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Exponential Pointwise Stabilization of Semi-linear
Parabolic Distributed Parameter Systems via the
Takagi-Sugeno Fuzzy PDE Model
Jun-Wei Wang*† and Huai-Ning Wu‡
†School of Automation and Electrical Engineering
University of Science and Technology Beijing
Beijing 100083, P. R. China
‡Science and Technology on Aircraft Control Laboratory
School of Automation Science and Electrical Engineering
Beihang University, Beijing 100191, P. R. China
Abstract
This paper deals with the problem of exponential stabilization for nonlinear parabolic
distributed parameter systems using the Takagi-Sugeno (T-S) fuzzy partial differential
equation (PDE) model, where a finite number of actuators are active only at some specified
points of the spatial domain (these actuators are referred to as pointwise actuators). Three
cases are considered in this study as follows: full state feedback, piecewise state feedback,
and collocated pointwise state feedback. It is initially assumed that a T-S fuzzy PDE model
obtained via the sector nonlinearity approach is employed to accurately represent the
semi-linear parabolic PDE system. Based on the obtained T-S fuzzy PDE model,
Lyapunov-based design methodologies of fuzzy feedback control laws are subsequently
derived for above three state feedback cases by using the vector-valued Wirtinger’s
inequality to guarantee locally exponential pointwise stabilization of the semi-linear PDE
system, and presented in terms of standard linear matrix inequalities (LMIs). Moreover, the
favorable property offered by sharing all the same premises in the T-S fuzzy PDE models
and fuzzy controllers is not applicable for the case of collocated pointwise state feedback.
A parameterized LMI is introduced for this case to enhance the stabilization ability of the
fuzzy controller. Finally, the merit and effectiveness of the proposed design methods are
demonstrated by numerical simulation results of two examples.
Keywords: Exponential pointwise stabilization; State feedback control; Partial differential
equation; Takagi-Sugeno fuzzy model; Linear matrix inequality.
* Corresponding author: Jun-Wei Wang, E-mail: junweiwang@ustb.edu.cn
- 2 -
1. Introduction
Over the past few decades, fuzzy control approach based on the so-called Takagi- Sugeno (T-S)
fuzzy model [1] has become increasingly popular (see e.g., [2]-[4] and the references therein for a
survey of recent developments), since it can combine the merits of both fuzzy logic theory and
linear system theory. Fuzzy logic theory enables us to utilize qualitative, linguistic information
about a highly complex nonlinear system to decompose the task of modeling and control design into
a group of easier local tasks. At the same time, it also provides the mechanism to blend these local
tasks together to yield the overall model and control design. On the other hand, advances in linear
system theory have made a large number of powerful design tools available. As a consequence,
based on the T-S fuzzy model, the fruitful linear system theory can be applied to the analysis and
controller synthesis of nonlinear systems represented by ordinary differential equations (ODEs)
[5]-[8]. As a common belief, this fuzzy control technique is conceptually simple and effective for
controlling complex nonlinear systems. However, the existing results of stability analysis and
control design are mainly focused on fuzzy ODE systems or fuzzy systems modeled by delay
differential equations (DDEs) [2], [3] and [9]-[15].
In real world, most industrial processes are spatiotemporal in nature so that their behavior must
depend on time as well as spatial position, for example, thermal diffusion processes, fluid heat
exchangers, and chemical engineering, etc [16]-[20]. These processes can be modeled by distributed
parameter systems (DPSs). The mathematical models of DPSs are typically derived from the
dynamic conservation equations and take the form of partial differential equations (PDEs). In
particular, parabolic PDEs can be used to represent the dynamics of industrial processes involving
the diffusion-convection-reaction phenomenon, such as semiconductor thermal processes, chemical
processes, plasma reactors, crystal growth processes to name a few [16]-[18]. The T-S fuzzy ODE
model has been extended to solve the problem of finite-dimensional distributed control design for
semi-linear parabolic PDE (SLPPDE) systems [21]-[23]. For example, finite-dimensional
distributed fuzzy control design methods have been developed in [21] and [22] for a class of
SLPPDE systems based on the approximated ODE model derived from the singular perturbation
formulation of Galerkin method, where the closed-loop stability of original PDE system is
guaranteed using the singular perturbation theory. The methods reported in [21]-[23] are also
referred to as the “reduce-then-design (RTD)” approach. Due to the truncation before the controller
design, the RTD approach fails to take advantage of natural property of the systems.
To achieve better control performance and higher control precision, it is necessary to develop
control design methods on the basis of original PDE for nonlinear DPSs. More recently, fuzzy
control design methods based on the fuzzy PDE model directly have been proposed for a class of
- 3 -
SLPPDE systems [24]-[26]. Since the PDE model is employed, the methods in [24]-[26] allow the
designer to take full advantage of all the natural features of the systems and to understand the
system structure much more completely. The results in [24] and [25] address the distributed control
case and the result reported in [26] considers the boundary control case. Generally speaking, the
control actions of PDE systems are either distributed over the entire spatial domain (or part thereof),
applied at the boundary (or part thereof) of the spatial domain (boundary controls), or active only at
specified points of this domain (pointwise controls) [27]. In comparison to the distributed control
design methods in [24] and [25], pointwise control form is easily implemented in practice since it
needs only a few actuators distributed at some specified points of the spatial domain. Even though
the boundary control form considered in [26] is also easily implemented since only a few actuators
are required in the implementation, the control design method in [26] is conservative and only
applicable for the weak SLPPDE systems, where the sector bound of the nonlinear term is small.
Different from the boundary control form [26], pointwise control form can be applicable for the
SLPPDE system with a larger sector bound of the nonlinear term. A finite-dimensional robust H∞
fuzzy observer-based pointwise control design was presented in [28] for a class of quasi-linear
parabolic PDE systems, where the original PDE system is initially approximated by a T-S fuzzy
PDE model, the fuzzy PDE model is subsequently represented a finite-dimensional slow subsystem
coupled with an infinite-dimensional fast subsystem to be tolerated, and the small gain theorem is
employed to overcome the spillover phenomenon. But the design method developed in [28] belongs
to the category of “reduce-then-design (RTD)” for the control of PDE systems. The control design
of nonlinear DPSs is very difficult and requires more sophisticated mathematical techniques, as the
systems are infinite-dimension ones but only a finite number of actuators and sensors are available
for the control system implementation in practice. More recently, the authors of [29] have proposed
a simple but effective fuzzy control design method for nonlinear coupled parabolic PDE-ODE
systems, where the PDE subsystem is subject to only a few collocated pointwise actuator/sensor
pairs and is represented by a scalar SLPPDE model. To the best of authors’ knowledge, the result on
the design of fuzzy controllers via a finite number of pointwise actuators and sensors on the basis of
original PDE model has not been reported yet for nonlinear multi-dimensional parabolic DPSs,
which motivates this study.
In this paper, we will present exponentially stabilizing feedback controller design methods
based on the T-S fuzzy PDE model for a class of SLPPDE systems with a finite number of
pointwise control actuators. The following three cases are studied in this study: full state feedback,
piecewise state feedback, and collocated pointwise state feedback. A T-S fuzzy PDE model
constructed by using the sector nonlinearity approach is first employed to accurately describe the
SLPPDE. Then, based on the fuzzy PDE model, Lyapunov-based design methodologies of fuzzy
- 4 -
feedback controllers are developed for above three state feedback cases by utilizing the
vector-valued Wirtinger’s inequality to ensure locally exponential pointwise stabilization of the
SLPPDE system, and presented in term of standard linear matrix inequalities (LMIs), which are
directly solved via the polynomial-time interior-point method [5] and [30]. For the case of
collocated pointwise state feedback, a parameterized LMI is introduced to enhance the stabilization
ability of the fuzzy controller, since the favorable property offered by sharing all the same premises
in the T-S fuzzy PDE model and fuzzy controller cannot be employed. Finally, the proposed design
methods are applied to an exponential stabilization control of a Fisher equation and a FitzHugh-
Nagumo (FHN) equation, and the obtained simulation results show their merit and effectiveness.
The main contribution and novelty of this paper are threefold. First, the definition of locally
exponential pointwise stabilization is introduced for the SLPPDE system. Second, LMI-based fuzzy
control design methods are developed for three cases of state feedback: full state feedback,
piecewise state feedback, and collocated pointwise state feedback, to guarantee locally exponential
pointwise stabilization of the SLPPDE system. Third, a parameterized LMI is introduced to derive a
less conservative control design method for the case of collocated pointwise state feedback.
In comparison to the existing results [24]-[26], [29], [31], and [32], the advantages of proposed
control design methods of this paper can be summarized as follows: a) Comparing to the existing
fuzzy control design methods in [24] and [25] presented in terms of spatial differential linear matrix
inequalities, fuzzy control design methods proposed in this paper are presented in terms of standard
LMIs, which are simple but effective; b) Different from the fuzzy boundary control design in [26]
only applicable for the SLPPDE system with Neumann boundary conditions, the proposed fuzzy
pointwise control design of this paper is applicable for the SLPPDE system with Dirichlet boundary
conditions; c) Whereas [29] and [31] address the pointwise fuzzy control design and piecewise
fuzzy control design for nonlinear ODE systems coupled with scalar parabolic PDE systems and [32]
develops a robust piecewise control design for scalar SLPPDE systems, this paper proposes
LMI-based fuzzy pointwise control design methods for multi-dimensional SLPPDE systems.
The rest of this paper is organized as follows. Section 2 introduces preliminaries and problem
formulation. Section 3 gives LMI-based sufficient conditions on locally exponential pointwise
stabilization for three cases of full state feedback, piecewise state feedback, and collocated
pointwise state feedback. Numerical simulations are provided in Section 4 to show the merit and
effectiveness of the proposed design methods. Finally, Section 5 offers some concluding remarks.
