Control strategies for the Fokker−Planck equation
ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 2, pp. 741-763.

Using a projection-based decoupling of the Fokker−Planck equation, control strategies that allow to speed up the convergence to the stationary distribution are investigated. By means of an operator theoretic framework for a bilinear control system, two different feedback control laws are proposed. Projected Riccati and Lyapunov equations are derived and properties of the associated solutions are given. The well-posedness of the closed loop systems is shown and local and global stabilization results, respectively, are obtained. An essential tool in the construction of the controls is the choice of appropriate control shape functions. Results for a two dimensional double well potential illustrate the theoretical findings in a numerical setup.

DOI : 10.1051/cocv/2017046
Classification : 35Q35, 49J20, 93D05, 93D15
Mots-clés : Fokker−Planck equation, bilinear control systems, Lyapunov functions, Riccati equation, Lyapunov equation
Breiten, Tobias 1 ; Kunisch, Karl 1 ; Pfeiffer, Laurent 1

1 
@article{COCV_2018__24_2_741_0,
     author = {Breiten, Tobias and Kunisch, Karl and Pfeiffer, Laurent},
     title = {Control strategies for the {Fokker\ensuremath{-}Planck} equation},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {741--763},
     publisher = {EDP-Sciences},
     volume = {24},
     number = {2},
     year = {2018},
     doi = {10.1051/cocv/2017046},
     zbl = {1403.35298},
     mrnumber = {3816413},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2017046/}
}
TY  - JOUR
AU  - Breiten, Tobias
AU  - Kunisch, Karl
AU  - Pfeiffer, Laurent
TI  - Control strategies for the Fokker−Planck equation
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2018
SP  - 741
EP  - 763
VL  - 24
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2017046/
DO  - 10.1051/cocv/2017046
LA  - en
ID  - COCV_2018__24_2_741_0
ER  - 
%0 Journal Article
%A Breiten, Tobias
%A Kunisch, Karl
%A Pfeiffer, Laurent
%T Control strategies for the Fokker−Planck equation
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2018
%P 741-763
%V 24
%N 2
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2017046/
%R 10.1051/cocv/2017046
%G en
%F COCV_2018__24_2_741_0
Breiten, Tobias; Kunisch, Karl; Pfeiffer, Laurent. Control strategies for the Fokker−Planck equation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 2, pp. 741-763. doi : 10.1051/cocv/2017046. http://www.numdam.org/articles/10.1051/cocv/2017046/

[1] R. Adams, Sobolev Spaces.New York Academic Press (1975) | MR | Zbl

[2] M. Annunziato and A. Borzì, A Fokker-Planck control framework for multidimensional stochastic processes. J. Comput. Appl. Math. (2013) 487–507 | DOI | MR | Zbl

[3] M. Badra and T. Takahashi, Feedback stabilization of a fluid-rigid body interaction system. Adv. Differ. Equ. 19 (2014) 1137–1184 | MR | Zbl

[4] J. Ball and M. Slemrod, Feedback stabilization of distributed semilinear control systems. Appl. Math. Optimiz. 5 (1979) 169–179 | DOI | MR | Zbl

[5] V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier–Stokes equations. Memoirs Amer. Math. Soc. 181 (2006) 1–128 | DOI | MR | Zbl

[6] A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, Representation and Control of Infinite Dimensional Systems. Birkhäuser Boston, Basel, Berlin (2007) | DOI | MR | Zbl

[7] V.I. Bogachev, G. Da Prato and M. Röckner, Fokker−Planck equations and maximal dissipativity for Kolmogorov operators with time dependent singular drifts in Hilbert spaces. J. Functional Anal. 256 (2009) 1269–1298 | DOI | MR | Zbl

[8] J. Chang and G. Cooper, A practical scheme for Fokker–Planck equations. J. Comput. Phys. 6 (1970) 1–16 | DOI | Zbl

[9] M. Chipot, Elements of Nonlinear Analysis. Birkhäuser (2000) | MR | Zbl

[10] R. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory. Springer Verlag (2005) | Zbl

