On the cost of fast controls for some families of dispersive or parabolic equations in one space dimension

TL;DR

This paper investigates the minimal energy required to control certain linear dispersive and parabolic equations in one dimension within small time frames, providing bounds that are proven to be optimal.

Contribution

It extends previous work to establish precise upper and lower bounds on control costs, demonstrating the optimality of these bounds for a broad class of equations.

Findings

Derived explicit upper bounds on control cost as time approaches zero.

Established lower bounds confirming the optimality of the control cost power law.

Applied results to specific equations like KdV and fractional Schrödinger equations.

Abstract

In this paper, we consider the cost of null controllability for a large class of linear equations of parabolic or dispersive type in one space dimension in small time. By extending the work of Tenenbaum and Tucsnak in "New blow-up rates for fast controls of Schr\"odinger and heat equations`", we are able to give precise upper bounds on the time-dependance of the cost of fast controls when the time of control T tends to 0. We also give a lower bound of the cost of fast controls for the same class of equations, which proves the optimality of the power of T involved in the cost of the control. These general results are then applied to treat notably the case of linear KdV equations and fractional heat or Schr\"odinger equations.