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

## TL;DR

This paper develops a framework for solving optimal control problems involving fractional derivatives on arbitrary time scales, providing existence, uniqueness, and optimality conditions using fractional calculus and fixed point theory.

## Contribution

It introduces new fractional integration by parts formulas on time scales and derives optimality conditions for fractional control problems in this setting.

## Key findings

- Established existence and uniqueness of solutions for fractional initial value problems.
- Derived an optimality system using fractional Euler-Lagrange equations.
- Proved new fractional integration by parts formulas 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.