  The DPG method with optimal test functions for solving linear quadratic
optimal control problems with control constraints is studied. We prove
existence of a unique optimal solution of the nonlinear discrete problem and
characterize it through first order optimality conditions. Furthermore, we
systematically develop a priori as well as a posteriori error estimates. Our
proposed method can be applied to a wide range of constrained optimal control
problems subject to, e.g., scalar second-order PDEs and the Stokes equations.
Numerical experiments that illustrate our theoretical findings are presented.
