An application of a conjecture due to Ervedoza and Zuazua concerning the observability of the heat equation in small time to a conjecture due to Coron and Guerrero concerning the uniform controllability of a convection-diffusion equation in the vanishing viscosity limit

Pierre Lissy (1)

1 LJLL - Laboratoire Jacques-Louis Lions

The aim of this short paper is to explore a new connection between a conjecture concerning sharp boundary observability estimates for the $1$-D heat equation in small time and a conjecture concerning the cost of null-controllability for a $1$-D convection-diffusion equation with constant coefficients controlled on the boundary in the vanishing viscosity limit, in the spirit of what is done in [Pierre Lissy, A link between the cost of fast controls for the $1$-D heat equation and the uniform controllability of a 1-D transport-diffusion equation, C. R. Math. Acad. Sci. Paris, Volume 352, 2012]. We notably establish that the first conjecture implies the second one as soon as the speed of the transport part is non-negative in the transport-diffusion equation.

Mots clés: Vanishing viscosity limit; Transport-diffusion equation; Null controllability; Vanishing viscosity limit.

Domaines: Optimisation et contrôle [math.OC]

Pierre Lissy. An application of a conjecture due to Ervedoza and Zuazua concerning the observability of the heat equation in small time to a conjecture due to Coron and Guerrero concerning the uniform controllability of a convection-diffusion equation in the vanishing viscosity limit. Systems and Control Letters, 2014, 69, http://www.sciencedirect.com/science/article/pii/S0167691114001029. ⟨10.1016/j.sysconle.2014.04.011⟩. ⟨hal-00873835⟩