Feedback Stabilization of Infinite-Dimensional Linear Systems with Constraints on Control and its Rate

Abstract

The feedback stabilization problem in the presence of nonsymmetrical constraints on both control and its rate is addressed for a class of distributed parameter systems.Necessary and sufficient conditions are established, under which the constraints are satisfied for autonomous infinitedimensional systems. The approach is developed using the semigroup formulation and the positive invariance concept. The main result allows us to determine some suitable positively invariant sets of admissible initial states,in which the constraints are satisfied, with respect to the asymptotically stable closed-loop system.

Full text

European Journal of Control (2011)2:183–190
© 2011 EUCA
DOI:10.3166/EJC.17.183–190
Feedback Stabilization of Infinite-Dimensional Linear Systems
with Constraints on Control and its Rateg
Bouchra Abouzaid1,∗, M. Elarbi Achhab2,∗∗, Vincent Wertz3,∗∗∗
1Laboratoire d’Ingénierie Mathématique, Département de Mathématique et Informatique, Université Chouaib Doukkali, B-20 El Jadida, Morocco;
2Ecole Supérieure de Technologie de Safi, Université Cadi Ayyad, BP 89 46000 Safi, Morocco;
3CESAME, université Catholique de Louvain, 4-6 avenue G. Lemaitre, 1348, Louvain-la-Neuve, Belgium
The feedback stabilization problem in the presence of non-
symmetrical constraints on both control and its rate is
addressed for a class of distributed parameter systems.
Necessary and sufficient conditions are established, under
which the constraints are satisfied for autonomous infinite-
dimensional systems. The approach is developed using
the semigroup formulation and the positive invariance
concept. The main result allows us to determine some suit-
able positively invariant sets of admissible initial states,
in which the constraints are satisfied, with respect to the
asymptotically stable closed-loop system.
Keywords: Distributed parameter systems, constrained
control, increment or rate constraints, positive invariance
1. Introduction
Several works are dedicated to the stabilizability problem
for infinite dimensional linear systems. Useful references
are the survey papers by Pritchard et al. [10], Russell et al.
[11], and Schumacher [12]. The feedback stabilization
problem for infinite-dimensional linear systems operat-
ing under control constraints has been studied by several
authors, using a state space framework which is based on
a semigroup formulation, see for example [6], [9]. Dif-
ferent approaches have been used and many results have
∗Correspondence to: B. Abouzaid, E-mail: bhabouzaid@yahoo.fr
∗∗E-mail: elarbi.achhab@menara.ma
∗∗∗E-mail: vincent.wertz@uclouvain.be
been obtained. Slemrod in [13] investigates the problem
for systems
˙x(t)=Ax(t)+Bu(t);x(0)=x0∈D(A),(1)
where Ais the infinitesimal generator of a linear
C0-semigroup on a real Hilbert space, and the control input
is subject to the symmetrical constraints u(t)≤ rfor
some fixed positive real number r. In the single input case,
he showed that the saturated control law stabilizes glob-
ally asymptotically the system (1), under some conditions.
The main one is that Ais the infinitesimal generator of a
linear C0-semigroup of contraction. In [4], the authors
generalized the previous result under the same conditions
to the multi-input case. The stabilizing state feedback they
used is a smooth nonlinear control law. Recently and in
contrast to the previous works, the paper [1] investigated
the stabilization problem for distributed parameter sys-
tems subject to nonsymmetrical control constraints. Also,
it is not assumed that the C0-semigroup generated by Ais
of contraction. The main result links the problem with the
positive invariance of an appropriate polyhedral set which
is, in fact, a stability domain for the closed-loop system
with control constraints. Such property was also studied
by Boulite et al. in [2], [3] for a class of delay differ-
ential equations, where they give a characterization of a
C0-semigroup which leaves particular sets of an ordered
Banach space positively invariant, using the notions of
positive semigroups and dissipative operators.
Received 14 September 2009; Accepted 23 August 2010
Recommended by D.A. Melchor Aguilar, D.W. Clarke

