Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2011, Article ID 867932, 19 pages
doi:10.1155/2011/867932
Research Article
About Robust Stability of Caputo Linear
Fractional Dynamic Systems with Time Delays
through Fixed Point Theory
M. De la Sen
Faculty of Science and Technology, University of the Basque Country,
644 de Bilbao, Leioa, 48080 Bilbao, Spain
Correspondence should be addressed to M. De la Sen, [email protected]
Received 9 November 2010; Accepted 31 January 2011
Academic Editor: Marl
`
ene Frigon
Copyright q 2011 M. De la Sen. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
This paper investigates the global stability and the global asymptotic stability independent of the
sizes of the delays of linear time-varying Caputo fractional dynamic systems of real fractional
order possessing internal point delays. The investigation is performed via fixed point theory
in a complete metric space by defining appropriate nonexpansive or contractive self-mappings
from initial conditions to points of the state-trajectory solution. The existence of a unique fixed
point leading to a globally asymptotically stable equilibrium point is investigated, in particular,
under easily testable sufficiency-type stability conditions. The study is performed for both the
uncontrolled case and the controlled case under a wide class of state feedback laws.
1. Introduction
Fractional calculus is concerned with the calculus of integrals and derivatives of any arbitrary
real or complex orders. In this sense, it may be considered as a generalization of classical
calculus which is included in the theory as a particular case. There is a good compendium
independent of eventual coincidence of some values of delays if those ones are, in particular,
multiple related to the associate matrices of dynamics. Most of the results are centred in
characterizations via Caputo fractional differentiation although some extensions presented
are concerned with the classical Riemann-Liouville differ-integration. It is proved that the
existence nonnegative solutions independent of the sizes of the delays and the stability
properties of linear time-invariant fractional dynamic differential systems subject to point
delays may be characterized with sets of precise mathematical results.
On the other hand, fixed point theory is a very powerful mathematical tool to be
used in many applications where stability knowledge is needed. For instance, the concepts of
contractive, weak contractive, asymptotic contractive and nonexpansive mappings have been
investigated in detail in many papers from several decades ago see, for instance, 32–34
and references therein. It has been found, for instance, that contractivity, weak contractivity
and asymptotic contractivity ensure the existence of a unique fixed pointing complete metric
or Banach spaces. Some theory and applications of some types of functional equations in
the context of fixed point theory have been investigated in
35, 36. Fixed point theory has
also been employed successfully in stability problems of dynamic systems such as time-delay
and continuous-time/digital hybrid systems and in those involving switches among different
parameterizations. This paper is concerned with the investigation of fixed points in Caputo
linear fractional dynamic systems of real order α which involved delayed dynamics subject
to a finite set of bounded point delays which can be of arbitrary sizes. The self-mapping
defined in the state space from initial conditions to points of the state—trajectory solution
are characterized either as nonexpansive or as contractive. The first case allows to establish
global stability results while the second one characterizes global asymptotic stability.
1.1. Notation
C , R,andZ are the sets of complex, real, and integer numbers, respectively.
Fixed Point Theory and Applications 3
R
and Z
−
∪{iω : ω ∈ R}, where i is the complex unity, R
0
: R
−
∪{0} and Z
0−
: Z
−
∪{0}.
N : {1, 2, ,N}⊂Z
0
,“∨” is the logic disjunction, and “∧” is the logic conjunction.
t/h is the integer part of the rational quotient t/h.
σM denotes the spectrum of the real or complex square matrix M i.e., its set of
distinct eigenvalues.
denotes any vector or induced matrix norm. Also, m
p
and M
p
are the
p
-norms of the vector m or induced real or complex matrix M,andμ
p
M denote
the
p
measure of the square matrix M, 20. The matrix measure μ
p
∞
denotes the supremum norm on R
0
, or its induced supremum metric, for
functions or vector and matrix functions without specification of any pointwise particular
vector or matrix norm for each t ∈ R
0
. If pointwise vector or matrix norms are specified, the
corresponding particular supremum norms are defined by using an extra subscript. Thus,
m
p∞
: sup
t∈R
0
mt
p
and M
p∞
: sup
t∈R
0
Mt
p
are, respectively, the supremum
norms on R
0
for vector and matrix functions of domains in R
0
×R
i−1
dom, codom and with the ith derivative is
bounded piecewise continuous, respectively, piecewise continuous in the definition domain.
