In this talk we discuss a point-wise state constraint problem for a general class of PDEs optimal control problems and sparsity optimization. We use the penalty formulation and derive the necessary optimality condition based on the Lagrange multiplier theory.The existence of Lagrange multiplier associated with  the point-wise state constraint as a measure is established. Also we develop a semi-smooth Newton method for the penalty formulation. Numerical tests are presented for parabolic, elliptic and wave equation control problems.