Existence of optimal controls for systems governed by second order linear parabolic partial delay-differential equations with first boundary conditions

In this paper, we consider a class of systems governed by second order linear parabolic delay-partial differential equations with first boundary conditions. Our main results are reported in Theorems 3.1 and 3.2. As in [9, Theorems 4.1 and 4.2], the coefficients and forcing terms of the system considered in Theorem 3.1 are linear in the control variables. On the other hand, the forcing terms of the system considered in Theorem 3.2 are allowed to be nonlinear in the control variables at the expense of dropping the control variables in the cost integrand.