Construction of gevrey functions with compact support using the bray-mandelbrojt iterative process and applications to the moment method in control theory

Pierre Lissy

doi:10.3934/mcrf.2017002

Abstract

In this paper, we construct some interesting Gevrey functions of order α for every α > 1 with compact support by a clever use of the Bray-Mandelbrojt iterative process. We then apply these results to the moment method, which will enable us to derive some upper bounds for the cost of fast boundary controls for a class of linear equations of parabolic or dispersive type that partially improve the existing results proved in [P. Lissy, On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension SIAM J. Control Optim., 52(4), 2651-2676]. However this construction fails to improve the results of [G. Tenenbaum and M. Tucsnak, New blow-up rates of fast controls for the Schrodinger and heat equations, Journal of Differential Equations, 243 (2007), 70-100] in the precise case of the usual heat and Schrodinger equation.

Highlights

The main motivation of this paper is to continue the study of [11] concerning the estimation for the cost of fast “boundary” controls for a class of linear equations of parabolic or dispersive type
We assume that −A generates on H a strongly continuous semigroup S : t → S(t) = e−tA
The cost of the control is proved to be bounded from above by exp K/(RT )1/(α−1) where K is some explicit constant depending on α

Summary

Introduction

The author proved precise upper bounds concerning the cost of the control for some large classes of linear parabolic or dispersive equations when the time T goes to 0, where the underlying “elliptic” operator was chosen to be self-adjoint or skewadjoint with eigenvalues roughly as Rkα or ±Rkα (only for (2)) for some R > 0 and α ≥ 2 when k → +∞. The cost of the control is proved to be bounded from above by exp K/(RT )1/(α−1) where K is some explicit constant depending on α This does not cover all possible cases, because it is well-known that equations like (1) and (2) are controllable in arbitrary small time if and only if α > 1.

We observe that for α

Let us remark that b

We remark every j