ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - April 2004 - Volume 10, Issue 02  

Research Article

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

Cabot, Alexandre

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, $\Phi_1: H\to \xR$ a convex function of class ${\mathcal C}^1$ 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  $\Phi_1+\delta_S$ . 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  $\Phi_0: H\to \xR$ whose critical points coincide with S and a control parameter $\varepsilon:\xR_+\to \xR_+$ tending to zero, we consider the “Steepest Descent and Control” system \[(SDC) \qquad \dot{x}(t)+\nabla \Phi_0(x(t))+\varepsilon(t)\, \nabla \Phi_1(x(t))=0,\] where the control ε satisfies $\int_0^{+\infty} \varepsilon(t)\, {\rm d}t =+\infty$ . This last condition ensures that ε “slowly” tends to 0. When H is finite dimensional, we then prove that $d(x(t), {\rm argmin}\kern 0.12em_S \Phi_1) \to 0 \quad (t\to +\infty),$ and we give sufficient conditions under which  $x(t) \to \bar{x}\in \,{\rm argmin}\kern 0.12em_S \Phi_1$ . We end the paper by numerical experiments allowing to compare the (SDC) system with the other systems already mentioned.

(Received February 17 2003)

(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