We review recent results obtained to solve fractional order optimal control problems with free terminal time and a dynamic constraint involving integer and fractional order derivatives. Some particular cases are studied in detail. A numerical scheme is given, based on expansion formulas for the fractional derivative. The efficiency of the method is illustrated through examples.

In a letter dated September 30, 1695 l’Hôpital posed the question to Leibniz: what would be the derivative of order

We begin with some basic definitions and properties about fractional operators [

Let

We remark that if

Some basic properties are useful, namely, a relationship between the Riemann-Liouville and the Caputo fractional derivatives and a fractional integration by parts formula.

The following conditions hold:

For numerical purposes, one of the most common procedures is to replace the fractional operators by a series that involves integer derivatives only. The usual one is given by

We mention the recent papers [

Let

If

the Hamiltonian system

the stationary condition

the transversality conditions

This theorem states the general condition that the optimal solution

Let

If

If

If

If the terminal point

If

Numerically, by using approximation (

Theorem

If

the Hamiltonian system

the stationary condition

the transversality conditions

We remark that when

Under some additional conditions, namely, convexity conditions over

Let

So far, we have provided a theoretical approach to fractional optimal control problems, which involves solving fractional differential equations. As it is known, solving such equations is in most cases impossible to do, and numerical methods are used to find approximated solutions for the problem (see, e.g., [

Let

Fix

The idea is to replace the fractional derivative with such expansions and to consider finite sums only. When we use the approximation

To see the accuracy of the method, we exemplify it by considering some functions and compare the exact expression of the fractional derivative with the approximated one. To start, consider

Analytic versus numerical approximation for a fixed

A different approach is to consider a fixed

Analytic versus numerical approximation for a fixed

We will see that applying the numerical method given in the previous section, we are able to solve fractional optimal control problems applying known techniques from the classical optimal control theory. First, consider the following optimal control problem:

Exact solution (solid line) for the fractional optimal control problem (

Another approach is to approximate the original problem by using approximation from (

Exact solution (solid line) for the fractional optimal control problem (

For our next example, we consider the final time

The fractional necessary optimality conditions, after approximating the fractional terms, result in

Numerical solutions to the free final time problem (

This work was supported by