184 B. Abouzaid et al.
This article deals with the stabilization problem for
infinite-dimensional linear systems in the presence of non-
symmetrical constraints on both control magnitude and its
rate. In industrial applications (nowadays often described
by distributed-parameter models), such constraints are
very common and usually related to physical limitations
of the actuators. They may be nonsymmetrical when the
process operating point is close to one of the bounds of the
actuator. Obviously, rate constraints can also exist since
valve opening or heating or cooling power can be rate lim-
ited. The main idea of this work is to relate the satisfaction
of the constraints to the positive invariance of an appropri-
ate domain and to find necessary and sufficient conditions
that guarantee such positive invariance. During the last
years, the concept of positive invariance of polyhedral
domains has been used as an interesting theoretical tool to
treat the problem of constraints satisfaction for controlled
linear systems. Broadly speaking, the control law must be
computed such that the state of the closed-loop system is
confined in an appropriate region, while the control sig-
nal does not violate the constraints. In finite-dimensional
linear systems, the regulator problem with constraints on
both control and its rate or increment has been investigated
in [8]. The proposed approach is based on determining a
feedback control law that ensures positive invariance of a
set included in a region of closed-loop linear behavior and
including all admissible initial states. This article extends
to infinite-dimensional linear systems the results obtained
only for finite-dimensional linear systems [8].
The article is organized as follows. First of all, the
notations used hereafter are introduced. In Section 2,
we present the problem statement. In Section 3, we
recall some preliminary results for finite-dimensional sys-
tems, together with an extension to discrete-time infinite-
dimensional case. The conditions of positive invari-
ance polyhedral domain with respect to continuous-time
infinite-dimensional system such that the rate constraints
are satisfied is given in Section 4. The application of such
conditions allows us to give conditions such that a stabiliz-
ing state feedback satisfies the constraints on both control
and its rate. An illustrative example is given in Section 5.
1.1. Notations
For a matrix Hin IR m×m, let H1and H2be the matrices in
IR m×mwhose components are respectively given by:
h1
i,j=hii if i=j
h+
ij if i= jand h2
i,j=0ifi=j
h−
ij if i= j
where h+
ij =max(hij ,0)and h−
ij =max(0, −hij ).
We define H+=max(H,0)=(h+
ij )and H−=
max(−H,0)=(h−
ij )
We denote
˜
Hd=H+H−
H−H+and ˜
Hc=H1H2
H2H1.
int(IR m)denotes the interior of IR m,Imdenotes the m×m
identity matrix and IR ∗
−={λ∈IR |λ<0}.Fortwo
vectors xand yin IR n,x≤ymeans that the inequalities
hold for each component.
For q1,q2∈int(IR m
+)and Fan m×nmatrix, we define
the following domains
DF={x∈IR n|−q2≤Fx≤q1}(2)
DIm={x∈IR m|−q2≤x≤q1}(3)
Finally, we denote q=qT
1qT
2Tand δ=δT
1δT
2T
for δ1,δ2∈int(IR m
+).
2. Problem Statement
Consider the infinite-dimensional linear system described
by the following abstract differential equation
˙x(t)=Ax(t)+Bu(t),
x(0)=x0∈Z,(4)
where Ais the infinitesimal generator of a C0-semigroup
(T(t))t≥0on a real separable Hilbert space Z.Bis a
bounded linear operator from IR minto Z.u∈IR mis
the input vector, which is constrained to evolve in the
following domain:
={u∈IR m|−q2≤u≤q1,q1,q2∈int(IR m
+)}.
(5)
The control rate is constrained as follows:
−δ2≤˙u≤δ1,δ1,δ2∈int(IR m
+). (6)
Furthermore, we assume that the pair (A,B)is stabilizable.
Recall that a family (T(t))t≥0in L(Z)(i.e., for all
t≥0, T(t)is a bounded linear operator on Z) is called a
C0-semigroup if
(i) T(0)=I, where Iis the identity operator on Z,
(ii) T(t+s)=T(t)T(s),∀t,s≥0, and
(iii) lim
t→0+T(t)x−x= 0, ∀x∈Z.
The infinitesimal generator Aof a C0-semigroup (T(t))t≥0
is defined by
Ax =lim
t−→0+
T(t)x−x
t,∀x∈D(A),
where D(A)=x∈Z|lim
t−→0+
T(t)x−x
texists in Z.