2. Caputo Fractional Linear Dynamic Systems with Point Constant
Delays and the Contraction Mapping Theorem
Consider the linear functional Caputo fractional dynamic system of order α with r delays:
D
α
0
x
t
:
1
Γ
k − α
t
0
x
k
τ
i 0
A
i
x
t − r
i
r
i 0
A
i
t
x
t − r
i
B
t
u
t
→ R
n×n
i ∈ r ∪{0} which are decomposable as a nonunique sum of a constant
matrix plus a bounded matrix function of time, that is,
A
i
tA
i
A
i
t, for all t ∈ R
0
,and
B : R
0
→ R
n×m
is the piecewise continuous bounded control matrix. The initial condition
is given by kn-real vector functions ϕ
j
: −h, 0 → R
n
,withj ∈ k − 1 ∪{0}, which are
absolutely continuous except eventually in a set of zero measure of −h, 0 ⊂ R of bounded
discontinuities with ϕ
j
0x
by zeroing them on the irrelevant
intervals of −h, 0 so that any solution for t ∈ R
0
of 2.1 is identical to the corresponding
one under the above given definition domains of vector functions of initial conditions and
controls.
Theorem 2.1. The linear and time- invariant differential functional fractional dynamic system 2.1
of any order α ∈ C
0
has a unique continuous solution on −h, 0 ∪ R
0
satisfying
a x ≡ ϕ ≡
k−1
j0
ϕ
j
on R
0
with ϕ
j
0x
j
0x
j
0x
j0
; j ∈ k − 1 ∪{0}; for all t ∈
−h, 0 for each given set of initial functions and ϕ
αj
t
x
j0
r
i 1
r
i
0
Φ
α
t − τ
A
i
ϕ
j
τ − r
i
dτ
r
t
r
i
Φ
α
t − τ
A
i
τ
x
α
τ − r
i
dτ
r
i 0
t
r
i
Φ
α
τ
dτ, t ∈ R
0
,
2.2
which is time-differentiable satisfying 2.1 in R
with k Re α1 if α/∈ Z
and k α if α ∈ Z
,
and
Φ
αj
t
: t
j
E
α,j1
A
0
t
α
, Φ
t
α
Γ
α j
,j∈
k − 1 ∪
{
0,α
}
,
2.3
for t ∈ R
0
and Φ
α0
tΦ
α
t0 for t<0,whereE
α,j
A
0
t
α
are the Mittag-Leffler functions.
Fixed Point Theory and Applications 5
A technical result about norm upper-bounding functions of the matrix functions 2.3-
,
Φ
αj
t
≤ K
Φαj
t
j
e
A
0
t
,j∈
k − 1 ∪
{
0,α
≥ 1.
2.4
ii If α∈ R
≥ 1 then
E
αj
A
0
t
α
∞
0
!
Γ
α j
A
e
A
0
t
α
,j∈
k − 1 ∪
{
0
}
,t∈ R
0
,
Φ
αj
t
≤ sup
∈Z
0
α
,j∈
k − 1 ∪
{
0
}
,t∈ R
0
,
Φ
α
t
≤ sup
∈Z
0
!
Γ
1
α
,t∈ R
0
.
2.5
If, in addition, A
0
is a stability matrix then e
A
0
t
≤Ke
− λt
and e
A
0
t
α
≤Ke
− λt
α
≤ Ke
−λt
; t ∈ R
0
for some real constants K ≥ 1,λ∈ R
. Then, one gets from 2.5
,
Φ
α
t
≤ t
α−1
e
−λt
2.6
for t ∈ R
0
, and the fractional dynamic system in the absence of delayed dynamics is exponentially
stable if the standard fractional system for α 1 is exponentially stable.
iii The following inequalities hold.
