SIAM Journal on Control and Optimization, Volume 59, Issue 1, Page 709-726, January 2021.
One proves the exact controllability of the Fokker--Planck equation $\rho_t-\frac12\ \Delta(u\rho)-{div}(b\rho)=0,$ $\rho(0)=\rho_0,$ $\rho(T)=\rho_1,$ on a bounded domain $\mathcal{O}$ with reflecting conditions on the boundary $\partial\mathcal{O}$. In particular, the exact controllability of stochastic differential equations with reflection to $\partial\mathcal{O}$ is derived, and it is achieved by a nonlinear multivalued feedback controller $u=\Phi(\cdot,\rho)$, which steers $\rho_0$ in $\rho_1$ in a sufficiently large time $T$.

中文翻译：

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具有扩散边界条件的具有反射边界条件的Fokker-Planck方程和控制器的可控性

SIAM控制与优化杂志，第59卷，第1期，第709-726页，2021年1月。
一个证明了Fokker-Planck方程$ \ rho_t- \ frac12 \ \ Delta（u \ rho）-{div }（b \ rho）= 0，$ $ \ rho（0）= \ rho_0，$ $ \ rho（T）= \ rho_1，$在有界$ \ partial \ mathcal {O} $。尤其是，推导了反映到$ \ partial \ mathcal {O} $的随机微分方程的精确可控性，这是通过非线性多值反馈控制器$ u = \ Phi（\ cdot，\ rho）$实现的。在足够长的时间$ T $中将$ \ rho_0 $引导到$ \ rho_1 $中。