  In this paper, we analyze optimal control problems governed by semilinear
parabolic equations. Box constraints for the controls are imposed and the cost
functional involves the state and possibly a sparsity-promoting term, but not a
Tikhonov regularization term. Unlike finite dimensional optimization or control
problems involving Tikhonov regularization, second order sufficient optimality
conditions for the control problems we deal with must be imposed in a cone
larger than the one used to obtain necessary conditions. Different extensions
of this cone have been proposed in the literature for different kinds of
minima: strong or weak minimizers for optimal control problems. After a
discussion on these extensions, we propose a new extended cone smaller than
those considered until now. We prove that a second order condition based on
this new cone is sufficient for a strong local minimum.
