Constrained fractional variational problems of variable order

TL;DR

This paper develops necessary optimality conditions for fractional isoperimetric and variational problems involving variable order derivatives, expanding the calculus of variations to more complex fractional systems.

Contribution

It introduces new fractional isoperimetric problems with variable order derivatives and derives optimality conditions, including free terminal conditions.

Findings

Derived necessary conditions for fractional variational problems with variable order derivatives.

Formulated boundary conditions for free terminal points in fractional problems.

Extended calculus of variations to include holonomic constraints and variable fractional orders.

Abstract

Isoperimetric problems consist in minimizing or maximizing a cost functional subject to an integral constraint. In this work, we present two fractional isoperimetric problems where the Lagrangian depends on a combined Caputo derivative of variable fractional order and we present a new variational problem subject to a holonomic constraint. We establish necessary optimality conditions in order to determine the minimizers of the fractional problems. The terminal point in the cost integral, as well the terminal state, are considered to be free, and we obtain corresponding natural boundary conditions.