# The reflection principle in the control problem of the heat equation

## TL;DR

This paper investigates how reflection symmetry in a domain affects the null-controllability of the heat equation, providing new insights and explicit control cost bounds for various geometric domains.

## Contribution

It establishes that null-controllability on a symmetric domain implies controllability on its parts with bounded control costs, extending to specific geometric shapes.

## Key findings

- Null-controllability on the whole domain implies controllability on symmetric parts.
- Control cost on parts does not exceed that of the whole domain.
- Explicit bounds for control costs on triangles and prisms.

## Abstract

We consider the control problem for the generalized heat equation for a Schroedinger operator on a domain with a reflection symmetry with respect to a hyperplane. We show that if this system is null-controllable, then so is the system on its respective parts and the corresponding control cost does not exceed the one on the whole domain. As an application, we obtain null-controllability results for the heat equation on half-spaces, orthants, and sectors of angle $\pi/2$. As a byproduct, we also obtain explicit control cost bounds for the heat equation on certain triangles and corresponding prisms in terms of geometric parameters of the control set.