Stabilization of Infinite-Dimensional Linear Systems with Constraints 185
Moreover, for every x0∈D(A), the function xgiven
by x(t)=T(t)x0,t≥0, is differentiable on [0, ∞)
and is the unique solution of the abstract cauchy problem
˙x(t)=Ax(t),x(0)=x0. The growth bound of (T(t))t≥0
is defined by
ω0=inf
t>0
1
tlog T(t)= lim
t−→∞
1
tlog T(t).(7)
It is well known (see, for example, [6], [9]) that there
exists a constant M≥1 such that
T(t)≤ Meωt,∀t≥0, ∀ω>ω
0.(8)
We assume, throughout the article, that the operator A
has the following additional property: the resolvent of A,
R(λ,A)=(λI−A)−1is a compact operator for all λ>ω
0.
Note that (see [6])
R(λ,A)x=+∞
0
e−λtT(t)xdt,∀x∈Z. (9)
Definition 2.1:
(1) The system (4), that is, the pair (A,B), is said to be
stabilizable if there exists a state feedback control law
F∈L(Z,IR m)such that the C0-semigroup generated
by A +BF is an asymptotically stable C0-semigroup.
(2) The system ˙x(t)=Ax(t), that is, (T(t))t≥0, is said to
be asymptotically stable if lim
t→∞ T(t)x=0, ∀x∈Z,
where (T(t))t≥0is the semigroup generated by A.
The main question consists of finding conditions of the
existence of a state feedback that stabilizes the system (4)
and satisfies the constraints on both control magnitude and
its rate given by (5) and (6).
Since (A,B)is stabilizable, there exists F∈L(Z,IR m)
such that A+BF is the infinitesimal generator of an asymp-
totically stable C0-semigroup. The next step should be
the constraints analysis. Here, we adopt the concept of
positive invariance. Let us define the polyhedral set
D(F,q1,q2)={x∈Z|−q2≤Fx ≤q1}. (10)
Observe that, if the state trajectories of the closed-loop sys-
tem do not leave the domain D(F,q1,q2), then the state
feedback u(t)=Fx(t)satisfies the control constraints.
Thus, the study of the constraints on both control and its
rate given by (5) and (6) comes down to the study of the
positive invariance of D(F,q1,q2)with respect to the tra-
jectories of the closed-loop system ˙x(t)=(A+BF )x(t)
such that the following rate constraints are satisfied:
−δ2≤F˙x≤δ1. (11)
Definition 2.2: A subset D of Z is said to be positively
invariant w.r.t. the trajectories of system ˙x(t)=Ax(t)
(respectively x(k+1)=Ax(k)), if for all initial con-
dition x0in D, x(t)=T(t)x0∈D,∀t≥0, i.e.
T(t)D⊂D,∀t≥0, (respectively AD ⊂D), where
(T(t))t≥0is the semigroup generated by A.
3. Preliminary Result
An important intermediate step to study the constraints
for continuous-time infinite-dimensional systems is to find
conditions which ensure the satisfaction of the constraints
in the discrete-time case, that is, to find necessary and
sufficient conditions that guarantee the positive invari-
ance of D(F,q1,q2)w.r.t. the trajectories of a discrete-
time infinite-dimensional system such that the increment
constraints are satisfied. This is the aim of this section.
3.1. Finite-Dimensional Case
First, let recall the result established for finite-dimensional
system. Let Abe an n×nmatrix and Ban n×mmatrix. In
the main result of [8], necessary and sufficient conditions
are given to check whether a stabilizing state feedback con-
trol law u(t)=Fx(t),F∈IR m×n, will ensure constraints
fulfillment for both the control and its rate or increment.
Assume that m≤n.
Theorem 3.1: [8] Let Fbe an m ×n matrix of full
rank. The domain DFgiven by (2) is positively invari-
ant with respect to the trajectories of the system ˙x(t)=
(A+BF)x(t)(respectively, x(k+1)=(A+BF)x(k))
and the rate constraints −δ2≤F˙x≤δ1(respectively,
−δ2≤F(x(k+1)−x(k)) ≤δ1) are satisfied if and only
if there exists a matrix H ∈IR m×msuch that
(i) F(A+BF)=HF,
(iia) for the discrete-time case

(H−Im)dq≤δ,

Hdq≤q,
(iib) for the continuous-time case

Hdq≤δ,

Hcq≤0.
3.2. Discrete-Time Problem
This subsection is devoted to study the constraints w.r.t.
the autonomous discrete-time infinite-dimensional system
described by:
x(k+1)=Kx(k),
x(0)=x0∈Z,(12)
where Kis a compact linear operator mapping Zinto Z.

