Boundary Control of Vlasov--Fokker--Planck Equations

TL;DR

This paper develops a new Lyapunov-based boundary control method for stabilizing linear Vlasov--Fokker--Planck equations using hypocoercivity, and analyzes the macroscopic limit to prevent boundary layers.

Contribution

It introduces a novel Lyapunov function and extends hypocoercivity techniques to design boundary feedback controls for stabilization.

Findings

Successfully stabilizes Vlasov--Fokker--Planck equations with boundary control.

Derives conditions to avoid boundary layers in the macroscopic limit.

Extends hypocoercivity analysis to boundary control design.

Abstract

We introduce a novel Lyapunov function for stabilization of linear Vlasov--Fokker--Planck type equations with stiff source term. Contrary to existing results relying on transport properties to obtain stabilization, we present results based on hypocoercivity analysis for the Fokker--Planck operator. The existing estimates are extended to derive suitable feedback boundary control to guarantee the exponential stabilization. Further, we study the associated macroscopic limit and derive conditions on the feedback boundary control such that in the formal limit no boundary layer exists.