A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit
نویسنده
چکیده
We revisit a result by Coron and Guerrero stating that the one-dimensional transport-diffusion equation ut + Mux − εuxx = 0 in (0, T )× (0, L), controlled by the left Dirichlet boundary value is zero-controllable at a bounded cost as ε → 0, when T > 4.3 L/M if M > 0 and when T > 57.2 L/|M | if M < 0. By a completely different method, relying on complex analysis, we prove that this still holds when T > 4.2 L/M if M > 0 and when T > 6.1 L/|M | if M < 0.
منابع مشابه
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
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 [Pier...
متن کاملA link between the cost of fast controls for the 1-D heat equation and the uniform controllability of a 1-D transport-di usion equation
In this note, we explain how results on the cost of the null-controllability of the one-dimensional heat equation in small time can be used to bound from above the cost of the null-controllability of a one-dimensional transport-di usion equation in the vanishing viscosity limit. We improve previous results about the minimal time needed to obtain the exponential decrease of the cost of the contr...
متن کاملSingular Optimal Control of a 1-D Parabolic-Hyperbolic Degenerate Equation
In this paper, we consider the controllability of a strongly degenerate parabolic equation with a degenerate one-order transport term. Despite the strong degeneracy, we prove a result of well-posedness and null controllability with a Dirichlet boundary control that acts on the degenerate part of the boundary. Then, we study the uniform controllability in the vanishing viscosity limit and prove ...
متن کاملApproximation of the controls for the beam equation with vanishing viscosity
We consider a finite difference semi-discrete scheme for the approximation of the boundary controls of a 1-D equation modelling the transversal vibrations of a hinged beam. It is known that, due to the high frequency numerical spurious oscillations, the uniform (with respect to the mesh-size) controllability property of the semi-discrete model fails in the natural setting. Consequently, the con...
متن کاملSmall Time Uniform Controllability of the Linear One-Dimensional Schrödinger Equation with Vanishing Viscosity
This article considers the linear 1-d Schrödinger equation in (0, π) perturbed by a vanishing viscosity term depending on a small parameter ε > 0. We study the boundary controllability properties of this perturbed equation and the behavior of its boundary controls vε as ε goes to zero. It is shown that, for any time T sufficiently large but independent of ε and for each initial datum in H−1(0, ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2009