Controllability Problems for the Heat Equation with Variable Coefficients on a Half-Axis Controlled by the Neumann Boundary Condition

TL;DR

This paper investigates the controllability and approximate controllability of a heat equation with variable coefficients on a half-axis, demonstrating that any initial state can be approximately driven to any target state within a finite time using boundary control.

Contribution

The paper establishes approximate controllability for a variable-coefficient heat equation on a half-axis using transformation operators, extending controllability results to more complex heat systems.

Findings

Any initial state can be approximately controlled to any target state within finite time.

Transformation operators are effective in analyzing controllability of variable-coefficient heat equations.

Theoretical results are supported by illustrative examples.

Abstract

In the paper, the problems of controllability and approximate controllability are studied for the control system $w_t=\frac{1}{\rho}\left(kw_x\right)_x+\gamma w$, $\left.\left(\sqrt{\frac{k}{\rho}}w_x\right)\right|_{x=0}=u$, $x>0$, $t\in(0,T)$, where $u$ is a control, $u\in L^\infty(0,T)$. It is proved that each initial state of the control system is approximately controllable to any target state in a given time $T>0$. To obtain this result, the transformation operator generated by the equation data $\rho$, $k$, $\gamma$ is applied. The results are illustrated by examples.