On the approximate controllability of Stackelberg–Nash strategies for linear heat equations in ℝ^N with potentials

Abstract

In this paper, we establish hierarchic control for the linear heat equation in ℝ^N with potentials. Our strategy is inspired by the techniques developed by Díaz and Lions [On the approximate controllability of Stackelberg-Nash strategies. In: Díaz JI, editor. Ocean circulation and pollution control mathematical and numerical investigations. Berlin: Springer; 2005. p. 17–27]; 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.

Keywords:

• hierarchic control
• Nash–Stackelberg strategy
• weighted Sobolev spaces
• unbounded domains

AMS Subject Classifications:

• Primary 35Q10
• 35B37
• Secondary 35B40

Acknowledgements

The authors want to express their gratitude to the anonymous reviewers for their questions and commentaries since they were very helpful in improving this article.