Electron. J. Differential Equations, Vol. 2024 (2024), No. 08, pp. 119.
Optimal mass of structure with motion described by SturmLiouville operator: design and predesign
Boris P. Belinskiy, Tanner A. Smith
Abstract:
We find an optimal design of a structure described by a SturmLiouville (SL)
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 SL 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 SL 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: SturmLiouville problem; vibrating rod; calculus of variations;
optimal design; boundary conditions with spectral parameter.
DOI: 10.58997/ejde.2024.08
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Boris P. Belinskiy
University of Tennessee at Chattanooga
Department of Mathematics
Dept 6956, 615 McCallie Ave.
Chattanooga, TN 374032598, USA
email: borisbelinskiy@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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