Simultaneous control for the heat equation with dirichlet and neumann boundary conditions

TL;DR

This paper demonstrates that it is possible to simultaneously control the heat equation with Dirichlet and Neumann boundary conditions to reach null state using a single control function, extending controllability results.

Contribution

It introduces a method to achieve simultaneous null controllability for both Dirichlet and Neumann heat equations with one control, which was not previously established.

Findings

Simultaneous null controllability for Dirichlet and Neumann heat equations.

A single control function can steer both systems to zero.

Extends controllability theory to coupled boundary condition problems.

Abstract

It is well known that both the heat equation with Dirichlet or Neumann boundary conditions are null controlable as soon as the control acts in a non trivial domain (i.e. a set of positive measure, see [10, 11, 12, 1, 6]. In this article, we show that for any couple of initial data (u0, v0) we can achieve the null control for both equations (Dirichlet and Neumann boundary conditions respectively) simultaneously with the same control function for both equations.