# Reachability and Controllability Problems for the Heat Equation on a Half-Axis

## TL;DR

This paper investigates controllability and reachability for the heat equation on a half-axis, establishing conditions for approximate reachability and controllability, and demonstrating the absence of null-controllability in finite time.

## Contribution

It provides necessary and sufficient conditions for reachability based on a Markov power moment problem, and clarifies the limits of controllability for this system.

## Key findings

- Every end state is approximately reachable in finite time.
- Every initial state is approximately controllable in finite time.
- No initial state is null-controllable in finite time.

## Abstract

In the paper, problems of controllability, approximate controllability, reachability and approximate reachability are studied for the control system $w_t=w_{xx}$, $w(0,\cdot)=u$, $x>0$, $t\in(0,T)$, where $u\in L^\infty(0,T)$ is a control. It is proved that each end state of this system is approximately reachable in a given time $T$, and each its initial state is approximately controllable in a given time $T$. A necessary and sufficient condition for reachability in a given time $T$ is obtained in terms of solvability a Markov power moment problem. It is also shown that there is no initial state that is null-controllable in a given time $T$. The results are illustrated by examples.