A non-classical class of variational problems

TL;DR

This paper introduces a novel non-classical variational problem class motivated by economic revenue models, characterized by unknown final states and a unique boundary condition approach, solved via numerical and symbolic methods.

Contribution

It formulates a new class of variational problems with unknown final states and applies innovative boundary conditions, expanding the scope of classical calculus of variations.

Findings

Successfully solved sample problems using shooting method

Demonstrated equivalence with symbolic algebra solutions

Extended variational problem framework to include unknown final states

Abstract

We study a new non-classical class of variational problems that is motivated by some recent research on the non-linear revenue problem in the field of economics. This class of problem can be set up as a maximising problem in the Calculus of Variations (CoV) or Optimal Control. However, the state value at the final fixed time, y(T), is a priori unknown and the integrand is a function of the unknown y(T). This is a non-standard CoV problem. In this paper we apply the new costate boundary conditions p(T) in the formulation of the CoV problem. We solve some sample examples in this problem class using the numerical shooting method to solve the resulting TPBVP, and incorporate the free y(T) as an additional unknown. Essentially the same results are obtained using symbolic algebra software.