Controllability properties for the one-dimensional Heat equation under multiplicative or nonnegative additive controls with local mobile support

TL;DR

This paper investigates the approximate controllability of the one-dimensional Heat equation using nonnegative controls, including both multiplicative and additive types, with static and mobile supports, revealing new links and properties.

Contribution

It introduces new results connecting multiplicative and additive controls for the Heat equation and explores controllability with mobile control supports.

Findings

Approximate controllability achieved with static supports.

Controllability established with mobile supports.

Links between multiplicative and additive control methods.

Abstract

We discuss several new results on nonnegative approximate controllability for the one-dimensional Heat equation governed by either multiplicative or nonnegative additive control, acting within a proper subset of the space domain at every moment of time. Our methods allow us to link these two types of controls to some extend. The main results include approximate controllability properties both for the static and mobile control supports.