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The steepest descent dynamical system with control. Applications to constrained minimization. (English) Zbl 1072.49004

Summary: Let H be a real Hilbert space, Φ 1 :H a convex function of class 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. [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 Φ 1 +δ S . 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 Φ 0 :H whose critical points coincide with S and a control parameter ε: + + tending to zero, we consider the “Steepest Descent and Control” system

x ˙(t)+Φ 0 x ( t )+ε(t)Φ 1 x ( t )=0,( SDC )

where the control ε satisfies 0 + ε(t)dt=+. This last condition ensures that ε “slowly” tends to 0. When H is finite dimensional, we then prove that d(x(t),argmin S Φ 1 )0 (t+), and we give sufficient conditions under which x(t)x ¯argmin S Φ 1 . We end the paper by numerical experiments allowing to compare the (SDC) system with the other systems already mentioned.

MSC:
49J24Optimal control problems with differential inclusions (existence) (MSC2000)
34D05Asymptotic stability of ODE
34G20Nonlinear ODE in abstract spaces
37N40Dynamical systems in optimization and economics
90C48Programming in abstract spaces