Controllability Problems for the Heat Equation in a Half-Plane Controlled by the Neumann Boundary Condition with a Point-Wise Control

TL;DR

This paper analyzes controllability and approximate controllability for the heat equation on a half-plane with a Neumann pointwise boundary control, formulating the problem as $w_t=\Delta w$ with $w_{x_1}(0,x_2,t)=u(t)\delta(x_2)$. By applying an even extension in $x_1$ and a Laguerre-based spectral approach, the authors reduce the 2D problem to a 1D framework and characterize end-states via reachability sets governed by a Markov power moment problem. They establish a complete picture: (i) necessary and sufficient conditions for controllability under a bound, (ii) that the end-states are dense in a Hilbert space $\mathcal{H}$ for approximate controllability with $u\in L^{\infty}(0,T)$, and (iii) that null-controllability to the origin is impossible unless the initial data vanish; plus an algorithmic method and an illustrative example. The work leverages transforms $\Psi$ and $\Phi$ to connect $L^2(\mathbb{R}_+)$ with $L^2(\mathbb{R}^2)$, enabling explicit moment-based criteria and practical numerical schemes.

Abstract

In the paper, the problems of controllability and approximate controllability are studied for the control system $w_t=Δw$, $w_{x_1}(0,x_2,t)=u(t)δ(x_2)$, $x_1>0$, $x_2\in\mathbb R$, $t\in(0,T)$, where $u\in L^\infty(0,T)$ is a control. To this aid, it is investigated the set $\mathcal{R}_T(0)\subset L^2((0,+\infty)\times\mathbb R)$ of its end states which are reachable from $0$. It is established that a function $f\in\mathcal{R}_T(0)$ can be represented in the form $f(x)=g\big(|x|^2\big)$ a.e. in $(0,+\infty)\times\mathbb R$ where $g\in L^2(0,+\infty)$. In fact, we reduce the problem dealing with functions from $L^2((0,+\infty)\times\mathbb R)$ to a problem dealing with functions from $L^2(0,+\infty)$. Both a necessary and sufficient condition for controllability and a sufficient condition for approximate controllability in a given time $T$ under a control $u$ bounded by a given constant are obtained in terms of solvability of a Markov power moment problem. Using the Laguerre functions (forming an orthonormal basis of $L^2(0,+\infty)$), necessary and sufficient conditions for approximate controllability and numerical solutions to the approximate controllability problem are obtained. It is also shown that there is no initial state that is null-controllable in a given time $T$. The results are illustrated by an example.

Figures (3)

• Figure I.5.1: The controls $\EuScript{U}^N_l$ defined by \ref{['contr3']}.
• Figure I.5.2: The influence of the controls $\EuScript{U}^N_l$ on the difference $W^T-\left(\EuScript W_0(\cdot,T)+\EuScript W_{\EuScript{U}^N_l}(\cdot,T)\right)$ with $T=3$.
• Figure I.5.3: The influence of the controls $\EuScript{U}^N_l$ on the difference $W^T-\left(\EuScript W_0(\cdot,T)+\EuScript W_{\EuScript{U}^N_l}(\cdot,T)\right)$ with $T=3$ (vertical section for $x_2=0$ and horizontal section for $x_1=0$).

Theorems & Definitions (30)

• Proposition I.2.1
• Proposition I.2.2
• Definition I.2.3
• Remark I.2.4
• Theorem I.2.5
• Theorem I.3.1
• Definition I.3.2
• Definition I.3.3
• Theorem I.3.4
• Theorem I.3.5