Eur. Phys. J. Plus (2021) 136: 896
https://doi.org/10.1140/epjp/s13360-021-01865-3

Regular Article

Asymptotic controllability of nonlinear Fokker–Planck equations

Al.I. Cuza University, and Octav Mayer Institute of Mathematics of the Romanian Academy, Iaşi, Romania

^a vb41@uaic.ro

Received: 18 January 2021
Accepted: 10 August 2021
Published online: 1 September 2021

Abstract

We discuss here the asymptotic controllability problem for the Fokker–Planck equations,

      ρ_t − Δβ(ρ) + div(uρ) = 0 in (0, ∞) × ℝ^d,

      ρ(0, x) = ρ0(x), x ∈ ℝ^d,

that is, the existence of a feedback controller $u\equiv u(x,\rho )$ such that ${lim}_{t\to \infty }\rho (t,x)={\rho }_1(x),$ a.e. $x\in {\mathbb{R}}^d$, where ${\rho }_0,{\rho }_1$ are given probability densities and $\beta \in C^2(\mathbb{R})$ is a monotonically increasing function. In this work, it is designed such a controller u for a certain class of final states ${\rho }_1$ which is identified. This problem is related to the controllability of McKean-Vlasov stochastic differential equations and the approach used here relies on the H-theorem established in (Barbu, Rockner in Indiana Univ Math J, 2020), Theorem 6.1, for nonlinear and nondegenerate Fokker–Planck equations.