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

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. For more information see https://ejde.math.txstate.edu/Volumes/2024/08/abstr.html