Notations: ,
, n
and mn
denote the set of all real numbers, the set of all positive
numbers, n-dimensional Euclidean space and the set of all mn
matrices, respectively.
2([0, ]; )
nn
L is a Hilbert space of square integrable vector functions ( ) : [0, ] n
xL
- 5 -
with 111
20
() () ()
LT
x
xdx
, where 1() n
. ,2 ([0, ]; )
ln
L
is a Sobolev space of
absolutely continuous vector functions ( ) :[0, ] n
xL
with square integrable derivatives
()
ll
dxdx
of the order 1l
and with ,2
() ()
0
0
() iT i
lii
Lldxdx
idx dx dx
. Identity matrix of
appropriate dimension will be denoted by
I
. For a symmetric matrix M, (,)0M means
that it is positive definite (negative definite, negative semi-definite, respectively). min ()
A
and
max ()
A
stand for the minimum and maximum eigenvalues of a square matrix
A
, respectively.
The superscript ‘T’ is used for the transpose of a vector or a matrix. The symbol ‘’ is used as an
ellipsis in matrix expressions that are induced by symmetry, e.g.,
[] []
TT
T
SMN X
S
MNM N X
Y
X
Y
.
The subscripts
x
and t stand for the partial derivatives of ( , )
x
ty with respect to
x
, t, i.e.,
(,) (,)
t
x
txtt yy, (,) (,)
x
x
txtx yy, and 22
(,) (,)
xx
x
txtx
yy, respectively.
2. Preliminaries and problem formulation
2.1 System description and preliminary
This paper considers the following SLPPDE system:
(,) (,) ( ( ,)) ( ) ()
txx
x
txtxtxt
yyfy
Gu, 0t, 0
(,0) ()
x
x
yy (1)
where 12
(, ) [ (, ) (, ) (, )]
T
n
tytyt yt y is the state variable (
,min
(, ) ni
t
y
,max
(, ) , {1,2, , } n
ii
yt i n
is a given local domain containing the equilibrium profile
(, ) 0ty, ,min 0
i
and ,max 0
i
, {1, 2, , }in
are real known scalars), [0, ]xL and
0t are the spatial position and time, respectively, nn
is a known constant matrix,
((,))
x
tfy is a sufficiently smooth nonlinear function in ( , )
x
ty satisfying (0) 0f.
Remark 1. Notice that the PDE model (1) has been widely utilized to describe the spatiotemporal
dynamics of chemical and biological systems [17], [33]. For example, if (,)
y
xt is the neutron
concentration profile of a nuclear reactor, (,) (,)
xx xx
x
tyxt
y
, and ((,)) (,)
T
x
tyxt
fy
2(,)
Uyxt
, the PDE model (1) describes the spatiotemporal evolution dynamics of neutron
population in a nuclear reactor [33]. If 1
(,) [ (,)
x
tyxty 2(,)]
T
yxt are the concentration and
temperature profiles in a chemical tubular reactor, 1
diag{ ( ) , }
ea p ma
CD
, ((,))
x
tfy
12
[((,)) ((,))]
T
f
xt f xtyy, 11
11, 1
( (,)) (,) 4 ( ) (,) ( ) ( ,)
xp p
f
xt y xt hd C y xt H C xt
y,
- 6 -
22,
( (,)) (,) (,)
x
f
xt y xt xt
y, with 02 1
(,) (,)exp( /( (,)))
x
tkyxt ERyxt
, the PDE model (1)
will be used to describe the spatiotemporal evolution dynamics of temperature and concentration
distribution profiles in the chemical tubular reactor [17].
Associated with the SLPPDE (1) are the appropriate boundary conditions. Without loss of
generality, the following Dirichlet boundary conditions are considered in this paper
(0, ) ( , ) 0tLtyy , 0t. (2)
()
x
G is a known square integrable matrix function of
x
, in which the i-th column describes the
i-th actuator’s spatial distribution. 12
() [ () () ()]
Tm
m
tutut ut
u is the control input,
which is provided by m-pointwise control actuators. To model these pointwise control actuators, the
function ()
x
G is chosen as 12
() [ () () ()] nm
m
xxx x
Ggg g , where () ( )
vvv
x
xx
gg ,
{1, 2 , , }vm, with v
x
denotes the location of control actuators and ( )
is a Dirac delta
function of the following form [34]
0, 0
() , 0
x
xx
(3)
and
2
1
() 1
l
l
x
dx
, if 12
0[,]ll
. (4)
The choice of ( )
x
G corresponds to pointwise control at specified positions v
x
, v. This
control form under consideration in this study is referred to as the spatial pointwise control [16],
which is illustrated in Fig. 1. According to the location of m-pointwise actuators, for the following
control design purpose, the spatial domain [0, ]
L
is divided into m subintervals 1
[, ]
vv
x
x
,
v such that 1
(, )
vvv
x
xx
, v, where 12
0xx
1mm
x
xL
.
()
m
ut
1()ut 2()ut
1
x
2
x
m
x
0L
1
x
2
x
3
x
m
x
1m
x
Fig. 1 Pointwise control form
As pointed out in [34], the function ( )
is of the following fundamental property:
2
1
()( ) ()
l
l
f
xxldxfl
, 12
[, ]lll, for any function ( )
f
x
. (5)
This property is sometimes referred to as the sifting property or the sampling property. Due to its
sifting property given by (5), ( )
is said to “sift out” the value at
x
l
and has been widely
used to describe the pointwise control/measurement problem in distributed parameter control theory
- 7 -
(e.g., see [16], [36] and the references therein) and applied to introduce the boundary control into
the differential equation of DPSs (e.g., see [16]). Clearly, the so-called Dirac delta ‘function’ ( )
is not compatible with the classical concept of a function. A rigorous definition of Dirac delta
function requires measure theory or the theory of distributions. It has been pointed out in [34] that a
possible explicit form of ()
x
l
is given by
1
22
0
0, otherwise
()
lim , if ( , )
xl xl l
.
Hence, from the numerical implementation point of view, we use the following approximation to
the Dirac delta function ()
x
l
[34]-[36]:
1
22
0, otherwise
() , if ( , )
xl xl l
where
is a sufficiently small positive constant.
For simplicity, when ( ) 0t
u, the SLPPDE system described by (1) and (2) is referred to as
an unforced system. We introduce the following definitions of locally exponential stability and
locally exponential pointwise stabilization in the sense of 2
on the Hilbert space n
:
Definition 1. The unforced system of (1) and (2) (i.e., () 0t
u) is said to be locally exponentially
stable in the sense of 2
, if there exist constants 1
and 0
such that the expression
0
22
( , ) ( ) exp( )tt
yy is fulfilled for ( , )t
y and 0t
.
Definition 2. The SLPPDE system described by (1) and (2) is said to be locally exponential
pointwise stabilization in the sense of 2
if the closed-loop system resulting from the controller
via pointwise control actuators applied to the SLPPDE system (1)-(2) is locally exponentially stable
in the sense of 2
.
Note that Definition 1 is borrowed and modified from the definition of exponential stability
provided in [26] and Definition 2 is modified from the results in [37]-[39].
2.2 T-S Fuzzy PDE model and problem formulation
By following the main idea of the sector nonlinearity method for the nonlinear ODE systems
[2], one can also derive an exact T-S fuzzy PDE model construction for the semi-linear PDE
systems. For example, we consider the semi-linear scalar PDE (,) (,) sin((,))
txx
y
xt y xt yxt
3(,)
y
xt , where (, )yt. Set 1(,) sin( (,))
x
tyxt
and 2
2(,) (,)
x
tyxt
be premise variables
and define { ( , ) 0.5 ( , ) 0.5 }yt yt
S. For the premise variable 1(,)
x
t
in the local
- 8 -
domain S, one can find the sector 1
[2 ,1]
that consists of two profiles 1
2(,)
y
xt
and (,)
y
xt .
sin( ( , ))
y
xt is represented as 1
11121
(,) sin( (,)) ( ( ( ,))1 ( (,))2 ) (,)
x
tyxtwxtwxt yxt
in
S, where
1
1
1
(,)2 (,)
1
(1 2 ) ( , )
11
(,) 0
((,))
1otherwise
xt y xt
yxt xt
wxt
and 21 11
((,)) 1 ((,))wxt wxt
. Similarly,
2(,)
y
xt is rewritten as 22
21222
(,) (,) ( ( (,))0.25 ( (,))0)xt y xt xt xt
in S, where
2
12 2
((,)) 4 (,)
x
txt
and 22 12
((,))1 ((,))
x
txt
. From above analysis, the semi-linear
PDE is represented as 4
1
(,) ( ,) ( (,)) ( ,)
txx i i
i
y
xt y xt h xt ayxt
, where (,)
x
t
12
[(,) (,)]
T
x
txt
, 11112
( (,)) ( (,)) ( ( ,))hxtwxt xt
, 21122
( (,)) ( (,)) ( ( ,))hxtwxt xt
,
32112
( (,)) ( (,)) ( (,))h xt w xt xt
, 42122
((,)) ((,)) ( (,))hxtwxt xt
, 2
110.25a
, 21a
,
12
320.25a
, and 1
42a
. These functions ((,))
i
hxt
, {1,2,3,4}i
are referred to as
membership ones and satisfy ((,)) 0
i
hxt
, {1,2,3,4}i
and 4
1((,)) 1
i
ihxt
in S for
[0, ]
x
L and 0t. The local domain S is referred to as the operating one. It is easily verified
that the resulting T-S fuzzy PDE model is equivalent to the semi-linear PDE model. That is, the
fuzzy PDE model exactly describes the dynamics of the semi-linear PDE in the operating domain
S. This idea can be extended to the semi-linear PDE (1). Therefore, we assume for brevity that the
following T-S fuzzy parabolic PDE model can be constructed to exactly describe the nonlinear
dynamics of the SLPPDE (1) in a given operating domain
,min ,max
ˆˆ
(, ) (, ) ,
nii i
tyt
yS
{1, 2 , , } n
in, with ,min
ˆ0
i
and ,max
ˆ0
i
, {1, 2, , }in
are real known scalars:
Plant Rule i:
IF 1(,)
x
t
is 1i
F and and ( , )
l
x
t
is il
F
THEN (,) (,) (,) () ()
txxi
x
txtxtxt
yy
A
y
Gu, 0t, {1,2, , }ir (6)
where ij
F, i, {1, 2, , }jl are fuzzy sets, nn
i
A, i
are known constant matrices, l
is the number of nonlinear elements in ((,))
x
tfy , and 2l
r
is the number of IF-THEN fuzzy
rules. 1(,)
x
t
, 2(,)
x
t
, , ( , )
l
x
t
are the premise variables that may be functions of ( , )
x
ty.
