<p>We consider linear quadratic optimal control problems for the heat equation with homogeneous Dirichlet or Neumann boundary conditions, initial data in Besov spaces, and bilateral control constraints in Lebesgue spaces. Generalizations of the classical results involving state and gradient observations will be established. Cost functionals defined on lower-dimensional subsets of the space-time domain will also be considered, for example, observations on the lateral boundary or on a finite set of points in the domain. It is well-known that the adjoint equations to such control problems are PDEs with Radon measure data. Our focus is to describe the regularity of the adjoint states for these problems. The analysis relies on the maximal regularity for parabolic equations, including suitable extensions, along with duality and interpolation methods.</p>