Bilinear local controllability to the trajectories of the Fokker–Planck equation with a localized control

Abstract

This work is devoted to the control of the Fokker-Planck equation, posed on a bounded domain of R^d (d>0). More precisely, the control is the drift force, localized on a small open subset. We prove that this system is locally controllable to regular nonzero trajectories. Moreover, under some conditions on the reference control, we explain how to reduce the number of controls around the reference control. The results are obtained thanks to a linearization method based on a standard inverse mapping procedure and the fictitious control method. The main novelties of the present article are twofold. Firstly, we propose an alternative strategy to the standard fictitious control method: the algebraic solvability is performed and used directly on the adjoint problem. Secondly, we prove a new Carleman inequality for the heat equation with one order space-varying coefficients: the right-hand side is the gradient of the solution localized on a subset (rather than the solution itself), and the left-hand side can contain arbitrary high derivatives of the solution.