On differentiability of solutions of fractional differential equations with respect to initial data

TL;DR

This paper investigates the differentiability of solutions to nonlinear fractional differential equations with respect to initial data, establishing the existence and formulas for fractional coinvariant derivatives related to the solution functional.

Contribution

It introduces new results on the differentiability properties and fractional coinvariant derivatives of solutions to fractional differential equations with respect to initial data.

Findings

Existence of fractional coinvariant derivatives for the solution functional.

Formulas for calculating these fractional derivatives.

Application to fractional optimal control problems.

Abstract

In this paper, we deal with a Cauchy problem for a nonlinear fractional differential equation with the Caputo derivative of order $\alpha \in (0, 1)$. As initial data, we consider a pair consisting of an initial point, which does not necessarily coincide with the inferior limit of the fractional derivative, and a function that determines the values of a solution on the interval from this inferior limit to the initial point. We study differentiability properties of the functional associating initial data with the endpoint of the corresponding solution of the Cauchy problem. Stimulated by recent results on the dynamic programming principle and Hamilton--Jacobi--Bellman equations for fractional optimal control problems, we examine so-called fractional coinvariant derivatives of order $\alpha$ of this functional. We prove that these derivatives exist and give formulas for their calculation.