  The present paper aims at providing a numerical strategy to deal with
PDE-constrained optimization problems solved with the adjoint method. It is
done through out a unified formulation of the constraint PDE and the adjoint
model. The resulting model is a non-conservative hyperbolic system and thus a
finite volume scheme is proposed to solve it. In this form, the scheme sets in
a single frame both constraint PDE and adjoint model. The forward and backward
evolutions are controlled by a single parameter $\eta$ and a stable time step
is obtained only once at each optimization iteration. The methodology requires
the complete eigenstructure of the system as well as the gradient of the cost
functional. Numerical tests evidence the applicability of the present technique
