Observability Inequalities from Measurable Sets for Some Abstract Evolution Equations

Abstract

This paper studies the observability from measurable sets in time for an evolution equation (in a Hilbert space): $u'=Au$ ($t\geq0)$, with an observation operator $B$. We obtain such an observability inequality in two different settings on $(A,B)$. In the first setting, $A$ generates an analytic semigroup, $B$ is an admissible observation operator for this semigroup, and $(A,B)$ satisfies an observability inequality from time intervals. By the propagation estimate of analytic functions and a telescoping series method (provided in the current paper), we build up the desired inequality for this setting. In the second setting, $A$ generates a $C_0$ semigroup, $B$ is a linear and bounded operator, and $(A, B)$ satisfies a spectral-like condition. By methods developed by Phung and Wang [J. Eur. Math. Soc. (JEMS), 15 (2013), pp. 681--703] and Apraiz et al. [J. Eur. Math. Soc. (JEMS), 16 (2014), pp. 2433--2475], we first obtain an interpolation inequality at one time point, and then derive the desired observability inequality for the second setting. This observability inequality is applied to get the bang-bang property of time optimal control problems for several kinds of differential equations.