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 1). 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.

Local null controllability of a quasi-linear system and related numerical experiments

“…To this respect, we apply a boot-strapping argument: after requiring higher regularity to the initial data, the controls can be found with higher regularity, thus guaranteeing more regularity to the state variables; in turn, this makes it possible to go further and get an even more regular control, a more regular state, etc. The authors of [18] employ a similar idea.…”

Turnpike in Lipschitz—nonlinear optimal control

“…This would be the somewhat true analog of the control-affine systems presented herein, and under suitable assumptions on the nonlinearity, one could expect that our methodology applies to such cases as well. We have not addressed such systems for the simplicity of presentation and due to the controllability assumptions we make, as the controllability theory for bilinear problems is not complete (albeit, see [4,5,10,34] for recent developments). Notwithstanding, our results should be applicable to a system of the form (see [4])…”

“…Since T − 2nτ 2T 0 , as in step 1, it can be shown that u T | [nτ ,T−nτ ] is a global minimizer of J nτ ,T−nτ defined in (5.2). Taking these facts into account, and noting that (5.27) holds 10 , we can apply lemma 5.3 on [nτ , T − nτ ] (noting (5.28)), and lemma 5.2 with τ 1 = nτ and τ 2 = T − nτ , to deduce that there exist a couple of times t 1 ∈ [nτ , (n + 1)τ ) and…”