In the second order analysis of infinite dimension optimization problems, we have to deal with the so-called two-norm discrepancy. As a consequence of this fact, the second order optimality conditions usually imply local optimality in the L ∞  sense. However, we have observed that the L² local optimality can be proved for many control problems of partial differential equations. This can be deduced from the standard second order conditions. To this end, we make some quite realistic assumptions on the second derivative of the cost functional. These assumptions do not hold if the control does not appear explicitly in the cost functional. In this case, the optimal control is usually of bang-bang type. For this type of problems we also formulate some new second order optimality conditions that lead to the strict L² local optimality of the bang-bang controls.