186 B. Abouzaid et al.
More precisely, the aim of this subsection is to state a
technical result similar the one established in Theorem 3.1
for the discrete-time case and which will be applied to the
special case K=λR(λ,A)in the next section.
Since Kis a compact linear operator on a separa-
ble Hilbert space Z, there exists an orthonormal basis
of Z, formed by the all eigenvectors of the operator
K∗K, where K∗is the adjoint operator of K(see [5]).
Let {φn}n≥1be such basis of Zand denote by Nthe
orthogonal projection of Zon the linear subspace ZN=
span{φ1,φ2,...,φN},N∈IN \{0}, where IN is the set of
all natural numbers. Put KN=KN, then KNis a bounded
linear operator of Z, which has a finite rank. Moreover, K
is a uniform limit of the sequence of operators KN, that is,
lim
N→∞ K−KN= 0.
Remark 3.2: Consider the finite-dimensional linear
system
xN(k+1)=KNxN(k),
xN(0)∈ZN,(13)
Define the set
DN(F,q1,q2)={xN∈ZN|−q2≤FxN≤q1}. (14)
If the set D(F,q1,q2)given by (10) is positively invariant
w.r.t. the trajectories of the system (12) and the following
increment constraints are satisfied
−δ2≤F(x(k+1)−x(k)) ≤δ1, (15)
then DN(F,q1,q2)is positively invariant w.r.t. the tra-
jectories of the system (13) and the following increment
constraints are satisfied
−δ2≤F(xN(k+1)−xN(k)) ≤δ1. (16)
Indeed: From the positive invariance of D(F,q1,q2)w.r.t.
the trajectories of the system (12) and the fact that KNxN=
KxN, it follows that DN(F,q1,q2)is positively invari-
ant w.r.t. the trajectories of the system (13). However,
since DN(F,q1,q2)⊂D(F,q1,q2), the increment con-
straints (15) are satisfied for all xN∈DN(F,q1,q2).
Consequently, DN(F,q1,q2)is positively invariant w.r.t.
the trajectories of the system (13) and the increment
constraints (16) are satisfied.
Lemma 3.3: Let ψbe a linear continuous mapping of Z
into IRmand S a dense subset of Z. Assume that ψ(Z)=
IR m, then ψ(S)=IR m.
Proof: Seeking contradiction, assume that ψ(S)= IR m,
then there exists p<msuch that ψ(S)=IR p. Since ψis
continuous, it follows that ψ(S)⊂ψ(S), where Sdenotes
the closure of subset S. Thus, ψ(Z)⊂IR pwhich leads to
a contradiction.
We shall end this subsection by the extension of
Theorem 3.1 to discrete-time infinite-dimensional sys-
tems.
Theorem 3.4: Let Z be a separable Hilbert space and
F∈L(Z,IR m)of full rank, then the polyhedral set (10)
is positively invariant with respect to the trajectories of
system (12) and the increment constraints (15) are satisfied
if and only if there exists an m ×m matrix H such that:
FK =HF. (17)

Hdq≤q. (18)

(H−Im)dq≤δ. (19)
Proof: Sufficiency: Assume that the conditions (17)–
(18) and (19) hold. By making the change of variables
z(k)=Fx(k), the set D(F,q1,q2)becomes the domain
DImgiven by (3) and zsatisfies the following equation
z(k+1)=Hz(k). Thus, the positive invariance of
D(F,q1,q2)w.r.t. the system (12) and the increment con-
straints (15) are equivalent to the positive invariance of
DImw.r.t. the system z(k+1)=Hz(k)and the incre-
ment constraints −δ2≤z(k+1)−z(k)≤δ1. Applying
Theorem 3.1 to system z(k+1)=Hz(k)and the domain
DIm, it follows from conditions (18) and (19) that DImis
positively invariant w.r.t. the system z(k+1)=Hz(k)and
the increment constraints −δ2≤z(k+1)−z(k)≤δ1are
satisfied. Consequently, the domain D(F,q1,q2)is pos-
itively invariant w.r.t. the system (12) and the increment
constraints (15) hold.
Necessity: Suppose that (10) is positively invariant with
respect to the trajectories of system (12) and the increment
constraints (15) are satisfied. From Remark 3.2, it follows
that DN(F,q1,q2)is positively invariant w.r.t. the trajecto-
ries of system (13) and the increment constraints (16) are
satisfied. However, recall that Fis a surjection and that the
set (∞
N=1ZN)is dense in Z(see [7], Part II). Since (ZN)
is a monotone increasing sequence of linear subspaces of
Zand according to Lemma 3.3, there exists an integer N0
such that F(ZN0)=IR mand F(ZN)=IR m, for all N≥N0.
Applying Theorem 3.1 to system (13) for all N≥N0,we
can deduce that for all N≥N0, there exists HN∈IR m×m
such that
FKN=HNF, (20)

(HN−Im)dq≤δ, (21)
(
HN)dq≤q, (22)

Stabilization of Infinite-Dimensional Linear Systems with Constraints 187
Fhas full rank implies that for all y∈IR m, there exists
x∈Zsuch that y=Fx. Thus FKNx=HNFx =HNy.
Since the sequence (KN)Nconverges uniformly to K, then
the sequence HNyconverges to an element Hy of IR m, and
we get FK =HF.
Now, let Ntends to +∞, from the inequalities (21)
and (22), we obtain

(H−Im)dq≤δ.

