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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Characterization of local quadratic growth for strong minima in the optimal control of semi-linear elliptic equations


Authors: Térence Bayen, J. Frédéric Bonnans and Francisco J. Silva
Journal: Trans. Amer. Math. Soc. 366 (2014), 2063-2087
MSC (2010): Primary 49J20, 49K20; Secondary 35J61
Published electronically: September 23, 2013
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Abstract: In this article we consider an optimal control problem of a semi-linear elliptic equation, with bound constraints on the control. Our aim is to characterize local quadratic growth for the cost function $ J$ in the sense of strong solutions. This means that the function $ J$ grows quadratically over all feasible controls whose associated state is close enough to the nominal one, in the uniform topology. The study of strong solutions, classical in the Calculus of Variations, seems to be new in the context of PDE optimization. Our analysis, based on a decomposition result for the variation of the cost, combines Pontryagin's principle and second order conditions. While these two ingredients are known, we use them in such a way that we do not need to assume that the Hessian of the Lagrangian of the problem is a Legendre form, or that it is uniformly positive on an extended set of critical directions.


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Additional Information

Térence Bayen
Affiliation: Département de Mathématiques, Université Montpellier 2, Case courrier 051, 34095 Montpellier cedex 5, France
Email: tbayen@math.univ-montp2.fr

J. Frédéric Bonnans
Affiliation: INRIA-Saclay and CMAP, École Polytechnique, 91128 Palaiseau, France
Email: Frederic.Bonnans@inria.fr

Francisco J. Silva
Affiliation: Dipartimento di Matematica Guido Castelnuovo, Università degli Studi di Roma “La Sapienza”, 00185 Rome, Italy
Address at time of publication: XLIM - UMR CNRS $7252$, Université de Limoges, 87060 Limoges, France
Email: francisco.silva@unilim.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05961-2
PII: S 0002-9947(2013)05961-2
Keywords: Optimal control of PDE, strong minima, Pontryagin's minimum principle, second order optimality conditions, control constraints, local quadratic growth.
Received by editor(s): October 16, 2011
Received by editor(s) in revised form: July 19, 2012
Published electronically: September 23, 2013
Additional Notes: The support of the Chair “Modélisation Mathématique et Simulation Numérique”, EADS-Polytechnique-INRIA and of the European Union under the “7th Framework Program FP7-PEOPLE-2010-ITN Grant agreement number 264735-SADCO” are gratefully acknowledged. The first author thanks the CNRS for providing him a research opportunity of one year at the Centro de Modelamiento Matematico (Chile).
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.