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In this paper, a stabilized finite element method for optimal control problems governed by a convection dominated diffusion equation is investigated. The state and the adjoint variables are approximated by piecewise linear continuous functions with bubble functions. The control variable either is approximated by piecewise linear functions (called the standard method) or is not discretized directly (called the variational discretization method). The stabilization term only depends on bubble functions, and the projection operator can be replaced by the difference of two local Gauss integrations. A priori error estimates for both methods are given and numerical examples are presented to illustrate the theoretical results. [ABSTRACT FROM PUBLISHER]