In this chapter, we analyze an optimal control problem involving equality constraints for the state variable at finitely many given points. The state equation and the cost functional are the same as in Chap. 4 . Considering state constraints leads to an optimality system involving Dirac measures on the right-hand side of the adjoint equation, which limits the regularity of the optimal control. We derive precise regularity results and provide optimal error estimates for the error in the control variable. Numerical results suggest that the state variable and the Lagrange multipliers converge at a better rate. We use a duality argument for the whole optimality system to prove the corresponding result.