Time-Fractional Optimal Control of Initial Value Problems on Time Scales

Abstract

We investigate Optimal Control Problems (OCP) for fractional systems involving fractional-time derivatives on time scales. The fractional-time derivatives and integrals are considered, on time scales, in the Riemann-Liouville sense. By using the Banach fixed point theorem, sufficient conditions for existence and uniqueness of solution to initial value problems described by fractional order differential equations on time scales are known. Here we consider a fractional OCP with a performance index given as a delta-integral function of both state and control variables, with time evolving on an arbitrarily given time scale. Interpreting the Euler-Lagrange first order optimality condition with an adjoint problem, defined by means of right Riemann-Liouville fractional delta derivatives, we obtain an optimality system for the considered fractional OCP. For that, we first prove new fractional integration by parts formulas on time scales.

Cauchy’s formula on nonempty closed sets and a new notion of Riemann–Liouville fractional integral on time scales

“…A different approach is proposed in [9], where the fractional integral on time scales is introduced by integrating on the time scale but making use of the classical Euler's gamma function Γ. Although such notion is now being used, with success, in several contexts and by different authors, see, e.g., [5,15,20,22,23], here we show that the definition of [9] is not the most natural one on time scales.…”

Fractional Sobolev Space on Time Scales and Its Application to a Fractional Boundary Value Problem on Time Scales

“…Theorem 19 (see [23]). Let α > 0, p, q ≥ 1, and 1/p + 1/q ≤ 1 + α, where p ≠ 1 and q ≠ 1 in the case when…”

A Stochastic Fractional Calculus with Applications to Variational Principles

“…Formulas of integration by parts play a fundamental role in the calculus of variations and optimal control [22,23]. Here we make use of Lemma 1 to prove in Section 4 a stochastic fractional Euler-Lagrange necessary optimality condition.…”