Fractional Optimal Control Fractional Optimal Control Problems: A Simple Problems: A Simple

Application in Fractional Application in Fractional KineticsKinetics

Vicente Rico-RamirezVicente Rico-Ramirez

Department of Chemical EngineeringDepartment of Chemical Engineering

Instituto Tecnologico de CelayaInstituto Tecnologico de Celaya

MexicoMexico

11 Introduction Introduction

What is Fractional What is Fractional Calculus?Calculus?

Fractional CalculusFractional Calculus

• Fractional calculus Fractional calculus is a generalization of ordinary is a generalization of ordinary differentiation and integration differentiation and integration to arbitrary to arbitrary NON NON INTEGER INTEGER order. order.

dx

fd

?21

21

dx

fd

n

n

dx

fdOrdinary differentiation:Ordinary differentiation:

Integer n=1 Integer n=1

Non-integer nNon-integer nFractional differentiationFractional differentiation

A Bit of History: 1695 (Igor A Bit of History: 1695 (Igor Podlubny)Podlubny)

It will lead to a It will lead to a paradox from which paradox from which one day useful one day useful consequences will be consequences will be drawndrawn

n

n

dt

fd

What if the order What if the order

will be will be n=1/2n=1/2 ? ?

L’HopitalL’Hopital(1661-1704)(1661-1704)LeibnizLeibniz

(1646-1716)(1646-1716)

??

A Bit of HistoryA Bit of History

XVII Century: Leibniz

XVIII Century: Euler

XIX Century Lagrange, Laplace, Fourier Riemann-LiouvilleRiemann-Liouville

Caputo, 1967

Several mathematicians have contributed with alternative Several mathematicians have contributed with alternative approaches to fractional order differentiation:approaches to fractional order differentiation:

mxnn

mxn

emdx

ed

nmn

mn

xnmmmdx

xd )1(1

Fractional IntegrationFractional Integration

)()(

tYdt

tFdn

n

t t t

nn

n

dtdtdttYtF0 0 0 1200

1 1

...)(...)(

t

nn dY

tntYJtF

0 1)(

)(

1

)!1(

1)()(

t

t dYt

tYD0 10 )(

)(

1

)(

1)(

Riemann-Riemann-

Liouville Liouville DefinitionDefinition

Using Laplace Using Laplace TransformTransform

F(t) is obtained back through nth-F(t) is obtained back through nth-integration of Y(t)integration of Y(t)

Non Non integer integer

values of values of n n

( renamed ( renamed as as ))

tYttYDt *

)()(

1

0

Fractional DerivationFractional Derivation

Riemann-Liouville Definition (Left)

t

a

ta dYtdt

dtYD

)(

1

1)(

Fractional differentiation or order Fractional differentiation or order is expected to be is expected to be the inverse operation of fractional integration: the inverse operation of fractional integration: