Applied Mathematics Letters 25 (2012) 682–686
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Applied Mathematics Letters
journal homepage: www.elsevier.com/locate/aml
Dissipativity for Lur’e distributed parameter control systems
Shuxian Lun
∗
, Zhixin Tai
Department of Automation, College of Engineering, Bohai University, 121013 Jinzhou, PR China
a r t i c l e i n f o
Article history:
Received 22 May 2011
Accepted 1 September 2011
Keywords:
Dissipativity
Distributed parameter systems
Heat equation
Linear operator inequalities (LOIs)
a b s t r a c t
In this work, dissipativity of Lur’e distributed parameter control systems has been
addressed. Delay-dependent sufficient conditions for the dissipativity with respect to the
infinite-dimensional version of energy supply rate (Q
1
, S
1
, R
1
) characterized exclusively
by unbounded operator Q
1
are established in terms of linear operator inequalities (LOIs).
Finally, the heat equation illustrates our result.
© 2011 Elsevier Ltd. All rights reserved.
1. Introduction
Dissipativity problem, which was initiated in [1], has been extensively investigated by many authors [2,3] and the
references therein. Problem of reliable dissipative control for a class of stochastic hybrid systems has been dealt with in [2]
by employing the linear matrix inequality (LMI) approach. Furthermore, dissipativity theory for switched systems has been
established in [3] using multiple storage functions and multiple supply rates. So far, however, dissipativity results are only
available for systems governed by ordinary differential equations (ODEs) rather than by partial differential equations (PDEs).
Motivated by the fact that distributed parameter systems described by PDEs are more in general, there is a realistic need to
discuss the dissipativity problem of such systems. Very recently, a linear operator inequality (LOI) approach [4] provides new
insights into the control theory of distributed parameter systems. In this paper, dissipativity problem of Lur’e distributed
parameter control systems will be presented for which delay-dependent sufficient conditions for the dissipativity are
established in terms of linear operator inequalities (LOIs).
2. Preliminaries
Consider the Lur’e nonlinear time-delay control systems in Hilbert spaces H and U described by
Σ
0
:
˙
x(t) = Ax(t) + Bx(t − h) + C w(t) + Du(t)
z(t) = Mx(t)
w(t) = −ϕ(t, z(t))
x(θ) = φ(θ), for ∀θ ∈ [−h, 0]
(1)
where x(t) ∈ H is the state, z(t) ∈ H is the controlled output, w(t), u(t) ∈ U are the control inputs, h denotes positive
constant delay, φ ∈ C([−h, 0], H) is the given initial state, and ϕ(t, z(t)) : R × H → H is an abstract nonlinear function
satisfying the following sector condition:
⟨
ϕ(t, z(t)) − K
1
z(t), ϕ(t, z(t)) − K
2
z(t)
⟩
≤ 0. (2)
∗
Corresponding author.
E-mail addresses: jzlunboda@yeah.net (S. Lun), taizhixin01@163.com (Z. Tai).
0893-9659/$ – see front matter © 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.aml.2011.09.003
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