This domain S is referred to as the operating one. The operating domain S, which is a bit larger
than the one (i.e., S), is in general chosen to guarantee the robustness of the T-S fuzzy
PDE model (7).
The overall dynamics of T-S fuzzy parabolic PDE (6) can be expressed as
1
( ,) ( ,) ( (,)) (,) ( ) ()
r
txx ii
i
x
txthxtxtxt
yy
A
y
Gu
, 0t (7)
- 9 -
where 12
(,) [ (,) (,) (,)]
T
l
x
txtxt xt
and
1
((,)) ( (,))
l
iijj
j
wxt F xt
, 1
((,)) ((,)) ((,))
r
ii i
i
hxt wxt wxt
, i.
((,))
ij j
Fxt
is the grade of the membership of (,)
j
x
t
in ij
F for i
. In this paper, we assume
((,)) 0
i
wxt
, i and 1((,)) 0
r
i
iwxt
for all [0, ]
x
L
and 0t. Then, we can obtain
the following conditions for all [0, ]
x
L and 0t:
((,)) 0
i
hxt
, i and 1((,)) 1
r
i
ihxt
. (8)
According to the above analysis, we know that the SLPPDE (1) in the operating domain S is
equivalent to the T-S fuzzy parabolic PDE (7), that is, for any (, )t
yS
1
((,)) ((,)) (,)
r
ii
i
x
t h xt xt
fy Ay
. (9)
Hence, the property of the SLPPDE (1) in the operating domain S is identical to the T-S fuzzy
PDE model (7). More recently, T-S fuzzy PDE models have been employed in [24]-[26] for
semi-linear parabolic PDE systems.
Based on the fuzzy PDE model (7), we consider the following fuzzy state feedback controller:
1
0
() ( ( , )) ( ) ( , )
Lr
ii
i
thxtxxtdx
uCKy
(10)
where 1, 2 , ,
[]
Tmn
iii mi
Kkk k , i
are control gain matrices to be determined and
12
() diag{ (), (), , ()} mm
m
xcxcxcx
C is a known matrix function with respect to
x
,
[0, ]
x
L. The matrix functional ( )
x
C is used to describe three cases of state feedback as follows
for the fuzzy controller (10): full state feedback, piecewise state feedback, and collocated pointwise
state feedback. That is, the fuzzy controller (10) is a full state feedback one (if ( ) 1
v
cx, [0, ]
x
L
,
v), a piecewise state feedback one (if 1
1if [, ]
() 0otherwise
vv
v
x
xx
cx
, v), or a collocated
pointwise state feedback one (if () ( )
vv
cx xx
, v
).
For the parabolic PDE system (1)-(2), the objective of this study is to develop LMI-based
design methods of the fuzzy controller (10) for the three cases: full state feedback, piecewise state
feedback, and collocated pointwise state feedback, such that the resulting closed-loop system is
locally exponentially stable in the sense of 2
. To do this, the following lemma, which gives a
vector-valued Wirtinger’s inequality, will be used in the development of the design method:
Lemma 1 (Vector-value Wirtinger’s inequality, [40]). Let 1,2
( ) ([0, ]; )
n
L
z be a vector
function with (0) 0
z or ( ) 0Lz. Then, for a matrix 0nn
S, we have
- 10 -
22
00
() () 4 () ()
LL
T
TssdsL dsds dsdsds
zSz z Sz . (11)
From the integral inequality (11) in Lemma 1, we can easily derive the following two inequalities
for 1,2
( ) ([0, ]; )
n
L z without the property (0) 0
z or ( ) 0L
z:
22
00
(() (0)) (() (0)) 4 () ()
LL
T
T
s s ds L dsds dsdsds
zzSzz z Sz ,
and
22
00
(() ( )) (() ( )) 4 () ()
LL
T
T
sL sLdsL dsdsdsdsds
zzSzz z Sz .
This is because () (0)s
zz satisfies (0) (0) 0
zz and (() (0)) ()d s ds d s ds
zz z (or
() ( )
sL
z
z satisfies () () 0LLzz and (() ( )) ()ds Ldsdsds
zz z ), respectively.
Remark 2. Notice that the integral inequality (11) has been reported for the case when 0S and
(0) 0z [41], [42], for the case when diagonal matrix 0S and (0) 0z or () 0L
z [26],
and for the case when 0S and (0) 0
z or () 0L
z [40]. The scalar version of the inequality
(11) has been recently used in [32] to address a robust collocated piecewise control design for a
class of scalar nonlinear parabolic DPSs.
3. LMI-based exponential pointwise stabilization
On the basis of the fuzzy PDE model (7), this section will provide LMI-based design methods
of fuzzy controller (10) guaranteeing locally exponential pointwise stabilization of the SLPPDE
system (1)-(2) for the three cases: full state feedback, piecewise state feedback and collocated
pointwise state feedback.
Let us consider the following Lyapunov function candidate for the SLPPDE system (1)-(2):
0
() ( ,) ( ,)
LT
Vt xt xtdxyPy (12)
where 0 nn
P is a constant Lyapunov matrix to be determined. Clearly, :V
, which
has a physical interpretation as the energy of the system (1)-(2). Using (9), the time derivative of
()Vt given in (12) along the solution to the SLPPDE (1) subject to the boundary conditions (2) is
given by
000
() 2 (,) (,) 2 (,) ( (,)) 2 (,) () ()
LLL
TTT
xx
Vt xt xtdx xt xt dx xt xdx t
y
P
yy
P
fy y
PG u
00
1
2 (,) (,) ((,)) (,)[ ](,)
r
LL
TT
xx i i
i
x
t xtdx h xt xt xtdx
yPy y PAy
0
2(,)()()
LT
x
txdxt
y
PG u . (13)
Integrating by parts and taking into account the boundary conditions (2), we have
- 11 -
0
0 0
(,) (,) (,) (,) (,) (,)
L L
xL
TT T
xx x x x
x
x
t xtdx xt xt xt xtdx
yPy yPy yPy
0(,) (,)
LT
xx
x
txtdx
yPy. (14)
Substituting (14) into (13), gives
1
00
() (,)[ ] ( ,) ( (,)) ( ,)[ ] ( ,)
LL
r
TT
xx i i
i
Vt xt xtdx h xt xt xtdx
y P y y PA y
0
2(,)()()
LT
x
txdxt
y
PG u . (15)
3.1 Full state feedback
We first consider fuzzy control design for the case of full state feedback. For this case (i.e.,
() 1
v
cx, [0, ]
x
L, v), the fuzzy controller (10) is written as
,
1
0
() ( (,)) (,)
LrT
vivi
i
ut h xt xtdx
ky
, v
, (16)
which is a fuzzy full state feedback controller.
Considering the definition of ( )
x
G and (16), we have
,
11
00
(,) ( ) () ( ,) ( (,)) ( ,)
LL
mr
TT T
vv i vi
vi
x
t x dx t x t h x t x t dx
yPGu y Pg ky
. (17)
Substituting (17) into (15) derives
1
00
() (,)[ ] ( ,) ( (,)) ( ,)[ ] ( ,)
LL
r
TT
xx i i
i
Vt xt xtdx h xt xt xtdx
y P y y PA y
,
11
0
2 ( ,) ( (,)) (,)
L
mr
TT
vv i vi
vi
x
thxtxtdx
yPg ky
. (18)
Hence, we have the following theorem:
Theorem 1. Consider the SLPPDE system (1)-(2) and the T-S fuzzy PDE model (7), if there exist a
matrix 0Q and vectors ,
n
viz, v
, i
such that the following LMIs are satisfied:
,
1
21, 22
[]
0
mT
ivvi
v
i
AQ g z , i
(19)
where 21, 1, 1 ,
[]
TTT
ii mim
zg
z
g
, and 21 1 1
22 1
0.25 diag{ [ ] [ ]}
m
m
QQ,
with 22
max{ , ( ) }
vvv
xLx
, v, then there exists a fuzzy full state feedback controller (16)
exponentially stabilizing the SLPPDE system (1)-(2) in the sense of 2
. In this case, the control
parameters ,vi
k, v, i are given by
1
,,
TT
vi vi
kzQ
, v, i. (20)
Proof. Assume the LMIs (19) are fulfilled. Set
1
QP, ,,
TT
vi vi
z
kQ, v, i
. (21)
- 12 -
By pre- and post-multiplying LMIs (19) with a block-diagonal matrix
11
QQ
1(1)(1)mnmn
Q and using (21), we get
,
1
21, 22
[]
0
mT
ivvi
v
i
i
PA Pg k
, i
(22)
where 21, 1, 1 ,
[]
TTT
ii mim
k
g
Pk
g
P, and
21 1 1
22 1
0.25 diag{ [ ] [ ]}
m
m
PP.
