We derive a framework to compute optimal controls for problems with states in the space of probability measures. Since many optimal control problems constrained by a system of ordinary differential equations modeling interacting particles converge to optimal control problems constrained by a partial differential equation in the mean-field limit, it is interesting to have a calculus directly on the mesoscopic level of probability measures which allows us to derive the corresponding first-order optimality system. In addition to this new calculus, we provide relations for the resulting system to the first-order optimality system derived on the particle level and the first-order optimality system based on $L^2$-calculus under additional regularity assumptions. We further justify the use of the $L^2$-adjoint in numerical simulations by establishing a link between the adjoint in the space of probability measures and the adjoint corresponding to $L^2$-calculus. Moreover, we prove a convergence rate for the convergence of the optimal controls corresponding to the particle formulation to the optimal controls of the mean-field problem as the number of particles tends to infinity.


  1. optimal control with ODE/PDE constraints
  2. interacting particle systems
  3. mean-field limits

MSC codes

  1. 49K15
  2. 49K20

Get full access to this article

View all available purchase options and get full access to this article.


G. Albi, Y.-P. Choi, M. Fornasier, and D. Kalise, Mean field control hierarchy, Appl. Math. Optim., 76 (2017), pp. 93--135, https://doi.org/10.1007/s00245-017-9429-x.
G. Albi and D. Kalise, (Sub)optimal feedback control of mean field multi-population dynamics, IFAC PapersOnLine, 51 (2018), pp. 86--91, https://doi.org/10.1016/j.ifacol.2018.06.020.
L. Ambrosio, N. Gigli, and G. Savaré, Gradient Flows: In Metric Spaces and in the Space of Probability Measures, Springer, New York, 2008.
R. Bailo, M. Bongini, J. Carrillo, and D. Kalise, Optimal consensus control of the Cucker-Smale model, IFAC PapersOnLine, 51 (2018), pp. 1--6, https://doi.org/10.1016/j.ifacol.2018.07.245.
M. Bongini and G. Buttazzo, Optimal control problems in transport dynamics, Math. Models Methods Appl. Sci., 27 (2017), pp. 427--451.
M. Bongini, M. Fornasier, F. Rossi, and F. Solombrino, Mean-field Pontryagin maximum principle, J. Optim. Theory Appl., 175 (2017), pp. 1--38, https://doi.org/10.1007/s10957-017-1149-5.
B. Bonnet and F. Rossi, The Pontryagin maximum principle in the Wasserstein space, Calc. Var., 58 (2019).
W. Braun and K. Hepp, The Vlasov dynamics and its fluctuations in the $1/N$ limit of interacting classical particles, Comm. Math. Phys., 56 (1977), pp. 101--113.
M. Burger, R. Pinnau, A. Roth, C. Totzeck, and O. Tse, Controlling a Self-Organizing System of Individuals Guided by a Few External Agents---Particle description and Mean-Field Limit, preprint, 2016.
J. Carrillo, Y.-P. Choi, C. Totzeck, and O. Tse, An analytical framework for a consensus-based global optimization method, accepted for publication in Math. Models Methods Appl. Sci., 28 (2018), pp. 1037--1066.
J. A. Carrillo, Y. Choi, and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, in Collective Dynamics from Bacteria to Crowds, CISM 553, Springer, Berlin, 2014, pp. 1--46.
J. A. Carrillo, M. Fornasier, G. Toscani, and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, G. Naldi, L. Pareschi, and G. Toscani, eds., Birkhäuser Boston, Boston, MA, 2010, pp. 297--336.
E. Casas and F. Tröltzsch, Second order optimality conditions and their role in PDE control, Jahresber. Dtsch. Math. Ver., 117 (2015), pp. 3--44.
R. L. Dobrushin, Vlasov equations, Funct. Anal. Appl., 13 (1979), pp. 115--123.
E. Emmrich, Gewöhnliche und Operator-Differentialgleichungen, Vieweg, Wiesbaden, 2004.
R. Escobedo, A. Iban͂ez, and E. Zuazua, Optimal strategies for driving a mobile agent in a “guidance by repulsion” model, Commun. Nonlinear Sci. Numer. Simul., 39 (2016), pp. 58--72.
M. Fornasier and F. Solombrino, Mean-field optimal control, ESAIM Control Optim. Calc. Var., 20 (2014), pp. 1123--1152.
F. Golse, The mean-field limit of the dynamics of large particle systems, J. Equations aux derivees partielles, 2003.
M. Herty and C. Ringhofer, Consistent mean field optimality conditions for interacting agent systems, Commun. Math. Sci., 17 (2019), pp. 1095--1108.
R. Jordan, D. Kinderlehrer, and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), pp. 1--17.
H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, in Kinetic Theories and the Boltzman Equation, Lect. Notes Math. 1048, Springer, New York, 1984, pp. 60--110.
F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), pp. 101--174.
H. Pham and X. Wei, Discrete time Mckean--Vlasov control problem: A dynamic programming approach, Appl. Math. Optim., 74 (2016), pp. 487--506.
R. Pinnau, C. Totzeck, O. Tse, and S. Martin, A consensus-based model for global optimization and its mean-field limit, Math Models Methods Appl. Sci., 27 (2017), pp. 183--204.
S. Roy, M. Annunziato, and A. Borzi, A Fokker-Planck feedback control-constrained approach for modeling crowd motion, J. Comput. Theor. Transp., 45 (2016), pp. 442--458.
S. Roy, M. Annunziato, A. Borz\`\i, and C. Klingenberg, A Fokker--Planck approach to control collective motion, Comput. Optim. Appl., 69 (2018), pp. 423--459.
C. Totzeck, Asymptotic Analysis of Optimal Control Problems and Global Optimization, Ph.D. Thesis, TU Kaiserslautern, Verlag Dr. Hut, 2017.
V. Varadarajan, On the convergence of sample probability distributions, Sankhya, (1933-1960), 19 (1958), pp. 23--26.

Information & Authors


Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 977 - 1006
ISSN (online): 1095-7138


Submitted: 12 March 2019
Accepted: 28 December 2020
Published online: 9 March 2021


  1. optimal control with ODE/PDE constraints
  2. interacting particle systems
  3. mean-field limits

MSC codes

  1. 49K15
  2. 49K20



Funding Information

Nederlandse Organisatie voor Wetenschappelijk Onderzoek https://doi.org/10.13039/501100003246 : 016.Vidi.189.102

Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

There are no citations for this item







Copy the content Link

Share with email

Email a colleague

Share on social media