Structure Preserving Approximation of Semiconcave Functions
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866917257394782208 |
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| author | Kunisch, Karl Vásquez-Varas, Donato |
| author_facet | Kunisch, Karl Vásquez-Varas, Donato |
| contents | This article addresses structure-preserving smooth approximation of semiconcave functions. semiconcave functions are of particular interest because they naturally arise in a variety of variational problems, including {optimal feedback control, game theory, and optimal transport}. We leverage the fact that any semiconcave function can be represented as the {infimum of a countable family of \(C^2\) functions}. This infimum is expressed in a form that allows {approximation by finitely many functions}, combined with {smoothing operations}, such that each element of the approximating sequence remains semiconcave. The {active sets of indices} contributing to the representation of the semiconcave function and its approximations are analyzed in detail. Moreover, we show that the {gradients of the elements in the expansion of the approximating functions form a probability distribution}, a property of particular interest for the {value function in optimal control}. Approximation results are established in \(C(\bar Ω)\) and in \(W^{1,p}(Ω)\) for \(p \in [1,\infty)\) and \(p = \infty\). Finally, {numerical results} are presented to illustrate the approach on a test example. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_07770 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Structure Preserving Approximation of Semiconcave Functions Kunisch, Karl Vásquez-Varas, Donato Optimization and Control 26B25, 49J52, 41A30, 49L25 This article addresses structure-preserving smooth approximation of semiconcave functions. semiconcave functions are of particular interest because they naturally arise in a variety of variational problems, including {optimal feedback control, game theory, and optimal transport}. We leverage the fact that any semiconcave function can be represented as the {infimum of a countable family of \(C^2\) functions}. This infimum is expressed in a form that allows {approximation by finitely many functions}, combined with {smoothing operations}, such that each element of the approximating sequence remains semiconcave. The {active sets of indices} contributing to the representation of the semiconcave function and its approximations are analyzed in detail. Moreover, we show that the {gradients of the elements in the expansion of the approximating functions form a probability distribution}, a property of particular interest for the {value function in optimal control}. Approximation results are established in \(C(\bar Ω)\) and in \(W^{1,p}(Ω)\) for \(p \in [1,\infty)\) and \(p = \infty\). Finally, {numerical results} are presented to illustrate the approach on a test example. |
| title | Structure Preserving Approximation of Semiconcave Functions |
| topic | Optimization and Control 26B25, 49J52, 41A30, 49L25 |
| url | https://arxiv.org/abs/2602.07770 |