Summary: Half century ago, Umberto Mosco was the ``relatore di tesi (tesi about the Mosco-convergence) di laurea'' of the first author; a quart of century ago, the first author was the ``relatore di tesi di laurea'' of the second author. The roots of this paper are the Mosco-convergence of convex sets and the minimization of integral functionals of the Calculus of Variations. We consider integral functionals of the type \[ J(v)=\int_\Omega j(x,Dv)-\int_\Omega f(x)v(x). \] We study the existence of T-minima (infinite energy minima) on convex sets of the Sobolev space \(W_0^{1,p} (\Omega)\) and the stability of the T-minima under the Mosco-convergence of the convex sets.