Hdq≤q.
This ends the proof.
4. Main Results
Consider the autonomous continuous-time infinite-
dimensional system
˙x(t)=Ax(t),
x(0)=x0∈D(A),(23)
where Ais the infinitesimal generator of a C0-semigroup
(T(t))t≥0on Z.
Let introduce a proposition that connects the
continuous-time case to the discrete-time one, to easily
extend the conditions (18) and (19) to the continuous-time
case.
Proposition 4.1: If the polyhedral set D(F,q1,q2)given
by (10) is positively invariant with respect to the trajec-
tories of system (23) and the rate constraints (11) are
satisfied, then it is also positively invariant with respect
to the trajectories of the discrete-time system x(k+1)=
λR(λ,A)x(k)and the increment constraints (15) hold, for
all λ>max(0, ω0).
Proof: Assume that D(F,q1,q2)is positively invariant
with respect to the trajectories of system (23) and the rate
constraints are satisfied.
Let (xn)nbe a convergent sequence in D(F,q1,q2)
D(A)such that −δ2≤F˙xn(t)≤δ1, then T(t)xn∈
D(F,q1,q2)and −δ2≤FdT(t)xn
dt ≤δ1,∀t≥0, that
implies −q2≤FT(t)xn≤q1and −δ2≤FAT (t)xn≤δ1.
Multiplying all terms by e−λtand integrating from 0
to +∞, we get −q2≤FλR(λ,A)xn≤q1and −δ2
λ≤
FA R (λ,A)xn≤δ1
λ,∀λ>max(0, ω0). Furthermore, we
have AR(λ,A)xn=λR(λ,A)xn−xn, that is,
−q2≤FλR(λ,A)xn≤q1, (24)
−δ2≤λF(λR(λ,A)xn−xn)≤δ1. (25)
Let ntends to +∞ in (24) and (25), then
−q2≤FλR(λ,A)x≤q1,∀λ>max(0, ω0), and
(26)
−δ2≤λF(λR(λ,A)x−x)≤δ1,∀λ>max(0, ω0).
(27)
where x=lim
n−→+∞ xn.
By making the change of variables z(k)=λx(k),we
can deduce from (26) and (27) that D(F,q1,q2)is posi-
tively invariant w.r.t. the trajectories of system z(k+1)=
λR(λ,A)z(k)and that the increment constraints −δ2≤
F(z(k+1)−z(k)) ≤δ1are satisfied.
Now, we are able to give necessary and sufficient condi-
tions that guarantee the positive invariance of D(F,q1,q2)
with respect to the trajectories of system (23) such that the
rate constraints are satisfied.
Theorem 4.2: Let Z be a separable Hilbert space and F ∈
L(Z,IR m)of full rank. Let A be the infinitesimal generator
ofaC
0-semigroup with a compact resolvent. Then, the
polyhedral set D(F,q1,q2)defined by (10) is positively
invariant with respect to the trajectories of system (23)
and the rate constraints (11) are satisfied if and only if
there exists an m ×m matrix H such that:
FA x =HFx,∀x∈D(A). (28)

Hdq≤δ. (29)

Hcq≤0. (30)
Proof: Sufficiency: Assume that the conditions (28)–(29)
and (30) hold. Using the change of variables z(t)=Fx(t),
the system (23) is transformed to the finite-dimensional
one
˙z(t)=Hz(t),
z(0)=Fx(0)∈IR m,(31)
Also, the set D(F,q1,q2)is transformed to DIm={z∈
IR m|−q2≤z(t)≤q1}and the rate constraints −δ2≤
F˙x(t)≤δ1become −δ2≤˙z(t)≤δ1.
According to Theorem 3.1, for the continuous-time
case, we deduce that the conditions (29)–(30) ensure the
positive invariance of the domain DImw.r.t. the trajec-
tories of system ˙z(t)=Hz(t)and the rate constraints
−δ2≤˙z(t)≤δ1satisfied, which is equivalent to the
positive invariance of D(F,q1,q2)w.r.t. the trajectories of
system (23) and the rate constraints (11) are satisfied.
Necessity: Suppose that D(F,q1,q2)is positively
invariant with respect to the trajectories of system (23)
and the rate constraints (11) are satisfied. According to