Φ
α, k−1
t
≤ t
k−α
Φ
α
for α ∈
k − 1,k
∩ R
,t∈ R
0
,
Φ
k
t
≡
Φ
k,k−1
t
for α k ∈ Z
,t∈ R
0
.
0
!
Γ
α j
A
0
t
α
!
∞
0
t
α−1
!
Γ
α j
A
!
τ
1−α
Γ
α j
e
A
0
t
,j∈
k − 1 ∪
{
0,α
}
,
t ∈
1, ∞
∩ R
∞
0
A
0
t
!
≤ lim sup
t →∞
e
A
0
t
1−α
Γ
α j 1
t
j
e
A
0
t
,j∈
k − 1 ∪
{
0,α
}
,
t ∈
1, ∞
∩ R
∞
0
A
0
t
j
!
≤ lim sup
t →∞
t
j
e
A
0
t
1
α
1
t
1−α
e
A
0
t
,t∈
1, ∞
∩ R
,
lim sup
t →∞
0
A
0
t
α−1
!
≤ lim sup
t →∞
1
t
1−α
e
A
0
t
≤ lim sup
Z
0
→∞
sup
∈Z
0
!
1−α
Γ
α j 1
0.
2.9
Fixed Point Theory and Applications 7
The inequalities 2.4 hold since the above matrix norms are bounded on the real interval
1, ∞ and their limit superior is upper-bounded by the given formulas and Property i is
proved. On the other hand, if R
α ≥ 1 then
E
αj
Γ
1
e
A
0
t
α
e
A
0
t
α
e
A
0
t
α
≤ t
j
e
A
0
t
α
,j∈
k − 1 ∪
{
0
}
,t∈ R
α
≤ t
α−1
e
A
0
t
α
,t∈ R
0
.
2.10
If, in addition, A
0
is a stability matrix then e
A
0
t
α
t
∞
0
A
0
t
αj
Γ
α j 1
≤ sup
∈Z
1
α
,t∈ R
0
,
2.11
j ∈
k − 1 ∪{0},sothatifk − 1 <α∈ R
≤ k, then
Φ
α,k−1
t
≤ sup
∈Z
0
Γ
≤ t
k−α
Φ
α
t
,t∈ R
0
2.12
Also,
Φ
α
t
Γ
1
α
∞
0
A
0
t
αj
Γ
α j 1
≤ t
t
for α ∈
k − 1,k
∩ R
,
Φ
k
t
≡
Φ
k,k−1
t
for α ∈
k − 1,k
and controls u ∈ BPC
0
R
0
, R
n
with ϕ
j
0x
j
0x
j
0x
j0
; for all j ∈ k − 1 ∪{0}.
A fixed point theorem is now given for the Caputo fractional system 2.1.
Theorem 3.2. Assume any set of r given finite delays 0 r
0
<r
1
≤ ≤ r
r
h<∞. The following
properties hold.
i Assume that Φ
αj
∈ L
∞
R
δ
0
Φ
α
δ − τ
dτ
A
0
∞
−1
×
⎛
⎝
dτ
r
i1
A
i
∞
⎞
⎠
≤ 1,δ∈ R
.
3.1
Then, the mapping f
h
then f
h
: −h, 0 × R
n
→ R
× R
n
is contractive and possesses a unique fixed point,
irrespective of the delays, in some bounded subset of R
n
. Such a fixed point is 0 ∈ R
n
which is also a
globally asymptotically stable equilibrium point.
