The Controllability of Fokker--Planck Equations with Reflecting Boundary Conditions and Controllers in Diffusion Term
Viorel Barbu · SIAM Journal on Control and Optimization · 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 $T$.