188 B. Abouzaid et al.
Proposition 4.1, it is also positively invariant w.r.t. the
trajectories of system x(k+1)=λR(λ,A)x(k),∀λ>
max(0, ω0)and the increments constraints (15) are sat-
isfied. From Theorem 3.4, it follows that there exists an
m×mmatrix Hλsuch that
FλR(λ,A)=HλF, (32)
(
Hλ)dq≤q, (33)
(

Hλ−Im)dq≤δ. (34)
Let Aλbe the approximation of Yosida, (see [9])
Aλ=λ2R(λ,A)−λIm. (35)
Combining (32) and (35), we obtain
FA λ=λ(Hλ−Im)F. (36)
It is known that lim
λ−→∞ Aλx=Ax,∀x∈D(A).
However, we have ∀y∈IR m,∃x∈D(A)such that
Fx =y. Hence, by (36) the limit limλ−→ ∞ λ(Hλ−Im)y
exists and is denoted by Hy ∈IR m. Using also the limit of
(36) while λtends to +∞, we obtain the identity (28).
Since Fis of full rank, multiplying (32) by (λI−A)and
using (28), we can deduce that
Hλ=λ(λIm−H)−1for all λ>max(ω0,H). (37)
So, the conditions (33) and (34) imply that DImis positively
invariant w.r.t. the trajectories of the discrete-time system
x(k+1)=Hλx(k)and the increment constraints
−δ2≤x(k+1)−x(k)≤δ1, (38)
are satisfied, for all λ>max(ω0,H), see Theorem 3.1.
Thus, for all z∈IR msuch that −q2≤z≤q1,wehave
−q2≤λ(λIm−H)−1≤q1for all λ>max(ω0,H).
Pick λ=n
t,fort>0 and nsufficiently large, we obtain
−q2≤n
tn
tIm−H−1n
≤q1.
Let ntend to +∞,weget
−q2≤etH z≤q1. (39)
However, let show that
−δ2≤˙z(t)≤δ1where ˙z(t)=Hz(t). (40)
Seeking a contradiction: Suppose that there exists
i0,j0∈IN ∗such that
⎧
⎪
⎨
⎪
⎩
(Hz)i0>δ
i0
1
or
(Hz)j0<−δj0
2
where δi0
1is the i0th component of δ1and δj0
2is the j0th
component of δ2.
We have Hz =limλ→+∞ λ(Hλ−Im)z
Hence, there exists λ0sufficiently large such that
⎧
⎪
⎨
⎪
⎩
(λ0(Hλ0−Im)z)i0≥δi0
1
or
(λ0(Hλ0−Im)z)j0≤−δj0
2
that is,
⎧
⎪
⎨
⎪
⎩
(λ0Hλ0z−λ0z)i0≥δi0
1
or
(λ0Hλ0z−λ0z)j0≤−δj0
2
By making the change of variables x(k)=λ0z(k), we get
⎧
⎪
⎨
⎪
⎩
(Hλ0x(k)−x(k))i0≥δi0
1
or
(Hλ0x(k)−x(k))j0≤−δj0
2
which leads to a contradiction with the condition (38).
Consequently, we have from (39) and (40) the positive
invariance of the domain DImw.r.t. the trajectories of sys-
tem ˙z(t)=Hz(t)and the rate constraints −δ2≤˙z(t)≤δ1
are satisfied.
Finally, from Theorem 3.1, we conclude that the
inequalities (29) and (30) are satisfied. This ends the proof.
As stated before, we are interested in the feedback stabi-
lization for a class of infinite-dimensional linear systems
operating under constraints on both control and its rate.
The satisfaction of the constraints is related to the positive
invariance of an appropriate polyhedral domain in which
the control constraints are satisfied. Applying Theorem 4.2
to the closed-loop system leads to the following result.
Theorem 4.3: Let Z be a separable Hilbert space. Let A
be the infinitesimal generator of a C0-semigroup with a
compact resolvent and F ∈L(Z,IR m)of full rank, such
that A +BF is the infinitesimal generator of an asymp-
totically stable C0-semigroup. Then, the polyhedral set
D(F,q1,q2)defined by (10) is positively invariant with
respect to the trajectories of system ˙x(t)=(A+BF)x(t)
and the rate constraints −δ2≤F˙x(t)≤δ1are satisfied
if and only if there exists an m ×m matrix H =(hij ),
1≤i,j≤m such that:
F(A+BF)x=HFx,∀x∈D(A). (41)