ii Assume that Φ
αj
∈ L
∞
R
0
, R
n×n
, Φ
α
∈ L
2
R
0
0
as follows
g
h
t, δ
:
1 −
δ
0
Φ
α
δ − τ
dτ
δ
0
A
δ
0
Φ
α
δ − τ
2
dτ
1/2
×
r
i1
δ
0
A
i
,txt − r
i
is injected to 2.1 where K
i
:
R
n
× R
0
→ R
m
is in BPCR
0
, R
m
,x
it
: max0,t− r
i
,t → R
n
, for all i ∈ r − 1 ∪{0}, for all
Fixed Point Theory and Applications 9
t ∈ R
0
is a strip of the s tate-trajectory solution of 2.1. Assume also that
K
i
R
0
, R
n×n
,
3.3
and define g
f
: R
→ R
0
as
g
h
δ
:
1 −
δ
0
Φ
α
k−1
j0
Φ
αj
δ
δ
0
Φ
K
0
i
⎞
⎠
≤ 1,
δ ∈ R
,
3.4
provided that
δ
0
Φ
α
δ − τdτ
A
0
∞
B
∞
K
0
0
< 1. Then, for any given set of finite delays, the
h
δ < 1 − ε; for all t ∈ R
0
. Then, state trajectory
solution 2.2 of the forced system from any initial conditions in the admissible set is defined by a
contractive self-mapping with a unique fixed point in some bounded subset of R
n
for all controls of the
form ut
r
i0
K
i
x
it
,txt − r
i
fulfilling K
i
x
it
,t
∞
≤ ε/r 1
δ
0
Φ
α
; for all i ∈ r − 1∪{0}, instead of the hypotheses 3.3, and define g
f
: R
0
×R
→ R
0
as:
g
f
t, δ
:
⎛
⎝
1 −
δ
0
Φ
α
δ − τ
dτ
×
t τ
K
0
x
tτ
,t τ
2
dτ
1/2
⎞
⎠
⎞
⎠
−1
×
⎛
⎝
k−1
j0
Φ
αj
0
A
i
t τ − r
i
2
1/2
dτ
δ
0
B
t τ
K
i
such
that g
h
δ < 1 − ε; for all t ∈ R
0
then the mapping f
f
: −h, 0 × R
n
× R
m
× R
0
→ R
× R
n
defining the state-trajectory solution from any set of admissible initial conditions and all controls
ut
r
i0
K
i
x
it
,txt − r
i
being subject to
r
Φ
α
δ − τ
2
dτ
1/2
B
∞
, ∀t ∈ R
3.6
is contractive with a unique fixed point, irrespective of the delays, which is 0 ∈ R
n
being a globally
asymptotically stable equilibrium point.
Proof. The pointwise difference between two solutions xt and zt of 2.1 subject to
respective piecewise continuous initial conditions ϕ
x
: −h, 0 → R
n
and ϕ
z
: −h, 0 → R
n
x
j0
− z
j0
r
i1
r
i
0
Φ
α
t − τ
A
i
ϕ
xj
τ − r
i
− ϕ
ϕ
xj
τ − r
i
− ϕ
zj
τ − r
i
dτ
r
i1
t
r
i
Φ
α
t − τ
A
i
τ
x
α
τ − r
i
− z
α
τ − r
i
dτ
t
0
Φ
α
t − τ
B
τ
u
∪ R
0
, R
n
: φ ∈ S
φ, u
,φ ≡
⎛
⎝
k−1
j0
φ
j
⎞
⎠
∈ BPC
0
−h, 0
, R
n
,
∀j ∈
,u∈ BPC
0
R
0
, R
n
, 3.9
where BPC
0
R, R
n
is the set of bounded continuous n-vector functions on R. Now, define
P : M → M as the subsequent piecewise bounded continuous function on −h, 0 ∪ R
0
,
which is bounded continuous on R
,thatis,φ ∈ PBC
0
−h, 0, R
n
, φ ∈ BC
0
R
, R
0
Φ
α
t − τ
A
0
τ
φ
τ
dτ
k−1
j0
r
i1
r
i
0
Φ
τ
φ
j
τ − r
i
dτ
t
0
Φ
α
t − τ
B
τ
u
φ
τ
dτ.
