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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, $\Phi_1:H\to\bbfR$ a convex function of class ${\cal C}^1$ 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. [{\it 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 {\it 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 $\Phi_0:H\to \bbfR$ whose critical points coincide with $S$ and a control parameter $\varepsilon:\bbfR_+\to\bbfR_+$ tending to zero, we consider the “Steepest Descent and Control” system $$\dot x(t)+\nabla \Phi_0 \bigl(x(t)\bigr)+ \varepsilon(t)\nabla\Phi_1\bigl(x(t)\bigr)=0,\tag SDC$$ where the control $\varepsilon$ satisfies $\int_0^{+\infty} \varepsilon (t)dt=+\infty$. This last condition ensures that $\varepsilon$ “slowly” tends to 0. When $H$ is finite dimensional, we then prove that $d(x(t), \text{argmin}_S\Phi_1)\to 0$ $(t\to+ \infty)$, and we give sufficient conditions under which $x(t)\to\overline x\in \text{argmin}_S \Phi_1$. We end the paper by numerical experiments allowing to compare the (SDC) system with the other systems already mentioned.

MSC:
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
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References:
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