Xiong. A second-order stochastic maximum principle for generalized mean-field singular control problem. Mathematical Control & Related Fields, 2018, 8 (2) : 451-473. doi: 10.3934/mcrf.2018018 |
[9] |
Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1891-1921. doi: 10.3934/dcds.2019080 |
[10] |
Seung-Yeal Ha, Jeongho Kim, Peter Pickl, Xiongtao Zhang. A probabilistic approach for the mean-field limit to the Cucker-Smale model with a singular communication. Kinetic & Related Models, 2019, 12 (5) : 1045-1067. doi: 10.3934/krm.2019039 |
[11] |
Seung-Yeal Ha, Jeongho Kim, Xiongtao Zhang. Uniform stability of the Cucker-Smale model and its application to the Mean-Field limit. Kinetic & Related Models, 2018, 11 (5) : 1157-1181. doi: 10.3934/krm.2018045 |
[12] |
Pierre-Emmanuel Jabin. A review of the mean field limits for Vlasov equations. Kinetic & Related Models, 2014, 7 (4) : 661-711. doi: 10.3934/krm.2014.7.661 |
[13] |
Adel Chala, Dahbia Hafayed. On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application. Evolution Equations & Control Theory, 2020, 9 (3) : 817-843. doi: 10.3934/eect.2020035 |
[14] |
Lars Grüne, Marleen Stieler. Multiobjective model predictive control for stabilizing cost criteria. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3905-3928. doi: 10.3934/dcdsb.2018336 |
[15] |
Rudy R. Negenborn, Peter-Jules van Overloop, Tamás Keviczky, Bart De Schutter. Distributed model predictive control of irrigation canals. Networks & Heterogeneous Media, 2009, 4 (2) : 359-380. doi: 10.3934/nhm.2009.4.359 |
[16] |
Torsten Trimborn, Lorenzo Pareschi, Martin Frank. Portfolio optimization and model predictive control: A kinetic approach. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6209-6238. doi: 10.3934/dcdsb.2019136 |
[17] |
Illés Horváth, Kristóf Attila Horváth, Péter Kovács, Miklós Telek. Mean-field analysis of a scaling MAC radio protocol. Journal of Industrial & Management Optimization, 2021, 17 (1) : 279-297. doi: 10.3934/jimo.2019111 |
[18] |
Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Mean-field backward stochastic Volterra integral equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1929-1967. doi: 10.3934/dcdsb.2013.18.1929 |
[19] |
Diogo A. Gomes, Gabriel E. Pires, Héctor Sánchez-Morgado. A-priori estimates for stationary mean-field games. Networks & Heterogeneous Media, 2012, 7 (2) : 303-314. doi: 10.3934/nhm.2012.7.303 |
[20] |
Gerasimenko Viktor. Heisenberg picture of quantum kinetic evolution in mean-field limit. Kinetic & Related Models, 2011, 4 (1) : 385-399. doi: 10.3934/krm.2011.4.385 |
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]