This paper extends the framework of Lyapunov–Krasovskii functional to address the problem of exponential stabilization for a class of linearly distributed parameter systems (DPSs) with continuous differentiable time-varying delay and a spatiotemporal control input, where the system model is described by parabolic partial differential-difference equations (PDdEs) subject to homogeneous Neumann or Dirichlet boundary conditions. By constructing an appropriate Lyapunov–Krasovskii functional candidate and using some inequality techniques (e.g., spatial integral form of Jensen's inequalities and vector-valued Wirtinger's inequalities), some delay-dependent exponential stabilization conditions are derived, and presented in terms of standard linear matrix inequalities (LMIs). These stabilization conditions are applicable to both slow-varying and fast-varying time delay cases. The detailed and rigorous proof of the closed-loop exponential stability is also provided in this paper. Moreover, the main results of this paper are reduced to the constant time delay case and extended to the stochastic time-varying delay case, and also extended to address the problem of exponential stabilization for linear parabolic PDdE systems with a temporal control input. The numerical simulation results of two examples show the effectiveness and merit of the main results.

References

References
1.
Nilsson
,
J.
,
1998
, “Real-Time Control Systems With Delays,” Ph.D. dissertation, Lund Institute of Technology, Lund, Sweden.
2.
Gu
,
K.
,
Kharitonov
,
V. L.
, and
Chen
,
J.
,
2003
,
Stability of Time-Delay Systems
,
Birkhauser
,
Boston, MA
.
3.
Zhong
,
Q.-C.
,
2006
,
Robust Control of Time-Delay Systems
,
Springer-Verlag
,
London
.
4.
Hespanha
,
J. P.
,
Naghshtabrizi
,
P.
, and
Xu
,
Y.
,
2007
, “
A Survey of Recent Results in Networked Control Systems
,”
Proc. IEEE
,
95
(
1
), pp.
138
162
.
5.
Wu
,
M.
,
He
,
Y.
,
She
,
J.-H.
, and
Liu
,
G.-P.
,
2004
, “
Delay-Dependent Criteria for Robust Stability of Time-Varying Delay Systems
,”
Automatica
,
40
(
8
), pp.
1435
1439
.
6.
He
,
Y.
,
Wang
,
Q.-G.
,
Xie
,
L. H.
, and
Lin
,
C.
,
2007
, “
Further Improvement of Free-Weighting Matrices Technique for Systems With Time-Varying Delay
,”
IEEE Trans. Autom. Control
,
52
(
2
), pp.
293
299
.
7.
Gouaisbaut
,
F.
, and
Peaucelle
,
D.
,
2006
, “
Delay-Dependent Robust Stability of Time Delay Systems
,”
Fifth IFAC Symposium on Robust Control Design
, Toulouse, France, ▪.
8.
Gouaisbaut
,
F.
, and
Peaucelle
,
D.
,
2006
, “
Delay-Dependent Stability Analysis of Linear Time Delay Systems
,”
Sixth IFAC Workshop on Time-Delay Systems, L'Aquila
, Italy, ▪.
9.
Han
,
Q.-L.
,
2005
, “
Absolute Stability of Time-Delay Systems With Sector-Bounded Nonlinearity
,”
Automatica
,
41
(
12
), pp.
2171
2176
.
10.
Liu
,
K.
,
Suplin
,
V.
, and
Fridman
,
E.
,
2010
, “
Stability of Linear Systems With General Sawtooth Delay
,”
IMA J. Math. Control Inf.
,
27
(
4
), pp.
419
436
.
11.
Seuret
,
A.
, and
Gouaisbaut
,
F.
,
2013
, “
Jensen's and Wirtinger's Inequalities for Time-Delay Systems
,”
11th IFAC Workshop on Time-Delay Systems, Grenoble
, France, ▪.
12.
Seuret
,
A.
, and
Gouaisbaut
,
F.
,
2013
, “
Wirtinger-Based Integral Inequality: Application to Time-Delay Systems
,”
Automatica
,
49
(
9
), pp.
2860
2866
.
13.
Wu
,
J.
,
1996
,
Theory and Applications of Partial Functional Differential Equations
,
Springer-Verlag
,
New York
.
14.
Allegretto
,
W.
, and
Papini
,
D.
,
2007
, “
Stability for Delayed Reaction-Diffusion Neural Networks
,”
Phys. Lett. A
,
360
(
6
), pp.
669
680
.
15.
Gaffney
,
E. A.
, and
Monk
,
N. A. M.
,
2006
, “
Gene Expression Time Delays and Turing Pattern Formation Systems
,”
Bull. Math. Biol.
,
68
(
1
), pp.
99
130
.
16.
Forys
,
U.
, and
Marciniak-Czochra
,
A.
,
2003
, “
Logistic Equations in Tumor Growth Modeling
,”
Int. J. Appl. Math. Comput. Sci.
,
13
(
3
), pp.
317
325
.
17.
Wang
,
P.
,
1963
, “
Asymptotic Stability of a Time-Delayed Diffusion System
,”
ASME J. Appl. Mech.
