Fractional Optimal Control Fractional Optimal Control Problems: A Simple Problems: A Simple Application in Fractional Application in Fractional Kinetics Kinetics Vicente Rico-Ramirez Vicente Rico-Ramirez Department of Chemical Engineering Department of Chemical Engineering Instituto Tecnologico de Celaya Instituto Tecnologico de Celaya Mexico Mexico
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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.
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
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: