On the Approximate Controllability of Stackelberg-Nash Strategies for Linear Heat Equations in $\mathbb{R}^N$ with Potentials

TL;DR

This paper develops hierarchic control strategies for linear heat equations with potentials in multi-dimensional spaces, overcoming challenges posed by non-compact Sobolev embeddings using advanced mathematical techniques.

Contribution

It introduces new methods employing similarity variables and weighted Sobolev spaces to establish approximate controllability for heat equations with potentials.

Findings

Established hierarchic control for heat equations with potentials in  space.

Overcame lack of compactness issues using similarity variables.

Extended control techniques inspired by Diaz and Lions.

Abstract

In this paper we establish hierarchic control for the linear heat equation in $\re^N$ with potentials. Our strategy is inspired by the techniques developed by J.I. D\'iaz and J.-L. Lions \cite{DL}; however many new difficulties arise due to lack of compactness of Sobolev embeddings. We manage these adversities by employing similarity variables and weighted Sobolev spaces.