We use recent improvements in the parametrizations of controllable linear multidimensional systems to show how to transform the study of a linear quadratic optimal problem into that of a variational problem without constraints. We give formal conditions on the differential module defined by the linear control system to pass from the Pontryagin approach to a purely Euler-Lagrange variational problem. This formal approach uses the cost function in order to link the locally exact sequence formed by the controllable system and its parametrizations with the sequence formed by their formal adjoint operators. In the case of partial differential equations, this scheme is typical for any problem of linear elasticity theory and electromagnetism.