In this article we study an optimal control problem subject to the Fokker-Planck equation
| $$ |
The control variable u is time-dependent and possibly multidimensional, and the function B depends on the space variable and the control. The cost functional is of tracking type and includes a quadratic regularization term on the control. For this problem, we prove existence of optimal controls and first order necessary conditions. Main emphasis is placed on second order necessary and sufficient conditions.
Keywords: optimal control, Fokker-Planck equation, existence of optimal control, first order optimality conditions, second order optimality conditions
@article{COCV_2021__27_1_A17_0,
author = {Soledad Aronna, M. and Tr\"oltzsch, Fredi},
title = {First and second order optimality conditions for the control of {Fokker-Planck} equations},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021014},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021014/}
}
TY - JOUR AU - Soledad Aronna, M. AU - Tröltzsch, Fredi TI - First and second order optimality conditions for the control of Fokker-Planck equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021014/ DO - 10.1051/cocv/2021014 LA - en ID - COCV_2021__27_1_A17_0 ER -
%0 Journal Article %A Soledad Aronna, M. %A Tröltzsch, Fredi %T First and second order optimality conditions for the control of Fokker-Planck equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021014/ %R 10.1051/cocv/2021014 %G en %F COCV_2021__27_1_A17_0
Soledad Aronna, M.; Tröltzsch, Fredi. First and second order optimality conditions for the control of Fokker-Planck equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 15. doi: 10.1051/cocv/2021014
[1] , , and , Mean field control hierarchy. Appl. Math. Optim. 76 (2017) 93–135.
[2] , and , Boltzmann-type control of opinion consensus through leaders. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372 (2014) 20140138.
[3] , Linear and quasilinear parabolic problems. Abstract linear theory. Vol. I, Vol. 89 of Monographs in Mathematics. Birkhäuser Boston, Inc., Boston, MA (1995).
[4] and , Optimal control of probability density functions of stochastic processes. Math. Model. Anal. 15 (2010) 393–407.
[5] and , A Fokker–Planck control framework for multidimensional stochastic processes. J. Comput. Appl. Math. 237 (2013) 487–507.
[6] and , A Fokker–Planck control framework for stochastic systems. EMS Surv. Math. Sci. 5 (2018) 65–98.
[7] , , and , Optimal control of infinite dimensional bilinear systems: application to the heat and wave equations. Math. Program. 168 (2018) 717–757.
[8] and , First and second order optimality conditions for the control of Fokker-Planck equations. Preprint (2020) . | arXiv
[9] and , First and second order optimality conditions for the control of Fokker-Planck equations. Preprint (2021) . | arXiv
[10] and , Well-posedness of parabolic difference equations. Vol. 69 of Operator Theory: Advances and Applications. Translated from the Russian by A. Iacob. Birkhäuser Verlag, Basel (1994).
[11] , Un théorème de compacité. CR Acad. Sci. Paris 256 (1963) 5042–5044.
[12] and , Stochastic model for tumor growth with immunization. Phys. Rev. E 79 (2009) 051903.
[13] , and , Control strategies for the Fokker- Planck equation. ESAIM: COCV 24 (2018) 741–763.
[14] , and , Infinite-horizon bilinear optimal control problems: sensitivity analysis and polynomial feedback laws. SIAM J. Control Optim. 56 (2018) 3184–3214.
[15] , Notes on mean field games. Technical report, (2010).
[16] and , Second order analysis for optimal control problems: Improving results expected from abstract theory. SIAM J. Optim. 22 (2012) 261–279.
[17] , and , Second-order analysis and numerical approximation for bang-bang bilinear control problems. SIAM J. Control Optim. 56 (2018) 4203–4227.
[18] , Nonlinear mean field Fokker-Planck equations. application to the chemotaxis of biological populations. Eur. Phys. J. B 62 (2008) 179–208.
[19] , Elements of nonlinear analysis. Birkhäuser (2012).
[20] and , Mathematical analysis and numerical methods for science and technology: Evolution Problems I, volume 5. Springer Science & Business Media (1992).
[21] , and , A kinetic flocking model with diffusion. Commun. Math. Phys. 300 (2010) 95–145.
[22] , Partial differential equations. American Mathematical Society (2010).
[23] and , Optimal control of the Fokker-Planck equation with space-dependent controls. J. Optim. Theory Appl. 174 (2017) 408–427.
[24] , , and , Fokker–Planck equations in the modeling of socio-economic phenomena. Math. Models Methods Appl. Sci. 27 (2017) 115–158.
[25] and , Finite element methods for Navier-Stokes equations: theory and algorithms. In vol. 5. Springer Science & Business Media (2012).
[26] and , Mean field games models—a brief survey. Dyn. Games Appl. 4 (2014) 110–154.
[27] , and , Optimization of a model Fokker-Planck equation. Kinet. Relat. Models 5 (2012).
[28] , Necessary and sufficient conditions for a local minimum 3: Second order conditions and augmented duality. SIAM J. Control Optim. 17 (1979) 266–288.
[29] , Perturbation theory for linear operators, Reprint of the corr. print of the 2nd edition. Classics in Mathematics. Springer-Verlag New York, Inc., New York (1980).
[30] , and , Linear and quasilinear equations of parabolic type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. American Mathematical Society, Providence, R.I. (1968).
[31] and , Mean field games. Jpn. J. Math. 2 (2007) 229–260.
[32] and , Problèmes aux limites non homogènes et applications. Vol. 1. Dunod, Paris (1968).
[33] , Analysis of heat equations on domains. Vol. 31 of London Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ (2005).
[34] , and , A Fokker–Planck feedback control-constrained approach for modelling crowd motion. J. Comput. Theor. Transport 45 (2016) 442–458.
[35] , Lectures notes (on mean field games). Technicalreport (2018) available at: link.
[36] and , Langevin equation, Fokker-Planck equation and cell migration. Bull. Math. Biol. 55 (1993) 585–608.
[37] , Interpolation theory, function spaces, differential operators. Vol. 18 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam-New York (1978).
[38] , Optimal control of partial differential equations. Theory, methods and applications, Vol. 112 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2010).
Cité par Sources :
The first author was supported by CAPES (Brazil) and by the Alexander von Humboldt Foundation (Germany).
