  In this paper, we carry out a rigorous analysis of four-dimensional
variational data assimilation problems for linear and semilinear parabolic
equations with control in the initial condition, coming from a Lebesgue space.
Due to the nature of the data assimilation problem, spatial pointwise
observations of the state have to be considered in the cost functional, which
requires continuity of the solution with respect to the spatial variable. To
obtain this for an initial condition in $L^{\beta}(\Omega)$, with $\beta>d$, we
make use of maximal parabolic regularity tools, jointly with real and complex
interpolation theory. We prove that the data assimilation problems admit an
optimal solution and derive a first-order necessary optimality system. For the
semilinear case, the situation is more involved. Indeed, due to the
differentiability properties of the control-to-state mapping, we can derive
first-order necessary optimality conditions just for values of $\beta>4$. Due
to the same reason, the study of the second-order sufficient optimality
conditions is carried out for $d=2$ and taking $\beta>6$.

Optimization and Control