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A Fokker–Planck approach to control collective motion

Abstract

A Fokker–Planck control strategy for collective motion is investigated. This strategy is formulated as the minimisation of an expectation objective with a bilinear optimal control problem governed by the Fokker–Planck equation modelling the evolution of the probability density function of the stochastic motion. Theoretical results on existence and regularity of optimal controls are provided. The resulting optimality system is discretized using an alternate-direction implicit Chang–Cooper scheme that guarantees conservativeness, positivity, \(L^1\) stability, and second-order accuracy of the forward solution. A projected non-linear conjugate gradient scheme is used to solve the optimality system. Results of numerical experiments validate the theoretical accuracy estimates and demonstrate the efficiency of the proposed control framework.

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References

  1. 1.

    Annunziato, M., Borzì, A.: Optimal control of probability density functions of stochastic processes. Math. Model. Anal. 15, 393–407 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Annunziato, M., Borzì, A.: A Fokker–Planck control framework for multidimensional stochastic process. J. Comput. Appl. Math. 237, 487–507 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Annunziato, M., Borzì, A., Nobile, F., Tempone, R.: On the connection between the Hamilton–Jacobi–Bellman and the Fokker–Planck control frameworks. Appl. Math. 5, 2476–2484 (2014)

    Article  Google Scholar 

  4. 4.

    Aronson, D.G.: Non-negative solutions of linear parabolic equations. Ann. della Scuola Normale Superiore di Pisa - Classe di Scienze 22(4), 607–694 (1968)

  5. 5.

    Bellomo, N., Bellouquid, A., Knopoff, D.: From the microscale to collective crowd dynamics. Multiscale Model. Simul. 11(3), 943–963 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Borzì, A., Schulz, V.: Computational Optimization of Systems Governed by Partial Differential Equations. SIAM Book Series on Computational Science and Engineering 08. SIAM, Philadelphia, PA (2012)

    MATH  Google Scholar 

  7. 7.

    Chang, J.S., Cooper, G.: A practical difference scheme for Fokker–Planck equations. J. Comput. Phys. 6, 1–16 (1970)

    Article  MATH  Google Scholar 

  8. 8.

    Cucker, F., Mordecki, E.: Flocking in noisy environments. J. Math. Pures Appl. 89, 278–296 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Dautray, R., Lions, J.-L.: Problèmes d’ évolution du premier ordre en temps, (Chap 18.) Masson, Paris (1984)

  10. 10.

    Deutsch, A., Theraulaz, G., Vicsek, T.: Collective motion in biological systems. Interface Focus 2(6), 689–692 (2012)

    Article  Google Scholar 

  11. 11.

    Douglas Jr., J.: On the numerical integration of \(u_{xx}+ u_{yy}= u_t\) by implicit methods. J. Soc. Ind. Appl. Math. 3, 42–65 (1955)

    Article  Google Scholar 

  12. 12.

    Douglas Jr., J.: Alternating direction methods for three space variables. Numer. Math. 4(1), 41–63 (1962)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Droniou, J., Vázquez, J.-L.: Noncoercive convection-diffusion elliptic problems with Neumann boundary conditions. Calc. Var. Partial Differ. Equ. 34, 413–434 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Evans, L.C.: Partial Differential Equations, Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence, RI (2002)

    Google Scholar 

  15. 15.

    Fleming, W., Soner, M.: Controlled Markov Processes and Viscosity Solutions. Springer, Berlin (2006)

    MATH  Google Scholar 

  16. 16.

    Flotron, S., Rappaz, J.: Conservation schemes for convection-diffusion equations with Robin boundary conditions. ESAIM 47, 1765–1781 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Gautrais, J., Ginelli, F., Fournier, R., Blanco, S., Soria, M., Chateé, H., Theraulaz, H.G.: Deciphering interactions in moving animal groups. PLoS Comput. Biol. 8(9), e1002 (2012)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Grosskinsky, S., Klingenberg, C., Oelschläger, K.: A rigorous derivation of Smoluchowski’s equation in the moderate limit. Stoch. Anal. Appl. 22(1), 113–141 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984)

    Book  MATH  Google Scholar 

  20. 20.

    Glowinski, R., Lions, J.-L., He, J.: Exact and approximate controllability for distributed parameter systems. Acta Numer. 3, 269–378 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Grüne, L., Pannek, J.: Nonlinear Model Predictive Control, Theory and Algorithms, Communications and Control Engineering. Springer, Berlin (2011)

    MATH  Google Scholar 

  22. 22.

