Crank--Nicolson Schemes for Optimal Control Problems with Evolution Equations

Crank--Nicolson methods are often used for the simulation of initial boundary value problems for parabolic partial differential equations. In this paper we present a family of discretizations for parabolic optimal control problems based on Crank--Nicolson schemes with different time discretizations for state y and adjoint state p so that discretization and optimization commute. One of these methods can also be explained as a Störmer--Verlet scheme in the context of geometric numerical integration of Hamiltonian systems. Two of the schemes may also be obtained as a Galerkin method with quadrature. Further we investigate the schemes for a variable time step size and prove second order convergence for this case if the time step size is chosen with respect to an arbitrary, sufficiently smooth mesh generating function.