## ESAIM: Control, Optimisation and Calculus of Variations

### The steepest descent dynamical system with control. Applications to constrained minimization

Laboratoire LACO, Faculté des Sciences, 123 avenue Albert Thomas, 87060 Limoges Cedex, France; alexandre.cabot@unilim.fr.

Abstract

Let H be a real Hilbert space, a convex function of class that we wish to minimize under the convex constraint S. A classical approach consists in following the trajectories of the generalized steepest descent system (cf.   Brézis [CITE]) applied to the non-smooth function  . Following Antipin [1], 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 S and a control parameter tending to zero, we consider the “Steepest Descent and Control” system where the control ε satisfies . This last condition ensures that ε “slowly” tends to 0. When H 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.

(Online publication March 15 2004)

Key Words:

• Dissipative dynamical system;
• steepest descent method;
• constrained optimization;
• convex minimization;
• asymptotic behaviour;
• non-linear oscillator.

Mathematics Subject Classification:

• 34A12;
• 34D05;
• 34G20;
• 34H05;
• 37N40