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We investigate $C^1$ finite element methods for one dimensional ellipticdistributed optimal control problems with pointwise constraints on thederivative of the state formulated as fourth order variational inequalities forthe state variable. For the problem with Dirichlet boundary conditions, we usean existing $H^{\frac52-\epsilon}$ regularity result for the optimal state toderive $O(h^{\frac12-\epsilon})$ convergence for the approximation of theoptimal state in the $H^2$ norm. For the problem with mixed Dirichlet andNeumann boundary conditions, we show that the optimal state belongs to $H^3$under appropriate assumptions on the data and obtain $O(h)$ convergence for theapproximation of the optimal state in the $H^2$ norm.

Finite Element Methods for One Dimensional Elliptic Distributed Optimal Control Problems with Pointwise Constraints on the Derivative of the State