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In this paper, we build up two observability inequalities from measurablesets in time for some evolution equations in Hilbert spaces from two differentsettings. The equation reads: $u'=Au,\; t>0$, and the observation operator isdenoted by $B$. In the first setting, we assume that $A$ generates an analyticsemigroup, $B$ is an admissible observation operator for this semigroup (cf.\cite{TG}), and the pair $(A,B)$ verifies some observability inequality fromtime intervals. With the help of the propagation estimate of analytic functions(cf. \cite{V}) and a telescoping series method provided in the current paper,we establish an observability inequality from measurable sets in time. In thesecond setting, we suppose that $A$ generates a $C_0$ semigroup, $B$ is alinear and bounded operator, and the pair $(A, B)$ verifies some spectral-likecondition. With the aid of methods developed in \cite{AEWZ} and \cite{PW2}respectively, we first obtain an interpolation inequality at one time, and thenderive an observability inequality from measurable sets in time. These twoobservability inequalities are applied to get the bang-bang property for sometime optimal control problems.

Observability inequalities from measurable sets for some evolution equations