The author considers the optimization problem Minimize \(f(x)\) subject to \(c(x)=0\), \(a\leq u\leq b\) componentwise, where \(x=(y,u)\in \mathbb{R}^{m+n}\) and \(f:\mathbb{R}^{m+n} \to \mathbb{R}\), \(c: \mathbb{R}^{m+n}\to \mathbb{R}^m\) are sufficiently smooth. Such problems frequently arise in the numerical solution of optimal control problems. In the paper they are solved by projected sequential quadratic programming (SQP) methods. Global and local convergence properties and the identification of active indices are discussed. Numerical examples for an optimal control problem governed by a nonlinear heat equation are presented.