Optimization for a Sturm--Liouville problem with the spectral parameter in the boundary condition

We find an optimal mass of a structure described by a Sturm-Liouville (S-L) problem with a spectral parameter in the boundary conditions. While previous work on the subject focused on a somewhat simplified model, we consider a more general S-L problem. We use the calculus of variations approach to determine a set of critical points of a corresponding mass functional, yet these critical points - which we call \textit{predesigns} - do not necessarily themselves represent meaningful solutions. It is natural to expect a mass to be real and positive. To this end, we additionally introduce a set of solvability conditions on the S-L problem data, confirming that these critical points represent meaningful solutions we refer to as \textit{designs}. We further present the analytic continuation of these predesigns in regards to the spectral parameter as well as a discussion of the stability of these (pre)designs. We present a code that allows us to for the given data of the S-L problem check conditions of solvability, plot the design, and calculate the value of the functional that represents the optimal mass.

Grantor dc:publisher

University of Tennessee at Chattanooga

Author dc:creator

• Smith, Tanner

Contributors dc:contributor

• Belinskiy, Boris P.
• Kong, Lingju; Cox, Christopher; Wang, Jin; Nichols, Roger
• College of Engineering and Computer Science

• Calculus of variations
• Differential equations

Licence

• http://rightsstatements.org/vocab/InC/1.0/

Language dc:language

English, eng

Repository record dc:identifier

https://scholar.utc.edu/theses/778

OAI identifier oai:identifier

oai:scholar.utc.edu:theses-1951