Finite element methods for optimal control problems governed by linear quasi-parabolic integro-differential equations (Q2865626)

The authors derive optimality conditions for the optimal control problem (OCP) governed by a linear quasi-parabolic integro-differential equation. Although there exists a huge literature about OCP problems governed by elliptic and parabolic equations, this class of OCPs was not studied before. The regularity and existence of the solution for the OCP are analyzed. A priori error estimates are derived for the linear finite element approximation of the OCP. Numerical experiments confirm the theoretical error estimates for the control variable \(\mathcal{O}(h)\) and for state and adjoint variables \(\mathcal{O}(h)^2\) in the \(L^2\)-norm with the mesh size \(h\) in space.