Electron. J. Differential Equations, Vol. 2024 (2024), No. 08, pp. 1-19.

Optimal mass of structure with motion described by Sturm-Liouville operator: design and predesign

Boris P. Belinskiy, Tanner A. Smith

We find an optimal design of a structure described by a Sturm-Liouville (S-L) problem with a spectral parameter in the boundary conditions. Using an approach from calculus of variations, we determine a set of critical points of a corresponding mass functional. However, these critical points - which we call predesigns - do not necessarily themselves represent meaningful solutions: it is of course natural to expect a mass to be real and positive. This represents a generalization of previous work on the topic in several ways. First, previous work considered only boundary conditions and S-L coefficients under certain simplifying assumptions. Principally, we do not assume that one of the coefficients vanishes as in the previous work. Finally, we introduce a set of solvability conditions on the S-L problem data, confirming that the corresponding critical points represent meaningful solutions we refer to as designs. Additionally, we present a natural schematic for testing these conditions, as well as suggesting a code and several numerical examples.

Submitted August 6, 2023. Published January 24, 2024.
Math Subject Classifications: 34L15, 74P05, 49K15, 49S05, 49R05.
Key Words: Sturm-Liouville problem; vibrating rod; calculus of variations; optimal design; boundary conditions with spectral parameter.
DOI: 10.58997/ejde.2023.08

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Boris P. Belinskiy
University of Tennessee at Chattanooga
Department of Mathematics
Dept 6956, 615 McCallie Ave.
Chattanooga, TN 37403-2598, USA
email: boris-belinskiy@utc.edu
Tanner A. Smith
University of Alabama at Birmingham
Department of Mathematics
Room 4005, 10th Ave S
Birmingham, AL 35294, USA
email: tsmith46@uab.edu

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