Observable set, observability, interpolation inequality and spectral inequality for the heat equation in ℝⁿ

2017/11/12 by Gengsheng Wang, Ming Wang, Wang, Gengsheng +5 · 4 citations
Engineering · Mathematics · Computer Science · #Stability and Controllability of Differential Equations #Numerical methods in inverse problems #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.1711.04279

Abstract

This paper studies connections among observable sets, the observability inequality, the Hölder-type interpolation inequality and the spectral inequality for the heat equation in \mathbb Rⁿ. We present a characteristic of observable sets for the heat equation. In more detail, we show that a measurable set in ℝⁿ satisfies the observability inequality if and only if it is γ-thick at scale L for some γ>0 and L>0.We also build up the equivalence among the above-mentioned three inequalities. More precisely, we obtain that if a measurable set E⊂ℝⁿ satisfies one of these inequalities, then it satisfies others. Finally, we get some weak observability inequalities and weak interpolation inequalities where observations are made over a ball.

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