Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 08, pp. 1-19.

Title: Optimal mass of structure with motion described by Sturm-Liouville operator: design and predesign
       
Authors: Boris P. Belinskiy (Univ. of Tennessee,Chattanooga, TN, USA)
         Tanner A. Smith (Univ. of Alabama, Birmingham, AL, USA)

Abstract:  
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.