A continuity result for Nemyckii Operators and some applications in PDE constrained optimal control

Please always quote using this URN: urn:nbn:de:0297-zib-9354
  • This work explores two applications of a classical result on the continuity of Nemyckii operators to optimal control with PDEs. First, we present an alternative approach to the analysis of Newton's method for function space problems involving semi-smooth Nemyckii operators. A concise proof for superlinear convergence is presented, and sharpened bounds on the rate of convergence are derived. Second, we derive second order sufficient conditions for problems, where the underlying PDE has poor regularity properties. We point out that the analytical structure in both topics is essentially the same.

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Metadaten
Author:Anton Schiela
Document Type:ZIB-Report
Tag:Newton methods in function space; continuity of Nemyckii Operators; optimal control; second order sufficient conditions
MSC-Classification:46-XX FUNCTIONAL ANALYSIS (For manifolds modeled on topological linear spaces, see 57Nxx, 58Bxx) / 46Nxx Miscellaneous applications of functional analysis [See also 47Nxx] / 46N40 Applications in numerical analysis [See also 65Jxx]
49-XX CALCULUS OF VARIATIONS AND OPTIMAL CONTROL; OPTIMIZATION [See also 34H05, 34K35, 65Kxx, 90Cxx, 93-XX] / 49Kxx Optimality conditions / 49K20 Problems involving partial differential equations
49-XX CALCULUS OF VARIATIONS AND OPTIMAL CONTROL; OPTIMIZATION [See also 34H05, 34K35, 65Kxx, 90Cxx, 93-XX] / 49Mxx Numerical methods [See also 90Cxx, 65Kxx] / 49M15 Newton-type methods
Date of first Publication:2006/08/01
Series (Serial Number):ZIB-Report (06-41)
Published in:A revised version appeared under the title: A Simplified Approach to Semismooth Newton Methods in Function Space in: SIAM J. Optimization, 19(3): 1417-1432, 2008