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April 2017, 37(4): 2023-2043. doi: 10.3934/dcds.2017086

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

* Corresponding author: Mattia Zanella

Received  August 2016 Revised  October 2016 Published  December 2016

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.

Citation: Michael Herty, Mattia Zanella. Performance bounds for the mean-field limit of constrained dynamics. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2023-2043. doi: 10.3934/dcds.2017086
References:
[1]

G. AlbiM. BonginiE. 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. AlbiM. 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. BalaguéJ. 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. BonginiM. 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.

[13]

M. CaponigroM. FornasierB. 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. CaponigroM. FornasierB. 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. CaponigroA. 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.

[17]

R. CarmonaJ.-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. ColomboM. 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. CordierL. 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. CouzinJ. KrauseN. 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).

[26]

P. DegondJ.-G. LiuS. 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. DüringP. MarkowichJ.-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. GalamY. Gefen and Y. Shapir, Sociophysics: A new approach of sociological collective behavior, Journal of Mathematical Sociology, 9 (1982), 1-13.

[33]

G. GrimmM. J. MessinaS. 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. GrüneJ. PannekM. 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. HertyS. 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.

[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. MayneJ. B. RawlingsC. 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.

[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. AlbiM. BonginiE. 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. AlbiM. 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. BalaguéJ. 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. BonginiM. 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.

[13]

M. CaponigroM. FornasierB. 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. CaponigroM. FornasierB. 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. CaponigroA. 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.

[17]

R. CarmonaJ.-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. ColomboM. 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. CordierL. 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. CouzinJ. KrauseN. 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).

[26]

P. DegondJ.-G. LiuS. 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. DüringP. MarkowichJ.-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. GalamY. Gefen and Y. Shapir, Sociophysics: A new approach of sociological collective behavior, Journal of Mathematical Sociology, 9 (1982), 1-13.

[33]

G. GrimmM. J. MessinaS. 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. GrüneJ. PannekM. 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. HertyS. 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.

[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. MayneJ. B. RawlingsC. 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.

[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.

Figure 1.  Computation of $\alpha_N$ for different values of the regularization parameter $\nu$.
Figure 2.  Value of the cost functional $J_T^{u^{MPC}_N}(X_0)$ for controls obtained using a MPC strategy with control horizon $N$ (red) and presentation of the optimal costs $V_T^{*}(X_0)$ multiplied by $\frac{1}{\alpha_N}$ where $\alpha_N$ is computed as in [35,Theorem 5.4]. For $N\leq 4$ no estimate of the type (10) could be established.
Figure 3.  Experimental results for the optimization problem with varying optimization horizon $N$ and regularization constant $\nu=10^2$.
Figure 4.  Experimental results for the optimization problem with varying optimization horizon $N$ and regularization constant $\nu=10^3$.
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