    Hager, W.W., Zhang, H.: A new conjugate gradient method with guaranteed descent and an efficient line search. SIAM J. Optim. 16(1), 170–192 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Hundsdorfer, W., Verwer, J.G.: Numerical Solution of Time-Dependent Advection–Diffusion–Reaction Equations. Springer Series in Computational Mathematics (2010)

  24. 24.

    Jovanović, B.S., Süli, E.: Analysis of Finite Difference Schemes For Linear Partial Differential Equations with Generalized Solutions. Springer Series in Computational Mathematics (2014)

  25. 25.

    Lions, J.-L.: Quelque Methodes de Résolution des Problemes aux Limites Non Linéaires. Dunod-Gauth. Vill, Paris (1969)

    MATH  Google Scholar 

  26. 26.

    Lions, J.-L.: Nonhomogeneous Boundary Value Problems and Applications. Springer, Berlin (1972)

    Book  Google Scholar 

  27. 27.

    Mohammadi, M., Borzì, A.: Analysis of the Chang–Cooper discretization scheme for a class of Fokker–Planck equations. J. Numer. Math. 23, 271–288 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Morale, D., Capasso, V., Oelschlaeger, K.: An interacting particle system modelling aggregation behavior: from individuals to populations. J. Math. Biol. 50(1), 49–66 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Neittaanmaki, P., Tiba, D.: Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications. Pure and Applied Mathematics. CRC Press, London (1994)

    MATH  Google Scholar 

  30. 30.

    Øksendal, B.K.: Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin (2003)

    Book  MATH  Google Scholar 

  31. 31.

    Peaceman, D.W., Rachford Jr., H.H.: The numerical solution of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Math. 3(1), 28–41 (1955)

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Petrelli, L., Kearsely, A.J.: Wasserstein metric convergence method for Fokker–Planck equations with point controls. Appl. Math. Lett. 22(7), 1130–1135 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer, Berlin (1994)

    MATH  Google Scholar 

  34. 34.

    Roy, S., Annunziato, M., Borzì, A.: A Fokker–Planck feedback control-constrained approach for modelling crowd motion. J. Comput. Theor. Transp. 45(6), 442–458 (2016)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Sepúlveda, N., Petitjean, L., Cochet, O., Grasland-Mongrain, E., Silberzan, P., Hakim, V.: Collective cell motion in an epithelial sheet can be quantitatively described by a stochastic interacting particle model. PLoS Comput. Biol. 9(3), e1002944 (2013)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Strandburg-Peshkin, A., Farine, D.R., Couzin, I.D., Crofoot, M.C.: Shared decision-making drives collective movement in wild baboons. Science 348(6241), 1358–1361 (2015)

    Article  Google Scholar 

  37. 37.

    Tao, T.: Nonlinear Dispersive Equations: Local and Global Analysis. American Mathematical Society, Providence, RI (2006)

    Book  MATH  Google Scholar 

  38. 38.

    Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications. American Mathematical Society, Providence, RI (2010)

    Book  MATH  Google Scholar 

  39. 39.

    Varga, R.S.: Matrix Iterative Analysis. Springer Series in Computational Mathematics. Springer, Berlin (2000)

    Book  Google Scholar 

  40. 40.

    Zienkiewicz, A., Barton, D.A.W., Porfiri, M., di Bernardo, M.: Data-driven stochastic modelling of zebrafish locomotion. J. Math. Biol. (2014). doi:10.1007/s00285-014-0843-2

    MATH  Google Scholar 

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Acknowledgements

The authors would like to gratefully acknowledge the comments by the referees which helped to improve this paper. S. Roy would like to thank A. S. Vasudeva Murthy and Praveen Chandrashekar for several fruitful discussions during the initial phases of this work. This work was supported in part by the European Union under Grant Agreement No. 304617 Marie Curie Research Training Network “Multi-ITN STRIKE—Novel Methods in Computational Finance” and the BMBF project “ROENOBIO”. S. Roy was also supported by the DAAD Passage to India Program and the AIRBUS Group Corporate Foundation Chair in Mathematics of Complex Systems established in TIFR/ICTS, Bangalore.

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Correspondence to Souvik Roy.