The inequalities (22) imply
[]0
P. (23)
Define (,) (,) ( ,)
vv
x
txtxtyyy
, [0, ]
x
L
, we obtain
(,) 0
vv
xty and ,(,) (,)
vx x
x
txtyy, v
, [0, ]
x
L
. (24)
Based on Lemma 1 and considering (23) and (24), we get
2
2
00
(,)[ ] (,) (,)[ ] (,)
4
v v
xx
TT
xx vv
v
x
txtdx xtxtdx
x
yPy yPy, (25)
and
2
2
( , )[ ] ( , ) ( , )[ ] ( , )
4( )
v v
LL
TT
xx vv
xx
v
x
t xtdx xt xtdx
Lx
yPy yPy. (26)
Using (25) and (26) as well as considering (0, )
v
x
L
, v
, we have
2
2
00
(,)[ ] (,) (,)[ ] (,)
4
v
Lx
TT
xx vv
v
x
t xtdx xt xtdx
x
yPy yPy
2
2(,)[ ] (,)
4( ) v
LT
vv
x
v
x
txtdx
Lx
yPy
21
0
0.25 ( , )[ ] ( , )
LT
vv v
x
txtdx
yPy. (27)
Substituting (27) into (18) and considering (24), we can get
211
10
( ) 0.25 ( , )[ ] ( , )
L
mT
vv v
v
Vt m xt xtdx
yPy
,
11
0((,)) (,)[ ](,)
Lrm
TT
iivvi
iv
hxt xt xtdx
yPA Pgky
,
11
0
2 ( ,) ( (,)) ( ,)
L
mr
TT
vvi vi
vi
x
t h xt xtdx
yPg ky
1
0((,)) (,) (,)
LrT
ii
ihxt xt xtdx
yy, (28)
where 1
(,) [ (,) (,) (,)]
TT TT
m
x
txtxt xt
yyy y and i
, i
are defined by (22).
It is immediate from the inequality (22) that one can find an appropriate scalar 0
such
- 13 -
that the following inequalities are fulfilled:
20
i
I, i. (29)
Hence, considering (8) and (29), the inequality (28) is written as
22
22
() 2 (,) 2 (,)Vt t t
yy, 0t. (30)
From (12), we can easily find two scalars 10
and 20
such that
22
12
22
(, ) ( ) (, )tVt t
yy
, 0t (31)
where 1min
()
P and 2max
()
P. From (30) and (31), we thus get 1
2
() 2 ()Vt Vt
, 0t,
which implies
1
2
() (0)exp( 2 )Vt V t
, 0t. (32)
Using (31) and (32), we have 22
11
120 2
22
(, ) () exp( 2 )tt
yy , 0t, which implies
11
120 2
22
( , ) ( ) exp( )tt
yy , 0t. Therefore, from Definition 2, the locally exponential
pointwise stabilization of the SLPPDE system (1)-(2) in the sense of 2
is ensured by the fuzzy
full state feedback controller (16). That is, from Definition 1, the closed-loop system resulting from
the fuzzy full state feedback controller (16) applied to the SLPPDE system (1)-(2) is exponentially
stable in the sense of 2
. From (21), we have (20). The proof is complete. □
Based on the T-S fuzzy PDE model (7), by using the Lyapunov technique, the vector-value
Wirtinger’s inequality (11) and integration by parts, Theorem 1 provides an LMI-based design
method of the fuzzy full state feedback controller (16) ensuring locally exponential pointwise
stabilization of the SLPPDE system (1)-(2). The corresponding control parameters ,vi
k, v
,
i are constructed as (20) via the feasible solutions to LMIs (19). As is well known, the
feasibility problem subject to LMI constraints is a convex optimization one. A lot of effective
algorithms have been reported to solve the LMI feasibility problem, such as polynomial-time
interior-point method [5]. The state-of-the-art interior-point LMI solvers (including feasp mincx,
and gevp) have been implemented in the Matlab’s LMI control toolbox [30]. The feasp solver is
utilized to verify and solve the LMIs (19).
3.2 Piecewise state feedback
For the piecewise state feedback case (i.e., 1
1if [, ]
() 0otherwise
vv
v
x
xx
cx
, v), the fuzzy
controller (10) is written as
1
,
1
() ( ( ,)) ( , )
v
v
xrT
vivi
i
x
ut h xt xtdx
ky
, v
, (33)
- 14 -
which is a fuzzy piecewise state feedback controller.
Consider the definition of ( )
x
G and using (33), we have
1
,
11
0(,) ( ) () ( ,) ( ( ,)) ( ,)
v
v
Lx
mr
TT T
vv i vi
vi
x
x
t xdx t x t h xt xtdx
yPGu y Pg ky
. (34)
Substituting (34) into (15) and considering 12 1
0mm
x
xxxL
, derive
1 1
11
1
() ( ,)[ ] ( ,) ( ( ,)) ( ,)[ ] ( ,)
v v
v v
r
xx
mm
T T
xx i i
vv
xx
i
Vt xt xtdx h xt xt xtdx
yPy yPAy
1
,
11
2 ( ,) ( ( ,)) ( ,)
v
v
x
mr
TT
vv i vi
vi
x
x
thxtxtdx
yPg ky
. (35)
From above analysis, we have the following theorem:
Theorem 2. Consider the parabolic PDE system (1)-(2) and the T-S fuzzy PDE model (7), if there
exist a matrix 0Q and vectors ,
n
viz, v
, i
satisfying the following LMIs:
,
21
,
[] 0
0.25 [ ]
T
ivvi
T
vvi v
AQ g z
gz Q, v
, i
(36)
where 22
1
max{( ) , ( ) }
vvvvv
xx x x
, v
, then there exists a fuzzy piecewise state feedback
controller (33) ensuring locally exponential pointwise stabilization of the PDE system (1)-(2) in the
sense of 2
. In this case, the control parameters ,vi
k, v
, i
are given by (20).
Proof. Assume the LMIs (36) are fulfilled. By pre- and post-multiplying LMIs (36) with a
block-diagonal matrix 11
diag{ }
QQ and using (21), we get
,
,21
,
[] 0
0.25 [ ]
T
ivvi
vi T
vvi v
PA Pg k
Pg k P, v
, i
(37)
which imply the LMI (23).
Define (,) (,) ( ,)
vv
x
txtxtyyy, 1
[, ]
vv
x
xx
, v
, we can obtain
(,) 0
vv
xty and ,(,) (,)
vx x
x
txtyy, 1
[, ]
vv
x
xx
, v
. (38)
Based on Lemma 1 and considering (23) and (38), we get
2
2
(,)[ ] (,) (,)[ ] (,)
4( )
v v
v v
xx
TT
xx vv
xx
vv
x
t xtdx xt xtdx
xx
yPy yPy, (39)
and
1 1
2
2
1
(,)[ ] (,) (,)[ ] (,)
4( )
v v
v v
xx
TT
xx vv
xx
vv
x
txtdx xtxtdx
xx
yPy yPy. (40)
Using (39) and (40) as well as considering 1
(, )
vvv
x
xx
, we have for any v
1
2
2
(,)[ ] (,) (,)[ ] (,)
4( )
v v
v v
xx
TT
xx vv
xx
vv
x
t xtdx xt xtdx
xx
yPy yPy
- 15 -
1
2
2
1
(,)[ ] (,)
4( )
v
v
xT
vv
x
vv
x
txtdx
xx
yPy
1
21
0.25 ( , )[ ] ( , )
v
v
xT
vv v
x
x
txtdx
yPy. (41)
From (8) and (41), the expression (35) can be written as
1
21
1
( ) 0.25 ( , )[ ] ( , )
v
v
x
mT
vv v
vx
Vt xt xtdx
yPy
1
11
( ( ,)) ( ,)[ ] ( ,)
v
v
x
mr T
ii
vi
xhxt xt xtdx
yPAy
1
,
11
2 ( ,) ( ( ,)) ( ,)
v
v
x
mr
TT
vv i vi
vi
x
x
thxtxtdx
yPg ky
1
,
11
((,)) (,) (,)
v
v
x
mr T
ivviv
vi
xhxt xt xtdx
yy
, (42)
where (,) [ (,) (,)]
TTT
vv
x
txtxtyyy and ,vi
, v
, i
are defined by (37).
From the inequalities (37), we can find an appropriate constant 0
such that the following
inequalities are fulfilled:
,20
vi
I, v, i. (43)
Substituting (43) into (42), we get
12
2
1
() 2 ( , ) ( , ) 2 (, )
v
v
x
mT
vv
vx
Vt xt xtdx t
yy y. (44)
Similar to the proof of Theorem 1, it can be easily shown from (44) that the closed-loop system
resulting from the SLPPDE system (1)-(2) plus the fuzzy piecewise state feedback controller (33) is
exponentially stable in the sense of 2
. The proof is complete. □
On the basis of the T-S fuzzy PDE model (7), Theorem 2 provides a simple but effective
LMI-based design method of the fuzzy piecewise state feedback controller (33) for the SLPPDE
system (1)-(2). The corresponding control gain parameters ,vi
k, v
, i are constructed as
(20) via the feasible solutions to LMIs (36), which can be directly verified and solved through the
feasp solver [30].
3.3 Collocated pointwise state feedback
In this subsection, we will present an LMI-based fuzzy control design for the case of
collocated pointwise state feedback. For this case (i.e., () ( )
vv
cx xx
, v), the fuzzy
controller (10) is written as
,
1
() ( ( ,)) ( , )
rT
vivviv
i
ut h xt xt
ky
, v
, (45)
which is a fuzzy collocated pointwise state feedback controller.
