Internal control of the transition kernel for stochastic lattice dynamics

In [4], we initiated the first study of control problems for kinetic equations arising from harmonic chains. Specifically, we developed impulsive and feedback control mechanisms for harmonic chains coupled with a point thermostat, effectively enabling control over the boundary conditions of the corresponding kinetic equations. However, the more intricate and fundamental challenge of internal control - namely, the design of control strategies that influence the collision operators within the kinetic framework - remained open.

In the present work, we address the internal control problem for stochastic lattice dynamics, with the objective of controlling the transition kernel of the limiting kinetic equation. A central innovation of our approach is the development of a novel geometric-combinatorial framework, which enables the systematic construction of control pathways within the microscopic dynamics. This methodology opens a new avenue for the internal control of kinetic equations.