Construction of Exact Control for a One-Dimensional Heat Equation with   Delay
  We prove an exact controllability result for a one-dimensional heat equation with delay in both lower and highest order terms and nonhomogeneous Dirichlet boundary conditions. Moreover, we give an explicit representation of the control function steering the system into a given final state. Under certain decay properties for corresponding Fourier coefficients which can be interpreted as a sufficiently high Sobolev regularity of the data, both control function and the solution are proved to be regular in the classical sense both with respect to time and space variables. 
Denys Khusainov, Michael Pokojovy, Elvin Azizbayov
math.OC