- 16 -
Note that the fuzzy controller (45) only utilizes the information of membership functions
((,))
i
hxt
, i used in the T-S fuzzy PDE model (7) at some specified points v
x
, v of the
spatial domain [0, ]
L and its membership functions are mismatched with the ones in the T-S fuzzy
PDE model with respect to the space variable. This mismatch prevents the direct use of the
parameterized LMI technique given in [24] to derive a less conservative design method for the
fuzzy controller (45). Motivated by the works [10] and [43]-[45], the following constraints
((,)) (( ,) 0
iviv
hxt hxt
, 1
[, ]
vv
x
xx
, 1
(, )
vvv
x
xx
, 0t, i
, v (46)
are imposed on the membership functions ((,))
i
hxt
, i
of the T-S fuzzy PDE model (7) for
some given constants v
, v. This paper will provide a new parameterized LMI technique by
using the parameterized LMI technique in [24] and the constraints (46) for the membership
functions ((,))
i
hxt
, i. To guarantee the constraints (46), we make the following assumption
for the membership functions ( ( , ))
i
hxt
, i
:
Assumption 1. For any i, 1
[, ]
vv
x
xx
, v
and 0t, the membership functions
((,))
i
hxt
, i satisfy ,min ,max
((,)) [ , ]
ivv
hxt
, where ,min ,max
01
vv
.
Obviously, the property assigned for the membership functions ( ( , ))
i
hxt
, i
by
Assumption 1 is rational and feasible, as the membership functions ( ( , ))
i
hxt
, i are chosen
by the control engineers and thus known in general. This property can be achieved by properly
building the fuzzy plant model for the nonlinear plant. For example, the operating domain S,
which is a bit larger than the one , i.e., S where n
S, is employed in the fuzzy
modeling process. ( ( , ))
i
hxt
, i will lie in the range of 0 and 1 but not reach the boundary
values due to the fact that the nonlinear plant operates in the domain which is slightly smaller
than the operating domain S considered in the fuzzy modeling process. In Section 4, we will
illustrate the detailed fuzzy modeling process for the Fisher equation, where the membership
functions ( ( , ))
i
hxt
, i satisfy Assumption 1. Lemma 2 gives a new parameterized LMI
technique based on Assumption 1 and the parameterized LMI technique reported in [24].
Lemma 2. Under Assumption 1, consider nn
parameterized matrices
,
11
((,)) (( ,))
rr
ijvvij
ij
hxthxt
, 1
[, ]
vv
x
xx
, 1
(, )
vvv
x
xx
, v
, 0t. Given the
constants v
, v satisfying 1
,min ,max
0vvv
, v
, the parameterized matrix
inequalities ,
11
((,)) (( ,)) 0
rr
ijvvij
ij
hxthxt
, v
are satisfied for any 1
[, ]
vv
x
xx
,
1
(, )
vvv
x
xx
, 0t, if there exist symmetric matrices ,
nn
vj
satisfying the following LMIs:
- 17 -
,,
0
vij v j
, v, ,ij (47)
,
,,,
0, ,
11 0, , , ,
12
vii
vii vij v ji
vi
vijij
r
(48)
where 1
,, ,,
0.5 (1 )( )
vij vij v v vi v j
, v
, ,ij
.
Proof. See Appendix.
Remark 3. The constraints (46) are derived from Assumption 1 for the membership functions
((,))
i
hxt
, i to obtain the less conservative results, i.e., the LMIs (47) and (48). These
constraints are inspired by the result reported in [10] to deal with fuzzy sampled-data control design
of nonlinear systems with time delay, in [43] to address stability analysis of fuzzy control systems
subject to uncertain grades of membership functions, and in [45] to discuss fuzzy guaranteed cost
sampled-data control design for nonlinear systems coupled with a scalar reaction-diffusion process.
Considering the definition of ( )
x
G and using (45), we have
,
11
0(,) () () ((,)) (,) (,)
Lmr
TTT
iv v vviv
vi
x
t xdx t h xt xt xt
yPGu y Pgky
. (49)
Similar to (35), substituting (49) into (15) and considering 12 1
0mm
x
xxxL
, we
obtain
1 1
111
() (,)[ ] (,) ( ( ,)) ( ,)[ ] (,)
v v
v v
xx
mmr
T T
xx i i
vvi
xx
Vt xt xtdx h xt xt xtdx
yPy yPAy
,
11
(( ,)) ( ,)[ ]( ,)
mr TT
iv v vvi v
vi
hxt xt xt
yPgky
. (50)
Henceforth, based on Lemma 2, the following theorem is derived in terms of LMIs:
Theorem 3. Consider the SLPPDE system (1)-(2) and the T-S fuzzy PDE model (7) with
Assumption 1. Given the scalars v
, v
satisfying 1
,min ,max
0vvv
, v, if there exist
a matrix 0
Q and vectors ,
n
vjz, v
,
j
and symmetric matrices 22
,
nn
vj
,
v, j such that the following LMIs are fulfilled:
,,
0
vij v j
, v, ,ij, (51)
,
,,,
0, ,
11 0, , , ,
12
vii
vii v ji vij
vi
vijij
r
(52)
where 1
,, ,,
0.5 (1 )( )
vij vij v v vi v j
, and
21
,21 21 1
1,
[ ] 0.25 [ ]
0.25 [ ] 0.25 [ ] ( ) [ ]
iv
vij T
vvvvvvj
xx
AQ Q
QQgz
, (53)
then there exists a fuzzy controller (45) applied to the SLPPDE system (1)-(2) such that the
- 18 -
resulting closed-loop system is exponentially stable in the sense of 2
. In this case, the control
gain parameters ,vi
k, v, i are given by (20).
Proof. Assume that the membership functions ( ( , ))
i
hxt
, i
satisfy Assumption 1 and LMIs
(51) and (52) are fulfilled. Based on Lemma 2, if the membership functions ( ( , ))
i
hxt
, i
satisfy Assumption 1 and the LMIs (51) and (52) are satisfied, we have the following inequalities:
,
11
((,)) (( ,)) 0
rr
ijvvij
ij
hxthxt
, 1
[, ]
vv
x
xx
, v
, 0t, (54)
where the definition of matrices ,vij
, v
, ,ij
is given in (53). Using (8) and (54), we get
the LMI (23). From (8), (23) and (41), the expression (50) is written as
1
21
1
( ) 0.25 ( , )[ ] ( , )
v
v
x
mT
vv v
vx
Vt xt xtdx
yPy
1
11
( ( ,)) ( ,)[ ] ( ,)
v
v
x
mr T
ii
vi
xhxt xt xtdx
yPAy
,
11
(( ,)) ( ,)[ ]( ,)
mr TT
iv v vvi v
vi
hxt xt xt
yPgky
1
,
11 1 ((,)) (( ,)) (,) (,)
v
v
x
mrr T
ijvvvijv
vi j
xh xt h x t xt xtdx
yy
(55)
where (,) [ (,) ( ,)]
TTT
vv
x
txtxt
yyy and
21
,21 21 1
1,
[ ] 0.25 [ ]
0.25 [ ] 0.25 [ ] ( ) [ ]
iv
vij T
vvvvvvj
xx
PA P
PPPgk
.
By pre- and post-multiplying the inequality (54) with the block-diagonal matrix and using (21),
we get
,
11
((,)) (( ,)) 0
rr
ijvvij
ij
hxthxt
, 1
[, ]
vv
x
xx
, v
, 0t. (56)
Similar to the proof of Theorems 1 and 2, it is easily shown from (56) that the closed-loop system
resulting from the SLPPDE system (1)-(2) plus the fuzzy controller (45) is exponentially stable in
the sense of 2
. The proof is complete. □
Based on the Lyapunov technique, integration by parts, Assumption 1, Lemma 1 and Lemma 2,
the design problem of the fuzzy controller (45) for the SLPPDE system (1)-(2) is formulated as the
feasibility problem of the LMIs (51) and (52). The corresponding control gains ,vi
k, v, i
are constructed as (20) via the feasible solutions to these LMIs, which can be directly verified and
solved via the feasp solver [30].
Before solving the LMIs (51) and (52) via the feasp solver, we need to set the value of the
scalars 0
v
, v. Assumption 1 in theory guarantees the existence of these scalars. Hence,
the value of these scalars is set on the basis of the constraints 1
,min ,max
0vvv
, v. On the
- 19 -
other hand, these scalars depend on the number and location of sensors, as the intervals 1
[, ]
vv
x
x
,
v are obtained by dividing the spatial domain [0, ]
L
according to the location of
m-pointwise actuators. Of course, as the number of sensors increases, the complexity of control
design problem addressed in Theorem 3 is high. But only a few sensors are in general required to
implement the fuzzy controller (45) from the control design in Theorem 3. Therefore, the
complexity of the control design in Theorem 3 can be tolerated in practice. According to the
aforementioned analysis, the design procedure for fuzzy controller (45) in Theorem 3 is
summarized as follows:
Design procedure for fuzzy controller (45):
Step 1. Construct the T-S fuzzy PDE model (7) for the SLPPDE model (1).
Step 2. Determine the spatial intervals 1
[, ]
vv
x
x
, v
according to the location of actuators
and find the upper bound and lower bounded of the membership functions ( ( , ))
i
hxt
, i
, i.e.,
the value of ,minv
and ,maxv
, v, according to Assumption 1.
Step 3. Set the value of v
according to the constraints 1
,min ,max
0vvv
, v.
Step 4. Solve the LMIs (51) and (52) to get the control gains of the fuzzy controller (45).
In fact, the above design procedure is a two-step one as the first step including Steps 1-3 is to set the
value of v
, v and the second one (i.e., Step 4) is to solve the LMIs (51) and (52) to get the
control gains.
Remark 4. Note that for the case when Assumption 1 is not fulfilled or removed, Lemma 2
established under Assumption 1 cannot be employed to derive Theorem 3. Following to lines of
(54)-(56) in the proof of Theorem 3, it can be shown that the design of fuzzy controller (45) for the
SLPPDE system (1) and (2) is formulated as the feasibility problem of following LMIs
,0
vij
, v, ,ij. (57)
Clearly, the LMIs (57) are more conservative than the LMIs (51) and (52) as their feasible region is
smaller than that of the LMIs (51) and (52). On the other hand, due to the fact that the feasible
solutions ,
{0, , , }
n
vj vjQz to the LMIs (57) are not unique, this conservativeness
may result in the same ,vj
z
, v, j
, i.e., ,1 , 2 ,vv vr
z
zz, v. Then, the fuzzy
controller (45) is reduced to a linear one, which will be verified in Section 4. To reduce this
conservativeness, an operational approach is to verify the constraints (46) for some given constants
v
, v through numerical simulation. Then, the same result (i.e., the LMIs (51) and (52)) can
be derived.
