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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, ${{\Phi }}_{1}:H\to ℝ$ 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 ${{\Phi }}_{1}+{\delta }_{S}$. 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 ${{\Phi }}_{0}:H\to ℝ$ whose critical points coincide with $S$ and a control parameter $\epsilon :{ℝ}_{+}\to {ℝ}_{+}$ tending to zero, we consider the “Steepest Descent and Control” system

$\stackrel{˙}{x}\left(t\right)+\nabla {{\Phi }}_{0}\left(x\left(t\right)\right)+\epsilon \left(t\right)\nabla {{\Phi }}_{1}\left(x\left(t\right)\right)=0,\phantom{\rule{2.em}{0ex}}\left(\mathrm{SDC}\right)$

where the control $\epsilon$ satisfies ${\int }_{0}^{+\infty }\epsilon \left(t\right)dt=+\infty$. This last condition ensures that $\epsilon$ “slowly” tends to 0. When $H$ is finite dimensional, we then prove that $d\left(x\left(t\right),{\text{argmin}}_{S}{{\Phi }}_{1}\right)\to 0$ $\left(t\to +\infty \right)$, and we give sufficient conditions under which $x\left(t\right)\to \overline{x}\in {\text{argmin}}_{S}{{\Phi }}_{1}$. We end the paper by numerical experiments allowing to compare the (SDC) system with the other systems already mentioned.

##### MSC:
 49J24 Optimal control problems with differential inclusions (existence) (MSC2000) 34D05 Asymptotic stability of ODE 34G20 Nonlinear ODE in abstract spaces 37N40 Dynamical systems in optimization and economics 90C48 Programming in abstract spaces