  We investigate the link between inverse problems and final state
observability for a general class of parabolic systems. We generalize a
stability result for initial data due to Garc\'ia and Takahashi [16], known for
the case of self-adjoint dissipative operators. More precisely, we consider a
system governed by an analytic semigroup. Under the assumption of final state
observability, we prove a logarithmic stability estimate depending on the
analyticity angle of the semigroup. This is done by using a general logarithmic
convexity result. The abstract result is illustrated by considering the
Ornstein-Uhlenbeck equation.
