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Journal of Convex Analysis 17 (2010), No. 2, 611--642
Copyright Heldermann Verlag 2010

Verification Theorem and Construction of ε-Optimal Controls for Control of Abstract Evolution Equations

Giorgio Fabbri
Dip. di Studi Economici, UniversitÓ di Napoli
and: School of Mathematics and Statistics, University of New South Wales, Sydney, Australia

Fausto Gozzi
Dip. di Scienze Economiche ed Aziendali, Libera UniversitÓ Internazionale degli Studi Sociali, Viale Pola 12, 00198 Roma, Italy

Andrzej Swiech
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, U.S.A.

We study several aspects of the dynamic programming approach to optimal control of abstract evolution equations, including a class of semilinear partial differential equations. We introduce and prove a verification theorem which provides a sufficient condition for optimality. Moreover we prove sub- and superoptimality principles of dynamic programming and present an explicit construction of ε-optimal controls.

Keywords: Optimal control of PDE, verification theorem, dynamic programming, epsilon-optimal controls, Hamilton-Jacobi-Bellman equations.

MSC: 35R15, 49L20, 49L25, 49K20

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