Appendix: Derivation of the numerical adjoint

Appendix: Derivation of the numerical adjoint

We derive the numerical scheme for the adjoint equation (14) using the discretize-before-optimize approach. The starting point of this derivation is the Lagrangian

$$\begin{aligned} L(f,u,p) = J(f,u) + \langle \partial _tf-\nabla \cdot {F},p\rangle . \end{aligned}$$
(81)

In order to obtain the discrete version of the adjoint equation, we need to consider a discrete version of the Lagrange function with the ADI-CC scheme for the time–space derivatives of f. Since the ADI-CC scheme has an intermediate time step \(t^{m+\frac{1}{2}}\), we define the Lagrangian on the following double grid

$$\begin{aligned} \begin{aligned} Q^d_{h,\delta {t}}&= \lbrace {(x,t_m):x\in \Omega _h,~t_m=m\delta {t},~0\le m\le N_t}\rbrace \\&\cup \lbrace {(x,t_{m+\frac{1}{2}}):x\in \Omega _h,~t_{m+\frac{1}{2}}=\left( {m+\frac{1}{2}}\right) \delta {t},~0\le m\le N_t-1}\rbrace . \end{aligned} \end{aligned}$$
(82)

The discrete Lagrangian is given by

$$\begin{aligned} \begin{aligned} \hat{L}(f,u,p)&= \alpha \sum _m\sum _{i,j=1}^{N_x-1}V\left( x_{i,j}-x_t^m\right) f_{i,j}^m~h^2\frac{{ dt}}{2}\\&\quad +\,\alpha \sum _m\sum _{i,j=1}^{N_x-1}V\left( x_{i,j}-x_t^{m+\frac{1}{2}}\right) f_{i,j}^{m+\frac{1}{2}}~h^2\frac{{ dt}}{2}\\&\quad +\,\beta \sum _{i,j=1}^{N_x-1}V\left( x_{i,j}-x_t^{N_t}\right) f_{i,j}^{N_t}~h^2 + \frac{\nu }{2}\sum _m\sum _{i,j=1}^{N_x-1} A(u_{i,j}^m)~h^2\frac{{ dt}}{2}\\&\quad \,+\frac{\nu }{2}\sum _m\sum _{i,j=1}^{N_x-1}A\left( u_{i,j}^{m+\frac{1}{2}}\right) ~h^2\frac{{ dt}}{2}\\&\quad \,+ \sum _m\sum _{i,j=1}^{N_x-1}\frac{f_{i,j}^{m+\frac{1}{2}}-f_{i,j}^m}{\delta {t}/2}p_{i,j}^m~h^2\frac{{ dt}}{2}\\&\quad \,+ \sum _m\sum _{i,j=1}^{N_x-1}\frac{f_{i,j}^{m+1}-f_{i,j}^{m+\frac{1}{2}}}{\delta {t}/2}p_{i,j}^{m+\frac{1}{2}}~h^2\frac{{ dt}}{2}\\&\quad -\,\sum _m\sum _{i,j=1}^{N_x-1}\left[ {\left( F_{i+\frac{1}{2},j}^{m+\frac{1}{2}}-F_{i-\frac{1}{2},j}^{m+\frac{1}{2}}\right) +\left( F_{i,j+\frac{1}{2}}^{m}-F_{i,j-\frac{1}{2}}^{m}\right) } \right] p_{i,j}^m~h\frac{{ dt}}{2}\\&\quad -\,\sum _m\sum _{i,j=1}^{N_x-1}\left[ {\left( F_{i+\frac{1}{2},j}^{m+\frac{1}{2}}-F_{i-\frac{1}{2},j}^{m+\frac{1}{2}}\right) +\left( F_{i,j+\frac{1}{2}}^{m+1}-F_{i,j-\frac{1}{2}}^{m+1}\right) } \right] p_{i,j}^{m+\frac{1}{2}}~h\frac{{ dt}}{2}. \end{aligned} \end{aligned}$$
(83)

We write the fluxes of the FP equation (13) in the following compact form

$$\begin{aligned} F_{i+\frac{1}{2},j}^m = K_{i+\frac{1}{2},j}^m f_{i+1,j}^m-R_{i+\frac{1}{2},j}^m f_{i,j}^m, \end{aligned}$$
(84)

where

$$\begin{aligned} \begin{aligned}&K_{i+\frac{1}{2},j}^m = (1-\delta _i)B_{i+\frac{1}{2},j}^m + \frac{\sigma ^2}{h},\\&R_{i+\frac{1}{2},j}^m = \frac{\sigma ^2}{h}-\delta _iB_{i+\frac{1}{2},j}^m. \\ \end{aligned} \end{aligned}$$
(85)

