Isoperimetric problems on time scales with nabla derivatives

TL;DR

This paper establishes a necessary optimality condition for isoperimetric problems involving nabla derivatives on time scales, extending recent unifying results in optimal control to a broader mathematical framework and demonstrating its application in economics.

Contribution

It introduces a new optimality condition for isoperimetric problems on time scales with nabla derivatives, generalizing previous results to a unified setting.

Findings

Extended recent isoperimetric results to generic time scales.

Provided an illustrative economic optimization example.

Established a necessary condition for nabla-differentiable curves.

Abstract

We prove a necessary optimality condition for isoperimetric problems under nabla-differentiable curves. As a consequence, the recent results of [M.R. Caputo, A unified view of ostensibly disparate isoperimetric variational problems, Appl. Math. Lett. (2008), doi:10.1016/j.aml.2008.04.004], that put together seemingly dissimilar optimal control problems in economics and physics, are extended to a generic time scale. We end with an illustrative example of application of our main result to a dynamic optimization problem from economics.