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

• Volume: 10, Issue: 2, page 243-258
• ISSN: 1292-8119

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## Abstract

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Let H be a real Hilbert space, ${\Phi }_{1}:H\to$ 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.   Brézis [CITE]) applied to the non-smooth function ${\Phi }_{1}+{\delta }_{S}$. Following Antipin , it is also possible to use a continuous gradient-projection system. We propose here an alternative method as follows: given a smooth convex function ${\Phi }_{0}:H\to$ whose critical points coincide with S and a control parameter $\epsilon {:}_{+}{\to }_{+}$ tending to zero, we consider the “Steepest Descent and Control” system $\left(SDC\right)\phantom{\rule{2.0em}{0ex}}\stackrel{˙}{x}\left(t\right)+\nabla {\Phi }_{0}\left(x\left(t\right)\right)+\epsilon \left(t\right)\phantom{\rule{0.166667em}{0ex}}\nabla {\Phi }_{1}\left(x\left(t\right)\right)=0,$ where the control ε satisfies ${\int }_{0}^{+\infty }\epsilon \left(t\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}t=+\infty$. This last condition ensures that ε “slowly” tends to 0. When H is finite dimensional, we then prove that $d\left(x\left(t\right),\mathrm{argmin}{\phantom{\rule{1.19995pt}{0ex}}}_{S}{\Phi }_{1}\right)\to 0\phantom{\rule{1.0em}{0ex}}\left(t\to +\infty \right),$ and we give sufficient conditions under which $x\left(t\right)\to \overline{x}\in \phantom{\rule{0.166667em}{0ex}}\mathrm{argmin}{\phantom{\rule{1.19995pt}{0ex}}}_{S}{\Phi }_{1}$. We end the paper by numerical experiments allowing to compare the (SDC) system with the other systems already mentioned.

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