A numerical method for linear quadratic optimal control problems with pure state constraints is analyzed. Using the virtual control concept introduced by Cherednichenko et al. (Inverse Probl. 24:1-21, 2008) and Krumbiegel and Rösch (Control Cybern. 37(2):369-392, 2008), the state constrained optimal control problem is embedded into a family of optimal control problems with mixed control-state constraints using a regularization parameter α > 0. It is shown that the solutions of the problems with mixed control-state constraints converge to the solution of the state constrained problem in the L2 norm as α tends to zero. The regularized problems can be solved by a semi-smooth Newton method for every α > 0 and thus the solution of the original state constrained problem can be approximated arbitrarily close as α approaches zero. Two numerical examples with benchmark problems are provided.