Error estimates for linear-quadratic control problems with control constraints

An abstract linear-quadratic optimal control problem is investigated with pointwise control constraints. This paper is concerned in discretization of the control by piecewise linear functions. Under the assumption that the optimal control and the optimal adjoint state is Lipschitz continuous and piecewise of class $C^2$ an approximation of order $h^{3/2}$ is proved for the solution of the control discretized problem with respect to the solution of the continuous one. Numerical tests are presented after the theoretical part