Similarly, we have

$$\begin{aligned} F_{i,j+\frac{1}{2}}^m = K_{i,j+\frac{1}{2}}^m f_{i,j+1}^m-R_{i,j+\frac{1}{2}}^m f_{i,j}^m. \end{aligned}$$
(86)

Therefore, we obtain

$$\begin{aligned} \begin{aligned} \sum _m\sum _{i,j=1}^{N_x-1}\left( F_{i+\frac{1}{2},j}^{m+\frac{1}{2}}-F_{i-\frac{1}{2},j}^{m+\frac{1}{2}}\right) p_{i,j}^m&=\sum _m\sum _{i,j=1}^{N_x-1} \left( K_{i+\frac{1}{2},j}^{m+\frac{1}{2}} f_{i+1,j}^{m+\frac{1}{2}}-R_{i+\frac{1}{2},j}^{m+\frac{1}{2}}f_{i,j}^{m+\frac{1}{2}}\right. \\&\quad \left. -\,K_{i-\frac{1}{2},j}^{m+\frac{1}{2}} f_{i,j}^{m+\frac{1}{2}}+R_{i-\frac{1}{2},j}^{m+\frac{1}{2}}f_{i-1,j}^{m+\frac{1}{2}}\right) p_{i,j}^{m}. \end{aligned} \end{aligned}$$
(87)

Rearranging the summation on the right-hand side of (87) to collect the terms \(f_{i,j}^{m+\frac{1}{2}}\) with same space index and using discrete flux zero (39), we have

$$\begin{aligned} \begin{aligned} \sum _m\sum _{i,j=1}^{N_x-1}\left( F_{i+\frac{1}{2},j}^{m+\frac{1}{2}}-F_{i-\frac{1}{2},j}^{m+\frac{1}{2}}\right) p_{i,j}^m&=\sum _m\sum _{i,j=1}^{N_x-1} \left( K_{i-\frac{1}{2},j}^{m+\frac{1}{2}} p_{i-1,j}^{m}-R_{i+\frac{1}{2},j}^{m+\frac{1}{2}}p_{i,j}^{m}\right. \\&\quad -\left. \,K_{i-\frac{1}{2},j}^{m+\frac{1}{2}} p_{i,j}^{m}+R_{i+\frac{1}{2},j}^{m+\frac{1}{2}}p_{i+1,j}^{m}\right) f_{i,j}^{m+\frac{1}{2}}.\\ \end{aligned} \end{aligned}$$
(88)

In a similar way, we have

$$\begin{aligned} \begin{aligned} \sum _m\sum _{i,j=1}^{N_x-1}\left( F_{i,j+\frac{1}{2}}^{m}-F_{i,j-\frac{1}{2}}^{m}\right) p_{i,j}^m&=\sum _m\sum _{i,j=1}^{N_x-1} \left( K_{i,j-\frac{1}{2}}^{m} p_{i,j-1}^{m}-\,R_{i,j+\frac{1}{2}}^{m}p_{i,j}^{m}\right. \\ {}&\quad \left. - K_{i,j-\frac{1}{2}}^{m} p_{i,j}^{m}+R_{i,j+\frac{1}{2}}^{m}p_{i,j+1}^{m}\right) f_{i,j}^{m},\\ \end{aligned} \end{aligned}$$
(89)
$$\begin{aligned} \begin{aligned} \sum _m\sum _{i,j=1}^{N_x-1}\left( F_{i+\frac{1}{2},j}^{m+\frac{1}{2}}-F_{i-\frac{1}{2},j}^{m+\frac{1}{2}}\right) p_{i,j}^{m+\frac{1}{2}}&=\sum _m\sum _{i,j=1}^{N_x-1} \left( K_{i-\frac{1}{2},j}^{m+\frac{1}{2}} p_{i-1,j}^{m+\frac{1}{2}}-R_{i+\frac{1}{2},j}^{m+\frac{1}{2}}p_{i,j}^{m+\frac{1}{2}}\right. \\ {}&\quad \left. -\, K_{i-\frac{1}{2},j}^{m+\frac{1}{2}} p_{i,j}^{m+\frac{1}{2}}+R_{i+\frac{1}{2},j}^{m+\frac{1}{2}}p_{i+1,j}^{m+\frac{1}{2}}\right) f_{i,j}^{m+\frac{1}{2}},\\ \end{aligned} \end{aligned}$$
(90)

