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    Second order optimality conditions and their role in PDE control

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    SecondOrderOptimality.pdf (436.9Kb)
    Identificadores
    URI: http://hdl.handle.net/10902/9398
    DOI: 10.1365/s13291-014-0109-3
    ISSN: 0012-0456
    ISSN: 1869-7135
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    Author
    Casas Rentería, EduardoAutoridad Unican; Tröltzsch, FrediAutoridad Unican
    Date
    2015-03
    Derechos
    © Springer. The final publication is available at Springer via http://dx.doi.org/10.1365/s13291-014-0109-3
    Publicado en
    Jahresbericht der Deutschen Mathematiker-Vereinigung, 2015, 117(1), 3-44
    Publisher
    Springer Verlag
    Enlace a la publicación
    http://dx.doi.org/10.1365/s13291-014-0109-3
    Palabras clave
    Nonlinear optimization
    Infinite dimensional space
    Second order optimality condition
    Critical cone
    Optimal control of partial differential equations
    Stability analysis
    Abstract:
    If f : Rn R is twice continuously differentiable, f’(u) = 0 and f’’(u) is positive definite, then u is a local minimizer of f. This paper surveys the extension of this well known second order suffcient optimality condition to the case f : U R, where U is an infinite-dimensional linear normed space. The reader will be guided from the case of finite-dimensions via a brief discussion of the calculus of variations and the optimal control of ordinary differential equations to the control of nonlinear partial differential equations, where U is a function space. In particular, the following questions will be addressed: Is the extension to infinite dimensions straightforward or will unexpected difficulties occur? How second order sufficient optimality conditions must be modified, if simple inequality constraints are imposed on u? Why do we need second order conditions and how can they be applied? If they are important, are we able to check if they are fulfilled order sufficient optimality condition to the case f : U R, where U is an infinite-dimensional linear normed space. The reader will be guided from the case of finite-dimensions via a brief discussion of the calculus of variations and the optimal control of ordinary differential equations to the control of nonlinear partial differential equations, where U is a function space. In particular, the following questions will be addressed: Is the extension to infinite dimensions straightforward or will unexpected difficulties occur? How second order sufficient optimality conditions must be modified, if simple inequality constraints are imposed on u? Why do we need second order conditions and how can they be applied? If they are important, are we able to check if they are fulfilled? It turns out that infinite dimensions cause new difficulties that do not occur in finite dimensions. We will be faced with the surprising fact that the space, where f’’(u) exists can be useless to ensure positive definiteness of the quadratic form v f’’(u)v2. In this context, the famous two-norm discrepancy, its consequences, and techniques for overcoming this difficulty are explained. To keep the presentation simple, the theory is developed for problems in function spaces with simple box constraints of the form a = u = ß. The theory of second order conditions in the control of partial differential equations is presented exemplarily for the nonlinear heat equation. Different types of critical cones are introduced, where the positivity of f’’(u) must be required. Their form depends on whether a so-called Tikhonov regularization term is part of the functional f or not. In this context, the paper contains also new results that lead to quadratic growth conditions in the strong sense. As a first application of second-order sufficient conditions, the stability of optimal solutions with respect to perturbations of the data of the control problem is discussed. Second, their use in analyzing the discretization of control problems by finite elements is studied. A survey on further related topics, open questions, and relevant literature concludes the paper.
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    • D20 Proyectos de Investigación [197]

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    UNIVERSIDAD DE CANTABRIA

    Repositorio realizado por la Biblioteca Universitaria utilizando DSpace software
    Contact Us | Send Feedback
    Metadatos sujetos a:licencia de Creative Commons Reconocimiento 3.0 España