,
30
(
4
), pp.
500
504
.
18.
Wang
,
P.
,
1964
, “
Optimum Control of Distributed Parameter Systems With Time Delays
,”
IEEE Trans. Autom. Control
,
9
(
1
), pp.
13
22
.
19.
Kim
,
M.
,
1974
, “
Synthesis of Linear Optimum Distributed Parameter Systems With Time Delays
,”
Proc. IEEE
,
62
(
8
), pp.
1177
1179
.
20.
Wang
,
T.
,
1994
, “
Stability in Abstract Functional-Differential Equations—I: General Theorems
,”
J. Math. Anal. Appl.
,
186
(
2
), pp.
534
558
.
21.
Wang
,
T.
,
1994
, “
Stability in Abstract Functional-Differential Equations—II: Applications
,”
J. Math. Anal. Appl.
,
186
(
3
), pp.
835
861
.
22.
Nicaise
,
S.
, and
Pignotti
,
C.
,
2006
, “
Stability and Instability Results of the Wave Equation With a Delay Term in the Boundary or Internal Feedback
,”
SIAM J. Control Optim.
,
45
(
5
), pp.
1561
1585
.
23.
Gurevich
,
S. V.
,
2013
, “
Dynamics of Localized Structures in Reaction-Diffusion Systems Induced by Delayed Feedback
,”
Phys. Rev. E
,
87
, p.
052922
.
24.
Fridman
,
E.
, and
Orlov
,
Y.
,
2009
, “
Exponential Stability of Linear Distributed Parameter Systems With Time-Varying Delays
,”
Automatica
,
45
(
1
), pp.
194
201
.
25.
Luo
,
Y.-P.
,
Xia
,
W.-H.
,
Liu
,
G.-R.
, and
Deng
,
F.-Q.
,
2009
, “
LMI Approach to Exponential Stabilization of Distributed Parameter Control Systems With Delay
,”
Acta Autom. Sin.
,
35
(
3
), pp.
299
304
.
26.
Wang
,
J.-W.
,
Sun
,
C.-Y.
,
Xin
,
X.
, and
Mu
,
C.-X.
,
2014
, “
Sufficient Conditions for Exponential Stabilization of Linear Distributed Parameter Systems With Time Delays
,”
IFAC Proc. Vol.
,
47
(
3
), pp.
6062
6067
.
27.
Solomon
,
O.
, and
Fridman
,
E.
,
2015
, “
Stability and Passivity Analysis of Semilinear Diffusion PDEs With Time-Delays
,”
Int. J. Control
,
88
(
1
), pp.
180
192
.
28.
Wang
,
J.-W.
, and
Wu
,
H.-N.
,
2015
, “
Some Extended Wirtinger's Inequalities and Distributed Proportional-Spatial Integral Control of Distributed Parameter Systems With Multi-Time Delays
,”
J. Franklin Inst.
,
352
(
10
), pp.
4423
4445
.
29.
Gahinet
,
P.
,
Nemirovskii
,
A.
,
Laub
,
A. J.
, and
Chilali
,
M.
,
1995
, “LMI Control Toolbox for Use With MATLAB,”
MathWorks
,
Natick, MA
.
30.
Liu
,
Z.
, and
Zheng
,
S.
,
1999
,
Semigroups Associated With Dissipative Systems
,
Chapman and Hall/CRC
,
Boca Raton, FL
.
31.
Curtain
,
R. F.
, and
Zwart
,
H. J.
,
1995
,
An Introduction to Infinite-Dimensional Linear Systems Theory
,
Springer-Verlag
,
New York
.
32.
Fridman
,
E.
, and
Blighovsky
,
A.
,
2012
, “
Robust Sampled-Data Control of a Class of Semilinear Parabolic Systems
,”
Automatica
,
48
(
5
), pp.
826
836
.
33.
Am
,
B. N.
, and
Fridman
,
E.
,
2014
, “
Network-Based Distributed H∞ Filtering of Parabolic Systems
,”
Automatica
,
50
(
12
), pp.
3139
3146
.
34.
Wang
,
J.-W.
,
Li
,
H.-X.
, and
Wu
,
H.-N.
,
2014
, “
Distributed Proportional Plus Second-Order Spatial Derivative Control for Distributed Parameter Systems Subject to Spatiotemporal Uncertainties
,”
Nonlinear Dyn.
,
76
(
4
), pp.
2041
2058
.
35.
Bode
,
M.
, and
Purwins
,
H.-G.
,
1995
, “
Pattern Formation in Reaction-Diffusion Systems-Dissipative Solitons in Physical Systems
,”
Physica D
,
86
(
1–2
), pp.
53
63
.
36.
Mao
,
X.
,
Koroleva
,
N.
, and
Rodkina
,
A.
,
1998
, “
Robust Stability of Uncertain Stochastic Differential Delay Equations
,”
Syst. Control Lett.
,
35
(
5
), pp.
325
336
.
You do not currently have access to this content.