3.10
t
∈ M is a simplified notation for the truncated φ ∈ M on 0, ∞. Norms without
subscripts mean, depending on context, vector or correspondingly induced matrix norms as,
for instance, the
2
-vector or induced matrix norms or pointwise values of such norms for
vector or matrix functions in the subsequent developments. Let M
t
be the space of truncated
functions φ
t
∈ M. Note that any truncated solution of 2.1 on any finite interval is always in
M so that one gets for any δ ∈ R
from 3.10 in the most general controlled case with control
ut
r
i0
K
i
x
t
,txt − r
i
P
φ
j
− η
j
t
δ
0
Φ
α
δ − τ
B
t τ
dτ
δ − τ
A
i
t τ
dτ
tδ−r
i
δ
0
Φ
α
δ − τ
A
αj
δ
r
i1
δ
0
Φ
α
δ − τ
B
t τ
K
i
i
t τ
dτ
φ − η
tδ−r
i
δ
0
Φ
α
δ − τ
A
k−1
j0
Φ
αj
δ
φ − η
tδ
r
i1
δ
K
0
i
φ − η
tδ−r
i
A
0
∞
B
∞
K
0
r
i1
δ
0
Φ
α
δ − τ
dτ
r
i1
A
i
0
⎞
⎠
×
φ − η
tδ
,
3.13
where the property that A
0
is constant has been used to rewrite the limits of the involved
integral is the most convenient fashion to simplify the related expressions. Equation 3.13
leads to
φ − η
t δ
≤
1 −
δ
0
∞
−1
×
⎛
⎝
k−1
j0
Φ
αj
δ
∞
B
∞
K
0
i
∞
φ − η
tδ−r
i
⎞
⎠
≤
1 −
K
0
0
∞
−1
×
⎛
⎝
k−1
j0
Φ
αj
δ
∞
B
∞
K
0
i
∞
φ − η
tδ−r
1
⎞
⎠
,
φ − η
t
≤
1 −
δ
0
Φ
α
δ − τ
dτ
A
0
∞
r
i1
δ
0
Φ
α
δ − τ
dτ
r
i1
A
i
g
h
δ
φ − η
t−r
1
; ∀δ ∈ R
, ∀t ∈ R
0
.
3.15
Then, the mapping f
h
: −h, 0 × R
n
→ R
× R
n
defining the state trajectory solution from
admissible initial conditions is nonexpansive if g
h
δ ≤ 1. Furthermore, the state trajectory
Z
0
k →∞,τ∈
0,r
1
∩R
0
φ − η
k1r
1
τ
≤
lim
Z
0
k →∞
K
k
c
δ
P
φ, u
φ
t
−
P
η, u
η
t
≤
⎛
⎝
t − τ
2
dτ
1/2
×
⎛
⎝
t
t−r
i
A
i
τ
2
1/2
1
A
0
τ
2
dτ
1/2
⎞
⎠
φ − η
t−r
1
, ∀δ ∈ R
, ∀t ∈ R
0
t − τ
A
i
τ
x
α
τ
dτ
t
0
Φ
α
t − τ
B
τ
u
τ
0≤j≤k−1
sup
t∈R
0
K
0j
t ≤
K
0
< ∞.
2
Φ
α
∈ L
1
R
0
, R
n×n
with sup
t∈R
0
Φ
α
t≤K
1
< ∞.
Then, the Caputo delay-free fractional dynamic system 2.1 of real order α has the following properties.
i It is globally stable under a control ut
K
0
i
, for all i ∈ r − 1{0}.
If, in addition, K
0j
t → 0 as t →∞; for all j ∈ k − 1 ∪{0} then the system is globally
asymptotically stable to the zero equilibrium point.
ii Property (i) holds if K
0
tK
0
0
is constant if K
1
< 1/
r
i0
A
i
∞
B
∞
r
t
≤
k−1
j0
Φ
αj
t
x
j0
r
i0
sup
τ∈
0,t
x
α
τ
≤
k−1
j0
K
0j
t
x
j0
t − τ
dτ
sup
τ∈
0,t
x
α
τ
≤
k−1
j0
K
0j
t
x
j0
τ∈
0,t
x
α
τ
3.19
≤
K
0
⎛
⎝
k−1
j0
x
j0
⎞
⎠
K
x
α
τ
,t∈ R
0
, 3.20
Fixed Point Theory and Applications 15
with sup
τ∈0,t
x
α
τ x
α
t
≤x
α
∞
. Thus, one gets from 3.20
x
α
t
∞
B
∞
K
0
i
−1
K
0
⎛
⎝
k−1
j0
x
j0
⎞
⎠
≤
α
t < ∞.Asaresult,the
Caputo fractional system of real order α is globally stable under zero delays since any state
trajectory solution generated from any admissible initial conditions is bounded for all time.
