NULL-CONTROLLABILITY OF THE KOLMOGOROV EQUATION IN THE WHOLE PHASE SPACE

Jérôme Le Rousseau (1, 2, 3) , Iván Moyano (4)

1 FDP - Fédération de recherche Denis Poisson
2 IUF - Institut universitaire de France
3 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
4 CMLS - Centre de Mathématiques Laurent Schwartz

We prove the null controllability, in arbitrary positive time, of the Kolmogorov equation ∂t + v · ∇x − ∆v with (x, v) ∈ R d × R d , with a control region of the form ω = ωx × ωv, where both ωx and ωv are open subsets of R d that are sufficiently spread out throughout the whole space R d. The proof is based on, on the one hand, a spectral inequality in R d with an observation on ωx, and, on the other hand, a Carleman-based observability inequality for a family of parabolic operators, ∂t − iv · ξ − ∆v, coupled with a knowledge of the decay rate of the free solutions of the Kolmogorov equation.

Mots clés: controllability; unbounded domain; Carleman estimates; spectral inequality

Domaines: Equations aux dérivées partielles [math.AP]

Jérôme Le Rousseau, Iván Moyano. NULL-CONTROLLABILITY OF THE KOLMOGOROV EQUATION IN THE WHOLE PHASE SPACE. 2014. ⟨hal-01134917v1⟩