and

$$\begin{aligned} \begin{aligned} \sum _m\sum _{i,j=1}^{N_x-1}\left( F_{i,j+\frac{1}{2}}^{m+1}-F_{i,j-\frac{1}{2}}^{m+1}\right) p_{i,j}^{m+\frac{1}{2}}&=\sum _m\sum _{i,j=1}^{N_x-1} \left( K_{i,j-\frac{1}{2}}^{m+1} p_{i,j-1}^{m+\frac{1}{2}}-R_{i,j+\frac{1}{2}}^{m+1}p_{i,j}^{m+\frac{1}{2}}\right. \\ {}&\quad \left. -\, K_{i,j-\frac{1}{2}}^{m+1} p_{i,j}^{m+\frac{1}{2}}+R_{i,j+\frac{1}{2}}^{m+1}p_{i,j+1}^{m+\frac{1}{2}}\right) f_{i,j}^{m+1}. \end{aligned} \end{aligned}$$
(91)

For our convenience, using (88)–(91) in (83), rearranging the time indices and collecting the terms \(f_{i,j}^{m+\frac{1}{2}}\) and \(f^{m+1}_{i,j}\), we obtain the Lagrange function in a different form as follows

$$\begin{aligned} \begin{aligned}&\hat{L}_1(f,u,p)\\&\quad =\alpha \sum _m\sum _{i,j=1}^{N_x-1}V(x_{i,j}-x_t^{m+1})f_{i,j}^{m+1}~h^2 \frac{{ dt}}{2} +\,\alpha \sum _m\sum _{i,j=1}^{N_x-1}V(x_{i,j}-x_t^{m+\frac{1}{2}})f_{i,j}^{m+\frac{1}{2}}~h^2\frac{{ dt}}{2}\\&\qquad +\,\beta \sum _{i,j=1}^{N_x-1}V(x_{i,j}-x_t^{N_t})f_{i,j}^{N_t}~h^2 + \frac{\nu }{2}\sum _m\sum _{i,j=1}^{N_x-1} A\left( u_{i,j}^m\right) ~h^2\frac{{ dt}}{2}+\,\frac{\nu }{2}\sum _{m=0}^{N_t-1}\sum _{i,j=1}^{N_x-1}A\left( u_{i,j}^{m+\frac{1}{2}}\right) ~h^2\frac{{ dt}}{2}\\&\qquad +\,\sum _m\sum _{i,j=1}^{N_x-1}\frac{p_{i,j}^{m}-p_{i,j}^{m+\frac{1}{2}}}{\delta {t}/2}f_{i,j}^{m+\frac{1}{2}}~h^2\frac{{ dt}}{2} +\,\sum _m\sum _{i,j=1}^{N_x-1}\frac{p_{i,j}^{m+\frac{1}{2}}-p_{i,j}^{m+1}}{\delta {t}/2}f_{i,j}^{m+1}~h^2\frac{{ dt}}{2}\\&\qquad -\,\sum _m\sum _{i,j}^{N_x-1} \left( K_{i-\frac{1}{2},j}^{m+\frac{1}{2}} p_{i-1,j}^{m}-R_{i+\frac{1}{2},j}^{m+\frac{1}{2}}p_{i,j}^{m}- K_{i-\frac{1}{2},j}^{m+\frac{1}{2}} p_{i,j}^{m}+R_{i+\frac{1}{2},j}^{m+\frac{1}{2}}p_{i+1,j}^{m}\right) f_{i,j}^{m+\frac{1}{2}}h\dfrac{{ dt}}{2}\\&\qquad -\,\sum _m\sum _{i,j}^{N_x-1} \left( K_{i,j-\frac{1}{2}}^{m+1} p_{i,j-1}^{m+1}-R_{i,j+\frac{1}{2}}^{m+1}p_{i,j}^{m+1}- K_{i,j-\frac{1}{2}}^{m+1} p_{i,j}^{m+1}+R_{i,j+\frac{1}{2}}^{m+1}p_{i,j+1}^{m+1}\right) f_{i,j}^{m+1}h\dfrac{{ dt}}{2}\\&\qquad -\,\sum _m\sum _{i,j}^{N_x-1} \left( K_{i-\frac{1}{2},j}^{m+\frac{1}{2}} p_{i-1,j}^{m+\frac{1}{2}}-R_{i+\frac{1}{2},j}^{m+\frac{1}{2}}p_{i,j}^{m+\frac{1}{2}}- K_{i-\frac{1}{2},j}^{m+\frac{1}{2}} p_{i,j}^{m+\frac{1}{2}}+R_{i+\frac{1}{2},j}^{m+\frac{1}{2}}p_{i+1,j}^{m+\frac{1}{2}}\right) f_{i,j}^{m+\frac{1}{2}}h\dfrac{{ dt}}{2}\\&\qquad -\,\sum _m\sum _{i,j}^{N_x-1} \left( K_{i,j-\frac{1}{2}}^{m+1} p_{i,j-1}^{m+\frac{1}{2}}-R_{i,j+\frac{1}{2}}^{m+1}p_{i,j}^{m+\frac{1}{2}}- K_{i,j-\frac{1}{2}}^{m+1} p_{i,j}^{m+\frac{1}{2}}+R_{i,j+\frac{1}{2}}^{m+1}p_{i,j+1}^{m+\frac{1}{2}}\right) f_{i,j}^{m+1}h\dfrac{{ dt}}{2}.\\ \end{aligned} \end{aligned}$$
(92)

