Study of the cost of the controllability of second order parabolic equations with small diffusion and a transport term

Jon Asier Bárcena-Petisco (1)

1 LJLL - Laboratoire Jacques-Louis Lions

In this paper we consider the heat equation with Neumann, Robin and mixed boundary conditions (with coefficients on the boundary which depend on the space variable). The main results concern the behaviour of the cost of null controllability with respect to the diffusion coefficient when the control acts in the interior. First, we prove that if we almost have Dirichlet boundary conditions in the part of the boundary in which the flux of the transport enters the cost of the controllability decays. Next, we show some examples of Neumann and mixed boundary conditions in which for any time T > 0 the cost explodes exponentially as the diffusion coefficient vanishes. Finally, we study some systems in which the control is located on the whole domain.

Mots clés: Heat equation; singular limits; spectral decomposition; transport equation; uniform controllability

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

Jon Asier Bárcena-Petisco. Study of the cost of the controllability of second order parabolic equations with small diffusion and a transport term. 2020. ⟨hal-02455632v1⟩