Remark 5. An obvious difference among fuzzy control design in Theorems 1-3 is that the fuzzy
- 20 -
control law (16) is designed for the entire spatial domain [0, ]L, whereas the fuzzy control laws (33)
and (45) are treated individually for each part of spatial domain 1
[, ]
vv
x
x
. The main difference
between the controller (16) and the one (33) lies in that each actuator of the controller (16) requires
the state information over the entire spatial domain while each actuator of the fuzzy controller (33)
requires the state information over its local area of the spatial domain. From the implementation
point of view, the fuzzy controller (16) (or (33)) requires a large number of sensors continuously
distributed over the entire spatial domain. Different from the fuzzy controller (16) (or (33)), the
fuzzy controller (45) is easily implemented since it requires only a few collocated pointwise sensors
to obtain the state information at the control actuators’ positions. On the other hand, from the
computation point of view, both the basic arithmetic operations and the numerical integration
computation are carried out for the controller (16) or (33) due to its integration form with respect to
the spatial variable. However, the fuzzy controller (45) only conducts the basic arithmetic
operations. Therefore, it can be concluded that the controller (45) is more feasible than the
controller (16) or (33) from the implementation and computation point of views.
Remark 6. Even though the proposed design methods (i.e., Theorems 1-3) are developed for the
case of homogeneous Dirichlet boundary conditions (2), the same design methods can be directly
obtained in a straightforward manner for the case of homogenous Neumann boundary conditions
(i.e., 0
(,) (,) 0
xx
xxL
xt xt
yy ), the case of homogenous mixed Dirichlet-Neumann boundary
conditions (i.e., (0, ) ( , ) 0
xxL
txt
yy ), the case of homogenous mixed Neumann-Dirichlet
boundary conditions (i.e., 0
(,) (,) 0
xx
xt Lt
yy), or the case of homogenous periodic boundary
conditions (i.e., (0, ) ( , )tLt
yy
, 0
(,) (,)
xx
x
xL
xt xt
yy). This is because the equation (14)
(derived by considering the Dirichlet boundary conditions (2)) used in the development of the
proposed design methods is also fulfilled for above four cases of boundary conditions.
4. Simulation study
In this section, to illustrate the merit and effectiveness of the proposed design methods, we
consider two SLPPDE systems: the first one is a Fisher equation and the second one is a FHN
equation, respectively. We utilize the finite difference method and the approximated Dirac delta
function ( )
to get the approximate solution of the SLPPDE systems. Set sample
t and sample
x
denote the sampling steps in time and space, respectively, and satisfy 12
sample max sample
0.5 ( )tx
. The
partial derive terms ( , )
t
x
ty and ( , )
xx
x
ty are respectively approximated by following forward
difference and central difference, i.e.,
- 21 -
1
sample
(,) (, )
(,) ph ph
t
x
txt
xt t
yy
y and 11
2
sample
(,)2(,)(,)
(,) ph ph ph
xx
x
txtxt
xt x
yyy
y,
where (,)
ph
x
ty stands for the discretization value of (,)
x
ty at point samplep
xx px and
sampleh
tt ht . As we all know, the aforementioned differences are stable if 1
sample max
0.5 ( )t
2
sample
x [46]. For brevity, we choose sample
x
to avoid the numerical problem.
Example 1. We consider the feedback control problem of a Fisher equation under the pointwise
state feedback control architecture. The Fisher equation is of the following form:
0
(,) (,) (,)(1 (,)) ()(),
(0, ) ( , ) 0, ( , 0) ( ),
T
txxy
y
xt y xt yxt yxt x t
yt yt yx yx
gu
(58)
where (,)yxt is the state variable, 0
y
is a known parameter. 0()
y
x is the initial value
of the system. 2
()xg and 12
() [ () ()]
T
tututu are the control distribution function and the
manipulated inputs, respectively. The control distribution function ( )
x
g
is of the form
12
() [ () ()]
T
x
gx gxg, where 1() ( 0.2 )gx x
and 2() ( 0.6 )gx x
. The Fisher equation
in (58) arises in heat and mass transfer, combustion theory, biology, and ecology. For example, it
describes the mass transfer in a two component medium at rest with a volume chemical reaction of
quasi-first order. The kinetic function (,)(1 (,))
yyxt yxt
also models an autocatalytic chain
reaction in combustion theory.
Fig. 2 Open-loop profiles of evolution of (,)yxt
Set 1.2
y
, 0() 0.1sin()
y
xx
, sample 0.05x
, and sample 0.001t
. Fig. 2 gives the open-loop
profile of evolution of (,)yxt. What is apparent from Fig. 2 is that the operating steady state
(, ) 0yt
is unstable (the open-loop state starting from the initial condition close to the steady state
(, ) 0yt
moves to another stable steady state). It is clear from Fig. 2 that ( , ) [0, 0.21]yt , 0t.
x t
y
- 22 -
It has also been verified through numerical simulation that ( , ) [0, 0.21]yt
if 0() [0,0.21]y for
any 0t. To ensure Assumption 1, the operating domain for the T-S fuzzy PDE modeling is set to
be min max
(, ) [ , ]yt
, where min 0
and max 0.21
. That is, the domain and the
operating domain S for the system (58) are chosen as
(, ) 0 (, ) 0.21yt yt and
min max
(, ) (, )yt yt
S, respectively.
Let ( , ) ( , )
x
tyxt
, the nonlinear term 2(,)
y
xt in (58) is rewritten as
2(,) (,)(,)
y
xt xtyxt
.
Since min max
(,) [ , ]yxt
, calculating the minimum and maximum values of ( , )
x
t
yields,
min max
min
(,)[ , ]
min ( , )
yxt xt
and
min max
max
(,)[ , ]
max ( , )
yxt xt
.
Using above minimum and maximum values, (,)
x
t
can be represented by
1max2 min
(,) ((,)) ((,))xt h xt h xt
(59)
where 1((,))hxt
, 2((,)) [0,1]hxt
and
12
((,)) ((,)) 1hxthxt
. (60)
By solving equations (59) and (60), one can obtain the following membership functions:
1
1 min max min
((,)) ((,) )( )hxt xt
and 21
((,)) 1 ((,))hxt hxt
. (61)
Define the fuzzy sets as “Big” and “Small”. Then, the Fisher equation in (58) can be exactly
represented by the following T-S fuzzy PDE model of two rules in the domain S:
Plant rule 1:
IF (,)
x
t
is “Big”
THEN max
(,) ( ,) (1 ) ( ,) () ()
T
txxy
yxt y xt yxt x t
g
u
Plant rule 2:
IF (,)
x
t
is “Small”
THEN min
(,) (,) (1 )(,) () ()
T
txxy
yxt y xt yxt x t
g
u.
Then, the overall fuzzy PDE model is given by
2
1
(,) ( ,) ( (,)) ( ,) ( ) ()
T
txx i i
i
yxt y xt h xt yxt x t
g
u (62)
where 1max
(1 )
y
and 2min
(1 )
y
. It can be concluded from the derivation of the
fuzzy PDE model (62) that 2
1
(,)(1 (,)) ((,)) (,)
yii
i
yxt yxt h xt yxt
for any (, )ytS. That
is, the PDE in (58) is equivalent to the fuzzy PDE model (62) for any (, )yt
S, which verifies the
equality (9) for any (, )yt
S.
We show the control performance of the fuzzy control design procedure in Theorem 3 for the
- 23 -
nonlinear system (58). Based on the T-S fuzzy PDE model (62), for the nonlinear system (58), the
fuzzy controller (45) is presented as
2
11,
1
() ( (0.2 ,)) (0.2 , )
ii
i
ut h tky t
and 2
22,
1
() ( (0.6 ,)) (0.6 ,)
ii
i
ut h tky t
, (63)
where 1, i
k and 2,i
k
, {1, 2}i are control gains to be determined.
Since
S, the membership functions ( ( , ))
i
hxt
, {1, 2}i
given in (61) satisfy
Assumption 1 with
min max
max min
min , 0.2 1
,min ()
v
and
min max
max min
max 0.21 ,
,max ()
v
, {1, 2}v
.
Set 10x, 20.4x
, 3
x
, min 0.2
and max 0.6
. We thus get 2
10.04
,
2
20.16
, ,min 0.25
v
, and ,max 0.75
v
, {1, 2}v
. According to the constraints
1
,min ,max
0vvv
, {1, 2}v, we choose 10.08
and 20.01
. Solving LMIs (51) and (52)
and using (20), the control gains of (63) are given as follows:
1,1 18.4429k , 1,1 18.6103k
, 2,1 60.8575k
, and 2,2 63.6564k
. (64)
Fig. 3 Closed-loop profiles of evolution of (,)yxt
00.5 11.5
-6
-4
-2
0
1
Fig. 4 Control input ()tu
Applying the fuzzy controller (63) with the control gains given in (64) to the semi-linear PDE
system (58), Fig. 3 provides the closed-loop profile of evolution of (,)yxt. It is observed from Fig.
3 that the fuzzy controller (63) with the control gains given in (64) can stabilize the system (58).
x t
y
t (sec.)
u1 (t)
u2 (t)
u(t)
- 24 -
The control input ( )tu is indicated in Fig. 4.