When the control cost A(u) is given by (C1), taking derivative with respect to \(f^{m+1}\), we obtain the following first integration step for the adjoint equation

$$\begin{aligned} \begin{aligned} \frac{p_{i,j}^{m+\frac{1}{2}}-p_{i,j}^{m+1}}{\delta {t}/2}&=\frac{1}{h}\left( K_{i,j-\frac{1}{2}}^{m+1} p_{i,j-1}^{m+\frac{1}{2}}- R_{i,j+\frac{1}{2}}^{m+1}p_{i,j}^{m+\frac{1}{2}}- K_{i,j-\frac{1}{2}}^{m+1} p_{i,j}^{m+\frac{1}{2}}+R_{i,j+\frac{1}{2}}^{m+1}p_{i,j+1}^{m+\frac{1}{2}}\right) \\&\quad +\,\frac{1}{h}\left( K_{i,j-\frac{1}{2}}^{m+1} p_{i,j-1}^{m+1}-R_{i,j+\frac{1}{2}}^{m+1}p_{i,j}^{m+1}- K_{i,j-\frac{1}{2}}^{m+1} p_{i,j}^{m+1}+R_{i,j+\frac{1}{2}}^{m+1}p_{i,j+1}^{m+1}\right) \\&\quad -\,\alpha V(x_{i,j}-x_t^{m+1}).\\ \end{aligned} \end{aligned}$$

Taking derivative with respect to \(f_{i,j}^{m+\frac{1}{2}}\), we obtain the following second integration step for the adjoint equation

$$\begin{aligned} \begin{aligned} \frac{p_{i,j}^{m}-p_{i,j}^{m+\frac{1}{2}}}{\delta {t}/2}&=\frac{1}{h}\left( K_{i-\frac{1}{2},j}^{m+\frac{1}{2}} p_{i-1,j}^{m}- R_{i+\frac{1}{2},j}^{m+\frac{1}{2}}p_{i,j}^{m}- K_{i-\frac{1}{2},j}^{m+\frac{1}{2}} p_{i,j}^{m}+R_{i+\frac{1}{2},j}^{m+\frac{1}{2}}p_{i+1,j}^{m}\right) \\&\quad +\,\frac{1}{h}\left( K_{i-\frac{1}{2},j}^{m+\frac{1}{2}} p_{i-1,j}^{m+\frac{1}{2}}-R_{i+\frac{1}{2},j}^{m+\frac{1}{2}}p_{i,j}^{m+\frac{1}{2}}- K_{i-\frac{1}{2},j}^{m+\frac{1}{2}} p_{i,j}^{m+\frac{1}{2}}+R_{i+\frac{1}{2},j}^{m+\frac{1}{2}}p_{i+1,j}^{m+\frac{1}{2}}\right) \\&\quad -\alpha V\left( x_{i,j}-x_t^{m+\frac{1}{2}}\right) ,\\ \end{aligned} \end{aligned}$$

