Some control and inverse problems for linear parabolic equations

Properties of solutions of control and inverse problems for one-dimensional parabolic equations with coefficients dependent on $(x,t)$ are studied. The proposed approach based on the duality principle allows one to generalize the known Lions' result on the density properties of averaged observations in control problems with a control function given in the initial conditions. It is shown that the significance of these density properties is not restricted to the control problems. Such properties are used to study inverse parabolic problems, in particular, to study the uniqueness conditions of their solutions.