1. Fractional Optimal Control
p
Problems: A Simple Application
in Fractional Kinetics
Vicente Rico-Ramirez
Vi t Rico-R i
Ri
Department of Chemical Engineering
p g g
Instituto Tecnologico de Celaya
Mexico
M i

2. 1 Introduction
What i Fractional
h is i l
Calculus?

3. Fractional Calculus
• Fractional calculus is a generalization of ordinary
differentiation and integration to arbitrary NON INTEGER
order.
order.
d f Ordinary differentiation:
O di diff ti ti
n dx Integer n=1
d f
n
dx Non-
Non-integer n
d1 2 f
12
= ? Fractional differentiation
dx

4. A Bit of History: 1695 (Igor Podlubny)
Podlubny)
dn f
dt n ?
What if the order
will be n=1/2 ?
It will lead to a paradox
from which one day
useful consequences will
be drawn
L’Hopital
p
Leibniz (1661-
(1661-1704)
(1646-
(1646-1716)

5. A Bit of History
Several mathematicians have contributed with alternative
approaches to fractional order differentiation:
differentiation:
n mx
XVII Century: Leibniz d e
= m n e mx
dx n
XVIII Century: Euler
d n xm
= m(m − 1)K (m − n + 1) x m − n
dx n
XIX Century
Lagrange, L l
L Laplace, F i
Fourier
Riemann-
Riemann-Liouville
Caputo, 1967

6. Fractional Integration
F(t) is obtained back through nth-integration
nth-
of Y(t)
d n F (t ) t t n −1 t1
dt n
= Y (t ) F (t ) = ∫∫
0 0
... ∫ Y ( t 0 ) dt 0 ... dt n − 2 dt n −1
0
1 t 1 Using Laplace
F (t ) = J Y (t ) = ∫0 (t − τ )1−n Y (τ )dτ
n
(n − 1)! Transform
Non
integer 1 t 1 Riemann-
Riemann-Liouville
∫0 (t − τ )1−α Y (τ )dτ
−α
values of n 0 D Y (t ) =
Γ(α ) Definition
t
( renamed
as α)
t α −1
−α
0 D Y (t ) = * Y (t )
Γ(α )
t

7. Fractional Derivation
Fractional differentiation or order α is expected to be the
inverse ope ation of f actional integ ation:
in e se operation fractional integration
integration:
Riemann-Liouville Definition (Left)
1 d t 
 ∫ (t − τ ) Y (τ )dτ 
α −α
a Dt Y (t ) =
Γ(1 − α ) dt  a 