Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation

Pierre Lissy (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

In this paper, we prove explicit lower bounds for the cost of fast boundary controls for a class of linear equations of parabolic or dispersive type involving the spectral fractional Laplace operator. We notably deduce the following striking result: in the case of the heat equation controlled on the boundary, the Miller's conjecture formulated in [Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time, J. Differential Equations, 204 (2004), pp. 202-226] is not verified. Moreover, we also give a new lower bound for the minimal time needed to ensure the uniform controllability of the one-dimensional convection-diffusion equation with negative speed controlled on the left boundary, proving that the conjecture formulated in [J.-M. Coron and S. Guerrero, Singular optimal control: A linear 1-D parabolic-hyperbolic example, Asymptot. Anal., 44 (2005), pp. 237-257] concerning this problem is also not verified at least for negative speeds. The proof is based on complex analysis, and more precisely on a representation formula for entire functions of exponential type, and is quite related to the moment method of Fattorini and Russell.

Mots clés: Uniform controllability in the vanishing viscosity limit; Parabolic equations; Dispersive equations; Cost of fast controls; Control of partial differential equations

Domaines: Optimisation et contrôle [math.OC]

Pierre Lissy. Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation. Journal of Differential Equations, 2015, 259 (10), pp.5331-5352. ⟨10.1016/j.jde.2015.06.031⟩. ⟨hal-01133526⟩