along with the terminal condition

$$\begin{aligned} p_{i,j}^{N_t} = -\beta V(x_{i,j}-x_T). \end{aligned}$$

When the control cost A(u) is given by (C2), taking derivative with respect to \(f^{m+1}\), we obtain the following first integration step for the adjoint equation

$$\begin{aligned} \begin{aligned} \frac{p_{i,j}^{m+\frac{1}{2}}-p_{i,j}^{m+1}}{\delta {t}/2}&=\frac{1}{h}\left( K_{i,j-\frac{1}{2}}^{m+1} p_{i,j-1}^{m+\frac{1}{2}}- R_{i,j+\frac{1}{2}}^{m+1}p_{i,j}^{m+\frac{1}{2}}- K_{i,j-\frac{1}{2}}^{m+1} p_{i,j}^{m+\frac{1}{2}}+R_{i,j+\frac{1}{2}}^{m+1}p_{i,j+1}^{m+\frac{1}{2}}\right) \\&\quad +\,\frac{1}{h}\left( K_{i,j-\frac{1}{2}}^{m+1} p_{i,j-1}^{m+1}-R_{i,j+\frac{1}{2}}^{m+1}p_{i,j}^{m+1}- K_{i,j-\frac{1}{2}}^{m+1} p_{i,j}^{m+1}+R_{i,j+\frac{1}{2}}^{m+1}p_{i,j+1}^{m+1}\right) \\&\quad -\,\alpha V\left( x_{i,j}-x_t^{m+1}\right) - \frac{\nu }{2}|u^{m+1}_{i,j}|^2 -\frac{\nu }{2} \Bigg |\frac{u^{m+1}_{i+1,j}-u^{m+1}_{i,j}}{h}\Bigg |^2 -\frac{\nu }{2} \Bigg |\frac{u^{m+1}_{i,j-1}-u^{m+1}_{i,j}}{h}\Bigg |^2.\\ \end{aligned} \end{aligned}$$

Taking derivative with respect to \(f_{i,j}^{m+\frac{1}{2}}\), we obtain the following second integration step for the adjoint equation

$$\begin{aligned} \begin{aligned} \frac{p_{i,j}^{m}-p_{i,j}^{m+\frac{1}{2}}}{\delta {t}/2}&=\frac{1}{h}\left( K_{i-\frac{1}{2},j}^{m+\frac{1}{2}} p_{i-1,j}^{m}- R_{i+\frac{1}{2},j}^{m+\frac{1}{2}}p_{i,j}^{m}- K_{i-\frac{1}{2},j}^{m+\frac{1}{2}} p_{i,j}^{m}+R_{i+\frac{1}{2},j}^{m+\frac{1}{2}}p_{i+1,j}^{m}\right) \\&\quad +\,\frac{1}{h}\left( K_{i-\frac{1}{2},j}^{m+\frac{1}{2}} p_{i-1,j}^{m+\frac{1}{2}}-R_{i+\frac{1}{2},j}^{m+\frac{1}{2}}p_{i,j}^{m+\frac{1}{2}}- K_{i-\frac{1}{2},j}^{m+\frac{1}{2}} p_{i,j}^{m+\frac{1}{2}}+R_{i+\frac{1}{2},j}^{m+\frac{1}{2}}p_{i+1,j}^{m+\frac{1}{2}}\right) \\&\quad -\,\alpha V\left( x_{i,j}-x_t^{m+\frac{1}{2}}\right) - \frac{\nu }{2}|u^{m+\frac{1}{2}}_{i,j}|^2 -\frac{\nu }{2} \Bigg |\frac{u^{m+\frac{1}{2}}_{i+1,j}-u^{m+\frac{1}{2}}_{i,j}}{h}\Bigg |^2 -\frac{\nu }{2} \Bigg |\frac{u^{m+\frac{1}{2}}_{i,j-1}-u^{m+\frac{1}{2}}_{i,j}}{h}\Bigg |^2.\\ \end{aligned} \end{aligned}$$

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Roy, S., Annunziato, M., Borzì, A. et al. A Fokker–Planck approach to control collective motion. Comput Optim Appl 69, 423–459 (2018). https://doi.org/10.1007/s10589-017-9944-3

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Keywords

  • Fokker–Planck equation
  • Alternate direction method
  • Chang–Cooper scheme
  • Projected gradient method
  • Control constrained PDE optimization

Mathematics Subject Classification

  • 35Q84
  • 35Q91
  • 35Q93
  • 49K20
  • 49J20
  • 65C20