Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An asymptotic viscosity selection result for the regularized Newton
  dynamic   Let $\Phi:\mathcal{H}\longrightarrow\mathbb{R\cup}\left\{ +\infty\right\} $
be a closed convex proper function on a real Hilbert space $\mathcal{H}$, and
$\partial\Phi:\mathcal{H}\rightrightarrows\mathcal{H}$ its subdifferential. For
any control function $\epsilon:\mathbb{R}_{+}\longrightarrow\mathbb{R}_{+}$
which tends to zero as $t$ goes to $+\infty$, and $\lambda$ a positive
parameter, we study the asymptotic behavior of the trajectories of the
regularized Newton dynamical system \begin{eqnarray*}
  & & \upsilon\left(t\right)\in\partial\Phi\left(x\left(t\right)\right)
  & &
\lambda\dot{x}\left(t\right)+\dot{\upsilon}\left(t\right)+\upsilon\left(t\right)+\varepsilon\left(t\right)x\left(t\right)=0.
\end{eqnarray*} Assuming that $\varepsilon\left(t\right)$ tends to zero
moderately as $t$ goes to $+\infty$, we show that the term
$\varepsilon\left(\cdot\right)x\left(\cdot\right)$ asymptotically acts as a
Tikhonov regularization, which forces the trajectories to converge to a
particular equilibrium. Precisely, when $C=\textrm{argmin}\Phi\neq\emptyset$,
and $\varepsilon (\cdot)$ is a ``slow'' control, i.e.,
$\int_{0}^{+\infty}\varepsilon\left(t\right)dt=+\infty$, then each trajectory
of the system converges weakly, as $t$ goes to $+\infty$, to the element of
minimal norm of the closed convex set $C.$ When $\Phi$ is a convex
differentiable function whose gradient is Lipschitz continuous,
  we show that the strong convergence property is satisfied. Then we examine
the effect of other types of regularizing methods.

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The categories of this paper are: math-ph
29
The categories of this paper are: funct-an
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The categories of this paper are: adap-org
1
The categories of this paper are: cs
34
The categories of this paper are: chem-ph
6
The categories of this paper are: astro-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: econ
0
The categories of this paper are: hep-ph
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The categories of this paper are: q-alg
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The categories of this paper are: solv-int
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The categories of this paper are: chao-dyn
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The categories of this paper are: stat
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The categories of this paper are: mtrl-th
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The categories of this paper are: cond-mat
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The categories of this paper are: q-fin