The aim of this paper is to deal with an output controllability problem. It consists in driving the state of a distributed parabolic system toward a state between two prescribed functions on a boundary subregion of the system evolution domain with minimum energy control. Two necessary conditions are derived. The first one is formulated in terms of subdifferential associated with a minimized functional. The second one is formulated as a system of equations for arguments of the Lagrange systems. Numerical illustrations show the efficiency of the second approach and lead to some conjectures.