Hdq≤δ. (42)

Hcq≤0. (43)

Stabilization of Infinite-Dimensional Linear Systems with Constraints 189
Remark 4.4: To apply Theorem 4.2 to the closed-loop
system, we must have a compact resolvent of the operator
A+BF.
Observe that the linear operator BF is bounded, that is,
BF ∈L(Z). Moreover,
R(λ,A+BF)=R(λ,A)(I−BFR(λ,A))−1.
If Ahas a compact resolvent, then A+BF is so, for each
λsuch that BFR(λ,A)<1, λ>ω
0. Consequently,
(A+BF) is the infinitesimal generator of a C0-semigroup
which has a compact resolvent.
5. Example
Consider the one-dimensional heat diffusion model with
Dirichlet boundary condition:
⎧
⎨
⎩
∂x
∂t=1
π2
∂2x
∂ξ2+4x+b1u1(t)+b2u2(t),
x(0, t)=x(1, t)=0,
(44)
This model can be described by the following abstract
differential equation, in the Hilbert separable state space
Z=L2(0, 1):
˙x(t)=Ax(t)+Bu(t),x(0)∈D(A), (45)
where Ais defined on its domain:
D(A)=h∈Z
dh
dξ,d2h
dξ2∈Zare absolutely
continuous and h(0)=h(1)=0
by A·=(1
π2)d2·
dξ2+4I,Bis a linear bounded operator
on IR 2, defined by Bu =b1u1(t)+b2u2(t), where u=
u1u2T,bi=1
ion the interval (xi−i,xi+i)and is
equal to 0 otherwise for i=1, 2, xi∈(0, 1)and i>0.
Assume that the control and its rate are subject to the
following constraints:
−3−5T≤u(t)≤23
T, (46)
−11 −6T≤˙u(t)≤10 7T. (47)
Recall that, see [6]
(1) Ahas the complete eigenset λn=(4−n2),φn(ξ ) =
√2 sin(nπξ),∀n≥1.
(2) for all x∈D(A),Ax =∞

n=1
λn<x,φn>φ
n.
(3) ∀λ∈ρ(A),R(λ,A)x=∞

n=1
1
λ−λn
<x,φn>φ
nis
a compact operator.
(4) {φn}n≥1is an orthonormal basis of Z.
To find Fand Hsatisfying the established conditions (41)–
(42) and (43), we adopt the inverse procedure: It consists
of giving a matrix Hsuch that the conditions (42) and (43)
hold and then computing Fsolution of (41).
Let Hbe the 2 ×2 matrix given by H=−21
0−1.
It is clear that the conditions (42) and (43) are satisfied:

Hdq−δ=−1−2−2−3T≤0.

Hcq=−1−3−1−5T≤0.
Now, we are interested in determining Fsolution of the
equation (41) such that
σ(A+BF)=(A)∪σ(H)⊂IR ∗
−, (48)
where (A)is the set of all stable eigenvalues of A, that
is, the control law u(t)=Fx(t)stabilizes system (45).
Consider λi∈σ(H), with the associated vectors θiand
ξisuch that Hθi=λiθiand (A+BF)ξi=λiξi,i=1, 2.
It follows from equation F(A+BF)=HF that
H(Fξi)=λi(Fξi),i=1, 2. Since λiis an eigenvalue
of the matrix H, we should obtain
Fξi=μθi,forμ∈IR ∗,i=1, 2. (49)
Without loss of generality, we can assume μ=1, so we
have
Fξi=θi. (50)
Replacing in Aξi+BFξi=λiξi,weget
ξi=(λiI−A)−1Bθi,i=1, 2. (51)
Now, consider λj∈(A)and ξjan associated eigenvector
of (A+BF), then
F(A+BF)ξj=HF ξj,∀j≥3,
which is equivalent to H(Fξj)=λj(Fξj).
Since λjis not an eigenvalue of the matrix H, this
implies that
Fξj=0, ∀j≥3. (52)
Hence, the eigenvectors ξjof (A+BF)for j≥3 are
eigenvectors of A.
Since (φn)n≥1is a basis of Zand Fφn=0 for n≥3,
then determining Freduces to compute Fφ1and Fφ2.

