Optimality conditions for fractional variational problems with free terminal time

Abstract

This paper provides necessary and sufficient conditions of optimality for a variational problem involving a fractional derivative with respect to another function. Fractional Euler-Lagrange equations are proven for the fundamental problem and when in presence of an integral constraint. A Legendre condition, which is a second-order necessary condition, is also obtained. Other cases, such as the infinite horizon problem, with delays in the Lagrangian, and with high-order derivatives, are considered. A necessary condition that the optimal fractional order must satisfy is proved.Mathematics Subject Classification 2010: 26A33, 49K05, 34A08.

Legendre’s Necessary Condition for Fractional Bolza Functionals with Mixed Initial/Final Constraints

“…Precisely, while η is indeed compactly supported, c D α a+ [η] is clearly not (in contrary to what is claimed in [45,Equality (12)]). Surprisingly the same mistake has been disseminated in a series of papers (see [5,6,8]). This discovery was the starting point of the present work.…”

“…Theorem 3.2. Let us assume that K is given by (6). If x ∈ K is a solution to Problem (P) and g is regular at (x(a), x(b)), then:…”

“…Soon after we found another contemporary work [36] in which a proof of such condition was also presented (in fact for a more general fractional optimal control problem but without final constraint) with, again, some erroneous assertions. Incredibly, more recently, in a series of papers [5,6,8], the author(s) presented a completely analogous (and thus incorrect) proof to the one in [45] for some variants of fractional variational problems. Therefore we became very eager to elaborate a correct proof of the Legendre condition in the fractional setting.…”

Variational problem containing psi‐RL complex‐order fractional derivatives

“…Similar to [20], which discusses necessary and sufficient conditions of optimality for a variational problem involving a fractional derivative, in this article, a complex‐order fractional variational problem with real‐valued optimums is considered. The complex‐order fractional derivative contained in the variational problem is based on the ψ‐Riemann‐Liouville (ψ‐RL) fractional derivative.…”

Variational Problems with Partial Fractional Derivative: Optimal Conditions and Noether’s Theorem

“…Considerable progress has been made to determine necessary and sufficient conditions that any extremal for the variational functional with fractional calculus must satisfy in recent years. R. Almeida [2] provides necessary and sufficient conditions of optimality for variational problems that deal with a fractional derivative with respect to another function. Almeida established the fractional Euler-Lagrange equations for the fundamental problem and when in presence of an integral constraint and Almeida obtained a Legendre condition.…”