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Problems involving the optimal control or ordinary of partial differential equations are infinite dimensional problems which are approximated by discretized problems for their numerical solution. Quasi Newton methods were applied to these finite dimensional problems and it was shown by analysis and numerical tests how the convergence rate could be predicted using information from the underlying infinite dimensional problem. For unconstrained optimal control problems with ordinary differential equations two approaches were studied: In one approach the control functions were used as unknowns whereas for the second route the control, state and costate functions were taken as unknowns. In the latter approach the quasi Newton update made extensive use of the special structure of the control problem and proved to be very effective. These algorithms were also studied for nonlinear elliptic boundary value problems and the optimal control of pseudoparabolic differential equations.