Cost evaluation of finite dimensional and approximate null-controllability for the one dimensional half-heat equation

Abstract

In this article we evaluate the cost of the finite dimensional and the approximate controllability of a one dimensional anomalous diffusion equation which is not null-controllable. The finite dimensional controllability property consists in driving to zero a finite number of frequencies of the corresponding solutions. To evaluate its cost, we construct an explicit biorthogonal sequence to a finite part of a family of real exponential functions. Since the exponents of these functions satisfy the so-called Müntz density condition, there is no biorthogonal sequence to the entire family which shows that the equation is not even spectrally controllable. For the approximate controllability problem, we combine the idea of finite dimensional controllability studied before and the intrinsic dissipative nature of the system. These arguments allow to show that the controlled solution can be made arbitrarily small after sufficiently large time and provide estimates for the corresponding approximate controls.