Observability Estimate for the Fractional Order Parabolic Equations on Measurable Sets

Abstract

We establish an observability estimate for the fractional order parabolic equations evolved in a bounded domain $\Omega$ of ${ℝ}^n$. The observation region is $F\times \omega$, where $\omega$ and $F$ are measurable subsets of $\Omega$ and (0,$T$), respectively, with positive measure. This inequality is equivalent to the null controllable property for a linear controlled fractional order parabolic equation. The building of this estimate is based on the Lebeau-Robbiano strategy and a delicate result in measure theory provided in Phung and Wang (2013).