Strong asymptotic convergence of evolution equations governed by maximal monotone operators with Tikhonov regularization

R. Cominetti, J. Peypouquet, S. Sorin

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30 Scopus citations

Abstract

We consider the Tikhonov-like dynamics - over(u, ̇) (t) ∈ A (u (t)) + ε (t) u (t) where A is a maximal monotone operator on a Hilbert space and the parameter function ε (t) tends to 0 as t → ∞ with ∫0 ε (t) d t = ∞. When A is the subdifferential of a closed proper convex function f, we establish strong convergence of u (t) towards the least-norm minimizer of f. In the general case we prove strong convergence towards the least-norm point in A-1 (0) provided that the function ε (t) has bounded variation, and provide a counterexample when this property fails.

Original languageEnglish
Pages (from-to)3753-3763
Number of pages11
JournalJournal of Differential Equations
Volume245
Issue number12
DOIs
StatePublished - 15 Dec 2008

Keywords

  • Maximal monotone operators
  • Tikhonov regularization

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