The proof of Property ii is similar to that of i under the modified constraints. Now, assume
that if, in addition, K
0j
t → 0ast →∞; for all j ∈ k − 1 ∪{0}, then
x
α
t
≤ min
⎛
⎝
1,
⎡
⎣
k−1
j0
K
0j
t
2
, 3.22
so that
lim sup
t →∞
x
α
t
≤ min
⎛
⎝
1, lim sup
t →∞
⎡
⎣
k−1
j0
K
0j
t
r
r
i0
A
i
∞
B
∞
K
0
i
K
1
K
2
< K
∈ R
0
. Thus, one gets from 3.19 that
lim
t →∞
x
α
t
≤ lim
t →∞
sup
τ∈
t
0
,t
x
α
τ
−1
⎛
⎝
k−1
j0
x
αjt
0
lim
t →∞
K
0j
t − t
0
⎞
⎠
0
3.24
provided that K
αj
t, Φ
α
t being
similar to Φ
αj
t from 2.3-2.4 by replacing A
0
→
r
i0
A
i
BK
0
0
.
16 Fixed Point Theory and Applications
The subsequent stability result is based on a transformation of the matrix A
0
to its
diagonal Jordan form which allows an easy computation of the
2
-matrix measure of its
diagonal part.
Theorem 3.4. Assume that J
A
0
J
A
0d
is negative, that is,
μ
2
J
1/α
A
0d
:
1
2
λ
max
J
1/α
A
0d
J
∗1/α
A
0d
max
k∈n
Re
0
T
−1
J
A
0
T,
1
β
1
T
−1
A
1
T,
1
β
2
T
−1
A
2
T, ,
1
β
r
T
−1
A
β
2
i
1. The fractional system is globally
asymptotically Lyapunov stable for one such set of real numbers if μ
2
J
1/α
A
0d
< 0, and
1
β
0
T
−1
J
A
0
T,
1
β
1
T
2
J
A
0d
1/α
. 3.27
ii A necessary condition for μ
2
J
1/α
A
0d
< 0 is that A
0
should be a stability matrix with
| argλ| < απ/2; for all λ ∈ σA
0
. Such a condition holds directly if α>2ϕ/π where
−ϕ, ϕ ⊆ −π/2,π/2 is the symmetric maximum real interval containing the arguments of all
λ ∈ σA
0
. It also holds, in particular, if A
0
is a stability matrix and α∈ R
≥ 1.
C
D
α
0
Tx
t
r
i0
A
i
Tx
t − h
i
⇐⇒
C
D
α
0
x
i
Tx
t − h
i
T
−1
J
A
0d
Tx
t
r
i0
T
−1
A
i
Tx
t − h
i
,
3.28
J
1/α
A
0d
1
2
λ
max
J
1/α
A
0d
J
1/α
A
0d
∗
Re λ
1/α
max
J
A
0d
Re λ
1/α
max
A
0d
0d
≥
1
2
λ
max
J
A
0d
J
∗
A
0d
1/α
if μ
2
J
1/α
A
0d
< 0sothat
0 >μ
2
J
A
0d
1/α
max
Re
λ :
λ ∈ σ
J
1/α
A
0d
≥
1
2
λ
max
J
A
0d
J
∗
A
0d
1/α
A
0d
is a stability matrix with μ
2
J
1/α
A
0d
< 0, then 1/α argλ ∈ −θ
1
/α, θ
2
/α ⊆
−π/2,π/2 so that | argλ| < απ/2; for all λ ∈ σA
0
, which is also a necessary condition
for the fulfillment of the sufficiency-type condition 3.27 for global asymptotic stability of
2.1, which implies the stability of the matrix J
1/α
A
0d
with the further constraint that μ
2
J
1/α
A
0d
<
0.
through Grant DPI2009-07197. He is also grateful to the Basque Government for its support
through Grants IT378-10 and SAIOTEK S-PE09UN12.
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