190 B. Abouzaid et al.
From equations (50) and (51), we have
θi=F(λiI−A)−1Bθi,i=1, 2. (53)
Let θ1=11
Tand θ2=10
Tthe eigenvectors of
Hassociated respectively to λ1=−1 and λ2=−2.
According to (53) and the fact that Fφn=0, ∀n≥3,
we obtain
⎧
⎪
⎨
⎪
⎩
−1
4Bθ1,φ1Fφ1−Bθ1,φ2Fφ2=θ1
−1
5Bθ2,φ1Fφ1−1
2Bθ2,φ2Fφ2=θ2
(54)
So, we get
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
Fφ1=5
2b1,φ2
Cb1,φ1θ1−5b1+b2,φ2
Cb1,φ1θ2,
Fφ2=−1
Cθ1+5
4b1+b2,φ1
Cb1,φ1θ2,
Fφn=0, ∀n≥3.
(55)
where C=b1+b2,φ2−5
8b1+b2,φ1b1,φ2
b1,φ1.
Note that the operator B(i.e. xi,i,i=1, 2) should be
chosen such that b1,φ1=0 and C= 0.
Consequently, the control law u(t)=Fx(t), where F
given by (48) and (55) is a stabilizing state feedback for
system (44) which obeys the constraints on control (46)
and its rate (47).
6. Conclusion
In this article, the feedback stabilization problem for
distributed parameter systems with constraints on both
control and its rate has been studied. The novelty of this
work is the consideration of the additional rate constraints
for infinite-dimensional systems. It was only assumed that
the operator Ahas a compact resolvent. The latter property
is satisfied by most parabolic partial differential equations
and also some delay systems. The main result links the
problem of constraints with the positive invariance of an
appropriate polyhedral set, including all admissible initial
states. Necessary and sufficient conditions were estab-
lished to ensure the positive invariance of an appropriate
set for the discrete-time and the continuous-time cases.
This positive invariance set is considered as a linear region
of stability for the closed-loop system operating under
constraints.
Acknowledgements
This article presents research results of the projet
de coopération scientifique inter-universitaire “Analyse,
Simulation et Commande des Systèmes Complexes” sup-
ported by the Agence Universitaires de la Francophonie.
The work is also supported by the Belgian Programme on
Interuniversity Poles of Attraction (PAI network DYSCO).
The scientific responsibility rests with the authors. The
work was completed while V. Wertz was on leave at the
ARC Center for Complex Dynamic Systems and Control,
University of Newcastle, Australia, whose support is also
gratefully acknowledged.
References
1. Achhab ME, Laabissi M. Feedback stabilization of a class of
distributed parameter systems with control constraints. Syst
Control Lett, 2002; 45(3):163–171.
2. Boulite S, Bouslous H. Positively invariant sets in Banach
spaces and applications to control by order constraints and
retarded differential equations. Diff Equ Dynam Syst, 2000;
8:257–280.
3. Boulite S, Bouslous H, Maniar L. Invariant sets in Banach
lattices and linear differential equations and delay. Positivity,
2004; 8:127–142.
4. Bounit H, Hammouri H. Bounded feedback stabilization and
global separation principal of distributed parameter systems.
IEEE Trans Autom Control, 1997; 40:414–419.
5. Brézis H. Analyse fonctionnelle, Theory et Application.
Paris: Masson, 1987.
6. Curtain RF, Zwart J. An introduction to infinite-dimensional
linear systems theory. New York: Springer, 1995.
7. Dunford N, Schwartz JT. Linear operators: Part II.New
York: Wiley-Interscience, 1951.
8. Mesquine F, Tadeo F, Benzaouia A. Regulator problem for
linear systems with constraints on control and its increments
or rate. Autom, 2004; 40(8):1387–1395.
9. Pazy A. Semigroups of linear operators and applications to
partial differential equations. Berlin: Springer, 1983.
10. Pritchard AJ, Zabczyk J. Stability and stabilizability of
infinitedimensional systems. SIAM Rev, 1981; 23:25–52.
11. Russell, DL. Controllability and stabilizability theory for
linear partial differential equations. SIAM Rev, 1978; 20:
639–739.
12. Schumacher, JM. Dynamical feedback in finite and infi-
nite dimensional linear systems. Amsterdam: Mathematical
Center Tracts, Mathematical Centrum, 1981.
13. Slemrod M. Feedback stabilization of a linear control system
in Hilbert space with a priori bounded control, Math Control
Signals Syst, 1989; 2:265–285.