Example 2. Consider the feedback control of the FHN equation, which is a widely used model of
wavy behavior in excitable media in biology [47] and chemistry [48]. The FHN equation with
pointwise control input can be written as the following form:
3
1, 1, 1 2 1
(,) (,) (,) (,) (,) ()()
T
txx
y xt y xt y xt y xt y xt x t
g
u,
2, 2, 1 2
( , ) ( , ) 0.45 ( , ) 0.1 ( , ) 0.2
txx
yxt y xt yxt yxt , (65)
subject to the homogeneous Neumann boundary conditions
1, 2,
00
(,) (,) 0
xx
xx
yxt yxt
, 1, 2,
11
(,) (,) 0
xx
xx
yxt yxt
, (66)
and the initial conditions
11,0
(,0) ()yx y x, 22,0
(,0) ()yx y x (67)
where (,)
i
yxt, {1,2}i are the state variables that are usually termed as the concentrations of
“activator” and “inhibitor”, 2
12
() [ () ()]tutut
u is a control input. 12
() [ () ()]
T
x
gx gxg
is the control distribution function describing the distribution of two pointwise control actuators, in
which 11
() ( )gx x x
and 22
() 2( )gx xx
. 1, 0 ()
y
x and 2,0 ()
y
x are the initial values.
Let ,()
di
yx, {1, 2 }i be the steady states of the unforced PDE system of (65)-(67).
Clearly, the steady states ,1 ()
d
yx and ,2 ()
d
yx satisfy the following algebraic equations:
3
,1, ,1 ,2 ,1
,2, ,1 ,2
() () () () 0
( ) 0.45 ( ) 0.1 ( ) 0.2 0
dxx d d d
dxx d d
y xyxyxyx
yx yx yx
(68)
subject to the following boundary conditions:
,1, , 2,
00
() () 0
dx dx
xx
yx yx
and ,1, , 2 ,
11
() () 0
dx dx
xx
yx yx
. (69)
Notice that the analytic solutions to algebraic equation (68) subject to (69) are difficultly derived.
An alternative option is to get numerical solutions via the finite difference method.
Set ,
(,) (,) ()
iidi
yxt yxt y x, {1,2}i be the steady error states. The spatiotemporal steady
error dynamics can be governed by the following parabolic PDE system:
3
1, 1, 1 2 1
(,) (,) (,) (,) (,) ()()
T
txx
yxt y xt yxt yxt yxt x t
g
u, (70)
2, 2, 1 2
(,) (,) 0.45 (,) 0.1 (,)
txx
yxt y xt yxt yxt , (71)
subject to the homogeneous Neumann boundary conditions
1, 2,
00
(,) (,) 0
xx
xx
yxt yxt
and 1, 2 ,
11
(,) (,) 0
xx
xx
yxt yxt
, (72)
and the initial conditions
11,0
(,0) ()
y
xyx and 22,0
(,0) ()
y
xyx (73)
- 25 -
where 1, 0 1,0 ,1
() () ()
d
yx yxyx and 2,0 2,0 ,2
() () ()
d
yx yxyx.
Set sample 0.05x and sample 0.001t. It will be verified by numerical simulation that the
steady states 1(, ) 0yt and 2(, ) 0yt of the PDE system (70)-(73) are unstable ones. The initial
conditions (73) are assumed to be 1, 0 () 0.6 0.5cos( )
y
xx
and 2,0 ( ) 0.1cos( )
y
xx
. Under
these initial values, Fig. 5 shows the open-loop profiles of evolution of 1(,)
y
xt and 2(,)
y
xt . It is
observed from Fig. 5 that the open-loop state starting from the initial conditions close to the steady
state move to the unstable solution of the system. Hence, the steady states 1(, ) 0yt and
2(, ) 0yt of the PDE system (70)-(73) are unstable.
Let 12
(,) [ (,) (,)]
T
x
tyxtyxty, 111
() ( )
x
xx
gg and 222
() ( )
x
xx
gg , where
1[1 0]T
g and 2[2 0]T
g, the system (70)-(73) can be written as the form (1), where
I
,
3
121 1 2
( ( ,)) [ (,) ( ,) (,) 0.45 (,) 0.1 (,)]
T
x
t y xt y xt y xt y xt y xt fy , and 12
() [ () ()]
x
xxGgg.
The boundary conditions (72) are rewritten as 0
(,)
xx
xt
y (,) 0
xxL
xt
y. According to Remark
6, fuzzy control design methods in Theorems 1-3 can be directly applied to the system (70)-(73).
Fig. 5 Open-loop profiles of evolution of 1(,)yxt and 2(,)yxt
For the SLPPDE system (70)-(73), a T-S fuzzy PDE modeling approach has been recently
proposed in Section IV [24]. Based on the result reported in Section IV [24], the nonlinear
dynamics of the PDE system (70)-(73) can be described by the following T-S fuzzy PDE model
subject to the boundary conditions 0
(,) (,) 0
xx
xxL
xt xt
yy :
Plant rule 1:
IF (,)
x
t
is “Big”
x t
y1
x t
y2
- 26 -
THEN 1
(,) (,) (,) ()()
txx
x
txt xtxt
yy
A
y
Gu
Plant rule 2:
IF (,)
x
t
is “Small”
THEN 2
(,) (,) (,) ()()
txx
x
txt xtxt
yy
A
y
Gu
where 2
1
(,) (,)
x
tyxt
, 1
11
0.45 0.1
A, and 2
11
0.45 0.1
A.
The overall fuzzy model is written as follows:
2
1
( ,) (,) ( (,)) ( ,) ( ) ()
txx i i
i
x
t xt h xt xt x t
yy
A
y
Gu, (74)
where
1
1((,)) (,)hxt xt
and 1
2((,)) 1 (,)hxt xt
, (75)
with
1
2
1
(,)
max ( , )
yxt
y
xt
. It is clear from Fig. 5 that 1( , ) [ 1.2,1.2]yt
and 2(, ) [ 1,1]yt , 0t. It
has also been verified through numerical simulation that 1( , ) [ 1.2,1.2]yt
and 2(, ) [ 1,1]yt if
1,0 ( ) [ 1.2,1.2]y and 2,0 () [ 1,1]y for any 0t. To improve the robustness of the fuzzy
controller, set 1( , ) [ 1.5,1.5]yt for the construction of the fuzzy PDE model (74). Hence, we get
2.25
. That is, the domain and the operating domain S for the system (70)-(73) are
chosen as
2
12
(, ) 1.2 (, ) 1.2, 1 (, ) 1tytyt y and
2
(, ) 1.5t yS
1(, ) 1.5yt , respectively. It can also be verified that the fuzzy model (74) is equivalent to the PDE
model (70) and (71) in the operating domain S, which verifies the equality (9) for any (, )t
yS.
4.1 Full state feedback
Set 10.5x and 20.8x, we thus have 10.25
and 20.64
. In this case, the fuzzy full
state feedback controller (16) is written as
12
11,
1
0
() ( ( ,)) ( ,)
T
ii
i
ut h xt xtdx
ky and 12
22,
1
0
() ( ( ,)) ( ,)
T
ii
i
ut h xt xtdx
ky . (76)
Solving the LMIs (19) yields the following feasible solutions:
0 1369 0 0024
0 0024 0 1389
..
..
Q= , 11
0 1504
0 0247
,
.
.
z= , 12
0 2736
0 0265
,
.
.
z= ,
21 [ 0 0752 0 0123]T
,..z and 22 [ 0 1368 0 0132]T
,..z.
By (20), we can obtain the following control gain parameters:
11
1 0962
0 1589
,
.
.
k= , 12
1 9962
0 1563
,
.
.
k= , 21
0 5481
0 0794
,
.
.
k= , and 22
0 9981
0 0781
,
.
.
k= .
- 27 -
Applying the fuzzy full state feedback controller (76) with above control gains, the closed-loop
profiles of evolution of the system (70)-(73) are shown in Fig. 6. It is observed from Fig. 6 that the
suggested fuzzy controller can stabilize the system. The corresponding control input ( )tu is
indicated in Fig. 7.
Fig. 6 Closed-loop profiles of evolution of 1(,)yxt and 2(,)yxt using the full state feedback controller
0 5 10 15
-1.2
-0.8
-0.4
0
0.2
Fig. 7 Trajectory of the full state feedback controller ()tu
0
510 15
0
0.25
0.5
0.75
Fuzzy controller (76)
Distributed fuzzy controller [24]
Fig. 8 Closed-loop trajectory of 2
(, )ty using the fuzzy controller (76) and fuzzy controller [24]
Moreover, a comparison between Theorem 1 and the control design in [24] is provided to
illustrate the benefit of the proposed design method. Fig. 8 gives the closed-loop trajectory of
x t
y1
x t
y2
t (sec.)
2
(, )ty
t (sec.)
u1 (t)
u2 (t)
u(t)
- 28 -
2
(, )ty by using the distributed fuzzy controller in [24] and the fuzzy controller (76). It is clear
from Fig. 8 that the fuzzy controller in [24] can provide a better performance in comparison to the
fuzzy controller (76). However, fuzzy controller (76) is easily implemented since only two actuators
are required to be active at the points 0.5 and 0.8 of the spatial domain [0,1] , respectively.
4.2 Piecewise state feedback
Set 10.3x, 20.8x, 10x, 20.6x
and 31x
, we thus have 10.09
and
20.04
. In this case, the fuzzy controller (33) can be written as
0.6 2
11,
1
0
() ( ( ,)) ( ,)
T
ii
i
ut h xt xtdx
ky and 12
22,
1
0.6
() ( ( ,)) ( ,)
T
ii
i
ut h xt xtdx
ky . (77)
Solving the LMIs (36) yields the following feasible solutions:
0 0573 0 0057
0 0057 0 0491
..
..
Q= , 11
1 3418
0 0078
,
.
.
z= , 12
1 4247
0 0192
,
.
.
z= ,
21 [ 0 7579 0 0065]T
,..z and 22 [ 0 8667 0 0103]T
,..z.
By (20), we can obtain the following control gain parameters:
11
23 6688
2 5801
,
.
