Control of the half-heat equation

TL;DR

This paper characterizes null-controllable states of the half-heat equation on a sub-arc of the circle, providing necessary and sufficient conditions based on frequency projections, and shows the controllable space is time-invariant.

Contribution

It introduces nearly sharp conditions for null-controllability of the half-heat equation using frequency projections and extends the analysis to the unprojected case via holomorphic function theory.

Findings

Necessary and sufficient conditions for null-controllability based on positive frequency projections.

The space of null-controllable functions is independent of time.

Controllability conditions are expressed through projections on positive frequencies supported on the control arc.

Abstract

In this paper we investigate null-controllable initial states of the half heat equation controlled from a sub-arc $\omega$ of the unit circle. We also study the projection on positive frequencies of the half-heat equation. For this projected half-heat equation, we obtain necessary as well as sufficient conditions for an initial condition to be null-controllable. These conditions, which are almost sharp, are expressed in term of projections on positive frequencies of functions supported on $\omega$. From these results, and with the help of classical results on sum of holomorphic and anti-holomorphic functions, we also treat the (unprojected) half-heat equation. Surprisingly, without using our conditions on null-controllable states, we are able to show that the space of null-controllable functions does not depend on time by using a result of separation of singularities for holomorphic functions.