[11] C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, vol. 13 of Springer Series in Synergetics. Springer Verlag, Berlin, 3rd edition (2004) | MR | Zbl

[12] J.J. Gorman, A. Balijepalli and T.W. Lebrun, Feedback Control of MEMS to Atoms. Springer US, Boston, MA, ch. Feedback Control of Optically Trapped Particles (2012) 141–177

[13] C. Hartmann, Balanced model reduction of partially-observed Langevin equations: an averaging principle. Math. Comput. Model. Dynamical Syst. 17 (2011) 463–490 | DOI | MR | Zbl

[14] C. Hartmann, B. Schäfer−Bung and A. Thöns-Zueva, Balanced averaging of bilinear systems with applications to stochastic control. SIAM J. Control Optimiz. 51 (2013) 2356–2378 | DOI | MR | Zbl

[15] W. Huang, M. Ji, Z. Liu and Y. Yi, Steady states of Fokker-Planck equations: I. existence. J. Dynamics Differ. Equ. 27 (2015) 721–742 | DOI | MR | Zbl

[16] P. H. Jones, M.M. Onofrio and G. Volpe, Optical Tweezers: Principles and Applications. Cambridge University Press (2015) | DOI

[17] T. Kato, Perturbation Theory for Linear Operators. Springer Verlag, Berlin/Heidelberg, Germany (1980) | MR | Zbl

[18] A. Khapalov, Controllability of partial differential equations governed by multiplicative controls, vol. 1995 of Lect. Notes Math., Springer Verlag, Berlin 2010 | MR | Zbl

[19] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Volume 1, Abstract Parabolic Systems: Continuous and Approximation Theories, vol. 1, Cambridge University Press (2000) | MR | Zbl

[20] C. Le Bris and P.-L. Lions, Existence and uniqueness of solutions to Fokker−Planck type equations with irregular coefficients. Commun. Partial Differ. Equ. 33 (2008) 1272–1317 | DOI | MR | Zbl

[21] J. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I/II. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Springer Verlag, Berlin (1972) | Zbl

[22] P.-L. Lions and A.-S. Sznitman, Stochastic differential equations with reflecting boundary conditions. Commun. Pure Appl. Math. 37(1984) 511–537 | DOI | MR | Zbl

[23] B.J. Matkowsky and Z. Schuss, Eigenvalues of the Fokker-Planck operator and the approach to equilibrium for diffusions in potential fields. SIAM J. Appl. Math. 40 (1981) 242–254 | DOI | MR | Zbl

[24] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer New York (1983) | DOI | MR | Zbl

[25] J. Raymond and L. Thevenet, Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers. Discrete Contin. Dyn. Syst. 27 (2010) 1159–1187 | DOI | MR | Zbl

[26] J.-P. Raymond, Feedback boundary stabilization of the two-dimensional Navier–Stokes equations. SIAM J. Control Optimiz. 45 (2006) 790–828 | DOI | MR | Zbl

[27] R. Risken, The Fokker−Planck Equation: Methods of Solutions and Applications. Springer Verlag Berlin (1996) | Zbl

[28] H. Tanabe, Equations of evolution, vol. 6 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, Mass.-London (1979). Translated from the Japanese by N. Mugibayashi and H. Haneda. | MR | Zbl

[29] L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces. Springer Berlin Heidelberg, Berlin, Heidelberg (2007) 99–101 | Zbl

[30] L. Thevenet, J.-M. Buchot and J.-P. Raymond, Nonlinear feedback stabilization of a two-dimensional Burgers equation. ESAIM: COCV 16 (2010) 929–955 | Numdam | MR | Zbl

[31] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators. North-Holland Publishing Company (1978) | MR | Zbl

[32] R. Triggiani, On the stabilizability problem in Banach space, J. Math. Anal. Appl. 52 (1975) 383–403 | DOI | MR | Zbl

[33] G. Troianiello, Elliptic Differential Equations and Obstacle Problems. The University Series in Mathematics, Plenum Press, New York (1987) | MR | Zbl

Cité par Sources :