.
k= , 12
25 1082
2 5138
,
.
.
k= , 21
13 3638
1 4131
,
.
.
k= , and 22
15 2771
1 5585
,
.
.
k= .
Applying the fuzzy controller (77) with above control gains, the closed-loop profiles of evolution of
the PDE system (70)-(73) are given in Fig. 9. It is obvious from Fig. 9 that the suggested fuzzy
controller can stabilize the system. The corresponding control input ( )tu is indicated in Fig. 10.
Fig. 9 Closed-loop profiles of evolution of 1(,)yxt and 2(,)yxt using the piecewise state feedback controller
x t
y1
x t
y2
- 29 -
00.5 11.5 2
-15
-10
-5
0
3
Fig. 10 Trajectory of the piecewise state feedback controller ()tu
4.3 Collocated pointwise state feedback
Set 10.25x, 20.8x
, 10x, 20.5x
and 31x
, we thus have 10.0625
and
20.09
. In this case, the fuzzy controller (45) can be written as
2
11,
1
( ) ( (0.25, )) (0.25, )
T
ii
i
ut h t t
ky and 2
22,
1
() ( (0.8,)) (0.8, )
T
ii
i
ut h t t
ky . (78)
Solving the LMIs (57), we get the following feasible solutions:
0 0216 0 0007
0 0007 0 0188
..
..
Q= , 11 1 2
0 6652
0 0022
,,
.
.
z=z= , and 21 2 2
0 3759
00025
,,
.
.
z=z= .
By (20), we get the control gains:
11 12 [ 30 8760 1 0821]T
,, .. kk , and 21 2 2 [ 17 4467 0 5464]T
,, ..k=k= .
The fuzzy controller (78) with above control gains is reduced to a linear one. Hence, the LMIs (57)
are conservative.
Note that the membership functions ( ( , ))
i
hxt
, {1, 2}i
given in (75) do not satisfy
Assumption 1 since the lower bound of 1((,))hxt
is zero. To reduce this conservativeness,
according to Remark 4, an operational approach is to verify the constraints
((,)) (( ,)) 0
iviv
hxt hxt
, 1
[, ]
vv
x
xx
, 0t, , {1, 2}iv
, where 10.08
and 20.01
.
For brevity, define ( , )
i
dxt
1
2
( ( , )) 0.08 ( ( , )) [0, 0.5]
(,) ( ( , )) 0.01 ( ( , )) [0.5,1]
ii
iii
hxt hxt x
dxt hxt hxt x
, {1, 2}i
.
Fig. 11 provides the open-loop profiles of evolution of ( , )
i
dxt, {1, 2}i
. It is easily observed from
Fig. 11 that ( , )
i
dxt, {1, 2}i
, i.e., the constraints ((,)) (( ,) 0
iviv
hxt hxt
, 1
[, ]
vv
x
xx
,
0t, ,{1,2}vi are fulfilled for the case of 10.08
, 20.01
, 10.25x
, 20.8x, 10x
,
20.5x and 31x.
Hence, the design problem of the fuzzy controller (78) can also be formulated as the feasibility
t (sec.)
u1 (t)
u2 (t)
u(t)
- 30 -
problem of the LMIs (51) and (52). Solving the LMIs (51) and (52) yields the following feasible
solutions:
0 1696 0 0868
0 0868 0 1859
..
..
Q= , 11
3 8561
0 0695
,
.
.
z= , 12
3 8280
0 0591
,
.
.
z= ,
21 [ 2 1813 0 0248]T
,..z, and 22 [ 2 1802 0 0242]T
,..z.
By (20), we can obtain the following control gain parameters:
11
29 6187
13 4512
,
.
.
k= , 12
29 4390
13 4235
,
.
.
k= , 21
16 8071
7 7119
,
.
.
k= , and 22
16 8009
7 7118
,
.
.
k= .
Fig. 11 Open-loop profiles of evolution of (,)
j
dxt, {1, 2}j
Fig. 12 Closed-loop profiles of evolution of 1(,)yxt and 2(,)yxt using the collocated pointwise state feedback
controller
x t
d1
x t
d2
x t
y1
x t
y2
- 31 -
00.3 0.6 0.9
-32
-24
-16
-8
0
6
Fig. 13 Trajectory of the collocated pointwise state feedback controller ()tu
Fig. 14 Closed-loop profiles of evolution of (,)
j
dxt, {1, 2}j
Applying the fuzzy controller (78) with above control gains, the closed-loop profiles of evolution of
the PDE system (70)-(73) are shown in Fig. 12. It is observed from Fig. 12 that the suggested fuzzy
controller can stabilize the system. The control input ()tu is indicated in Fig. 13. Fig. 14 shows
the closed-loop profiles of evolution of ( , )
i
dxt, {1,2}i
, which implies that the constraints
((,)) (( ,) 0
iviv
hxt hxt
, 1
[, ]
vv
x
xx
, 0t, , {1, 2}vi
are also satisfied for the case of
10.08
, 20.01
, 10.25x
, 20.8x, 10x
, 20.5x
and 31x
.
5. Conclusions
This paper has investigated the problem of fuzzy control design for semi-linear parabolic PDE
systems under the pointwise control form based on the T-S fuzzy PDE model. The main
contribution of this paper is that LMI-based fuzzy pointwise control design methods are developed
for the SLPPDE system. Three cases of state feedback, e.g., full state feedback, piecewise state
feedback, and collocated pointwise state feedback, are considered, respectively. Since only few
x t
d1
x t
d2
t (sec.)
u1 (t)
u2 (t)
u(t)
- 32 -
actuators are active at specified points of the spatial domain, the suggested controllers are easily
implemented. Using the Lyapunov technique and the vector-valued Wirtinger’s inequality, the
resulting closed-loop system is shown to be exponentially stable, and the corresponding sufficient
conditions are presented in terms of LMIs. For the collocated pointwise state feedback case, a
parameterized LMI is introduced to enhance stabilization ability of the fuzzy controller. Finally,
simulation results of pointwise control of the Fisher equation and the FHN equation indicate the
merit and effectiveness of the proposed design methods.
On the other hand, although the extension is very interesting, it is a pity that the design
methods proposed in this paper cannot be extended to nonlinear DPSs modeled by other PDEs, e.g.,
hyperbolic PDE, wave PDE or elliptic PDE. The reason is that the design methods utilize the
property given in the expression (14) of the parabolic spatial differential operator ( , )
xx
x
ty and
the technique of the vector-valued Wirtinger’s inequality (i.e., Lemma 1) which are very important
for the development of the control design, whereas the hyperbolic PDE, wave PDE or elliptic PDE
are not of the property given in the expression (14). The implementation of the control design in
Theorem 3 requires only a few collocated pointwise actuator/sensor pairs, i.e., the number and
location of sensors are the same as these of actuators. The suggested control design is economical
and feasible from the implementation and computation point of views and can be used to solve the
practical problems. Some experiments for practical problems will be provided in the future research
activity to illustrate the practical applicability of the design method proposed in Theorem 3.
Moreover, the proposed design methods are only applicable for the SLPPDE system in
one-dimensional spatial domain. In order to extend them for the two-dimensional spatial domain
case or even the n-dimensional spatial domain case, we need to modify the vector-valued
Wirtinger’s inequality (i.e., Lemma 1) for the case of two-dimensional spatial domain or
n-dimensional spatial domain (i.e., the vector-valued Poincare inequality). However, to the best of
authors’ knowledge, the modification of the vector-valued Poincare inequality is very difficult.
Therefore, it remains a challenge to address exponential pointwise stabilization of PDE systems in
two-dimensional spatial domain or even the n-dimensional spatial domain, which we leave for
future study.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under Grants
61403026 and 61473011, and in part by the Fundamental Research Fund for the Central University
under Grant FRF-TP-15-059A2. The authors also gratefully acknowledge the helpful comments and
suggestions of the Associate Editor and anonymous reviewers, which have improved the
presentation of this paper.
- 33 -
Appendix: Proof of Lemma 2
Proof. From Assumption 1, one can find scalars v
satisfying 1
,min ,max
0vvv
such that the
membership functions ( ( , ))
i
hxt
satisfy the constraints (46). Using (46) and considering the
property 111
((,)) ((,)) (( ,)) 1
rrr
jijv
jij
hxt hxthxt
, 1
[, ]
vv
x
xx
, 1
(, )
vvv
x
xx
,
v, 0t, we have for any 1
[, ]
vv
x
xx
, 1
(, )
vvv
x
xx
, v
, 0t
,
11
((,)) (( ,))
rr
ijvvij
ij
hxthxt
,, ,
11 11
[ ( ( , )) ( ( , ))] ( ( , ))[ ] ( ( , ))
rr rr
ivivjvvijvjvijvvij
ij ij
h xt h xt h xt h xt
, (A1)
where ( ( ,)) ( ( ,)) ( ( ,))
ij v i v j v
hxthxthxt
.
Obviously, the following inequalities hold:
,,
11
[ ( ( , )) ( ( , ))] ( ( , ))[ ] 0
rr
ivivjvvijvj
ij
hxt hxthxt
,
1
[, ]
vv
x
xx
, 1
(, )
vvv
x
xx
, v
, 0t (A2)
if the inequalities (46) and (47) are fulfilled. Applying Theorem 2.2 in [9], on the other hand, we
can get that the inequalities ,
11
(( ,)) 0
rr
ij v v ij
ij
hxt
, 1
(, )
vvv
x
xx
, v, 0t are
satisfied if the LMIs (48) hold.
Therefore, it can be derived from (A1) and (A2) that if LMIs (47) and (48) are satisfied, the
inequalities ,
11
((,)) (( ,)) 0
rr
ijvvij
ij
hxthxt
, v
are then fulfilled for any
1
[, ]
vv
x
xx
, 1
(, )
vvv
x
xx
, v, 0t. □
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