Second order optimality conditions and their role in PDE control
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....
| Authors: | , |
|---|---|
| Format: | article |
| Publication Date: | 2015 |
| Country: | España |
| Institution: | Universidad de Cantabria (UC) |
| Repository: | UCrea Repositorio Abierto de la Universidad de Cantabria |
| Language: | English |
| OAI Identifier: | oai:repositorio.unican.es:10902/9398 |
| Online Access: | http://hdl.handle.net/10902/9398 |
| Access Level: | Open access |
| Keyword: | Nonlinear optimization Infinite dimensional space Second order optimality condition Critical cone Optimal control of partial differential equations Stability analysis |
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Second order optimality conditions and their role in PDE control |
| title |
Second order optimality conditions and their role in PDE control |
| spellingShingle |
Second order optimality conditions and their role in PDE control Casas Rentería, Eduardo|||0000-0002-8364-9416 Nonlinear optimization Infinite dimensional space Second order optimality condition Critical cone Optimal control of partial differential equations Stability analysis |
| title_short |
Second order optimality conditions and their role in PDE control |
| title_full |
Second order optimality conditions and their role in PDE control |
| title_fullStr |
Second order optimality conditions and their role in PDE control |
| title_full_unstemmed |
Second order optimality conditions and their role in PDE control |
| title_sort |
Second order optimality conditions and their role in PDE control |
| dc.creator.none.fl_str_mv |
Casas Rentería, Eduardo|||0000-0002-8364-9416 Tröltzsch, Fredi |
| author |
Casas Rentería, Eduardo|||0000-0002-8364-9416 |
| author_facet |
Casas Rentería, Eduardo|||0000-0002-8364-9416 Tröltzsch, Fredi |
| author_role |
author |
| author2 |
Tröltzsch, Fredi |
| author2_role |
author |
| dc.contributor.none.fl_str_mv |
Universidad de Cantabria |
| dc.subject.none.fl_str_mv |
Nonlinear optimization Infinite dimensional space Second order optimality condition Critical cone Optimal control of partial differential equations Stability analysis |
| topic |
Nonlinear optimization Infinite dimensional space Second order optimality condition Critical cone Optimal control of partial differential equations Stability analysis |
| description |
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. |
| publishDate |
2015 |
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2015 2015-03-01 |
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journal article http://purl.org/coar/resource_type/c_6501 NA http://purl.org/coar/version/c_be7fb7dd8ff6fe43 |
| dc.type.openaire.fl_str_mv |
info:eu-repo/semantics/article |
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article |
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http://hdl.handle.net/10902/9398 |
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http://hdl.handle.net/10902/9398 |
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Inglés eng |
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Inglés |
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eng |
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open access http://purl.org/coar/access_right/c_abf2 |
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info:eu-repo/semantics/openAccess |
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open access http://purl.org/coar/access_right/c_abf2 |
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openAccess |
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Springer Verlag |
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Springer Verlag |
| dc.source.none.fl_str_mv |
Jahresbericht der Deutschen Mathematiker-Vereinigung, 2015, 117(1), 3-44 reponame:UCrea Repositorio Abierto de la Universidad de Cantabria instname:Universidad de Cantabria (UC) |
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Universidad de Cantabria (UC) |
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UCrea Repositorio Abierto de la Universidad de Cantabria |
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UCrea Repositorio Abierto de la Universidad de Cantabria |
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1869422457601916928 |
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Second order optimality conditions and their role in PDE controlCasas Rentería, Eduardo|||0000-0002-8364-9416Tröltzsch, FrediNonlinear optimizationInfinite dimensional spaceSecond order optimality conditionCritical coneOptimal control of partial differential equationsStability analysisIf 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.The first author was partially supported by the Spanish Ministerio de Economía y Competitividad under project MTM2011-22711, the second author by DFG in the framework of the Collaborative Research Center SFB 910, project B6.Springer VerlagUniversidad de Cantabria20152015-03-01journal articlehttp://purl.org/coar/resource_type/c_6501NAhttp://purl.org/coar/version/c_be7fb7dd8ff6fe43info:eu-repo/semantics/articlehttp://hdl.handle.net/10902/9398Jahresbericht der Deutschen Mathematiker-Vereinigung, 2015, 117(1), 3-44reponame:UCrea Repositorio Abierto de la Universidad de Cantabriainstname:Universidad de Cantabria (UC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:repositorio.unican.es:10902/93982026-06-02T12:39:31Z |
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