Pseudo-spectral optimal control of stochastic processes using Fokker Planck equation

Figures & data

Figure 1. Ornstein-Uhlenbleck process. (a) Desired PDF (Red) and Controlled PDF (Blue). (b) Control function Lagrange interpolation

Figure 2. Ornstein-Uhlenbleck process. (a) Mean of the state. (b) Variance of the state under control function

Figure 3. Ornstein-Uhlenbleck process. Time evolution of probability density function of state under control function

Figure 4. Typical nonlinear system

Table 1. Parameters of the simulated stochastic processes

Parameters	OU process	The typical nonlinear system
$t\in \left[t_0,t_f\right]$	$t_0=0,t_f=5$	$t_0=0,t_f=10$
$X\in \mathrm{\Omega }\in [X_0,X_f]$	$X_0=-5,X_f=5$	$X_0=-5,X_f=5$
$r$	1	1
$\sigma$	0.8	1
$P_{ref}\left(x,y\right)$	$\frac{\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{{\left[x-2sin\left(\frac{\pi t}{5}\right)\right]}^2}{2{\left(0.5\right)}^2}\right)}{\sqrt{2\pi {\left(0.5\right)}^2}}$	$\frac{\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{{\left[x-t/5\right]}^2}{2{\left(0.5\right)}^2}\right)}{\sqrt{2\pi {\left(0.5\right)}^2}}$
$\rho \left(x\right)$	$\frac{1}{\sqrt{2\pi {\left(0.5\right)}^2}}\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{x^2}{2{\left(0.5\right)}^2}\right)$	$\frac{1}{\sqrt{2\pi {\left(1\right)}^2}}\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{x^2}{2{\left(1\right)}^2}\right)$
$U_{ad}\in \left[u_a,u_b\right]$	$u_a=-4,u_b=4$	$u_a=-6,u_b=6$
$N_t$	10	10
$N_X$	100	100

Figure 5. Typical nonlinear system

Figure 6. Typical nonlinear system. Time evolution of probability density function of state under control function

Figure 7. Typical nonlinear system with two stable equilibrium points with initial condition $\mathit{p}(x,t)=N(0,1)\mathbf{a}\mathbf{n}\mathbf{d}r=1;$