Some obstacle problems for partially hinged plates and related optimization issues

Elvise Berchio; Filomena Feo; Antonio Giuseppe Grimaldi; Elvise Berchio; Filomena Feo; Antonio Giuseppe Grimaldi

doi:10.3934/mine.2026002

Mathematics in Engineering

2026, Volume 8, Issue 1: 43-69. doi: 10.3934/mine.2026002

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Research article

Some obstacle problems for partially hinged plates and related optimization issues

1.
Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129, Torino, Italy
2.
Dipartimento di Ingegneria, Università degli Studi di Napoli "Parthenope", Centro Direzionale Isola C4, Napoli, 80143, Italy

Received: 05 November 2025 Revised: 12 January 2026 Accepted: 15 January 2026 Published: 21 January 2026

We study optimization problems for partially hinged rectangular plates, modeling bridge roadways, in the presence of real and artificial obstacles. Real obstacles represent structural constraints to avoid, while artificial ones are introduced to enhance stability. For the former, aiming to prevent collisions, we set up a worst-case optimization problem in which we minimize the amplitude of oscillations with respect to the density distribution; for the latter, aiming to improve the torsional stability, we minimize, with respect to the obstacles, the maximum of a gap function quantifying the displacement between the long edges of the plate. For both problems, existence results are provided, along with a discussion about qualitative properties of optimal density distributions and obstacles.
- partially hinged plates,
- obstacle problems,
- worst-case optimization
Citation: Elvise Berchio, Filomena Feo, Antonio Giuseppe Grimaldi. Some obstacle problems for partially hinged plates and related optimization issues[J]. Mathematics in Engineering, 2026, 8(1): 43-69. doi: 10.3934/mine.2026002

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Abstract

We study optimization problems for partially hinged rectangular plates, modeling bridge roadways, in the presence of real and artificial obstacles. Real obstacles represent structural constraints to avoid, while artificial ones are introduced to enhance stability. For the former, aiming to prevent collisions, we set up a worst-case optimization problem in which we minimize the amplitude of oscillations with respect to the density distribution; for the latter, aiming to improve the torsional stability, we minimize, with respect to the obstacles, the maximum of a gap function quantifying the displacement between the long edges of the plate. For both problems, existence results are provided, along with a discussion about qualitative properties of optimal density distributions and obstacles.

References

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