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.

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

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