Summary: Let be a real Hilbert space, a convex function of class that we wish to minimize under the convex constraint . A classical approach consists in following the trajectories of the generalized steepest descent system (cf. [H Brézis, “Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert” (1999; Zbl 0252.47055)]) applied to the non-smooth function . Following A. S. Antipin [Differ. Equations 30, No. 9, 1365–1375 (1994); translation from Differ. Uravn. 30, No. 9, 1475–1486 (1994; Zbl 0852.49021)] it is also possible to use a continuous gradient-projection system. We propose here an alternative method as follows: given a smooth convex function whose critical points coincide with and a control parameter tending to zero, we consider the “Steepest Descent and Control” system
where the control satisfies . This last condition ensures that “slowly” tends to 0. When is finite dimensional, we then prove that , and we give sufficient conditions under which . We end the paper by numerical experiments allowing to compare the (SDC) system with the other systems already mentioned.