Compensator design via the separation principle for a class of semilinear evolution equations
UDC 517.9 We establish a compensator design via the separation principle in the practical sense for a class of semilinear evolution equations in Hilbert spaces. Under a restriction imposed on the perturbation, which is bounded by an integrable function, we propose a nonlinear time-varying practical...
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Institute of Mathematics, NAS of Ukraine
2022
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512277421096960 |
|---|---|
| author | Damak, H. Damak, H. |
| author_facet | Damak, H. Damak, H. |
| author_sort | Damak, H. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2022-10-24T09:23:18Z |
| description | UDC 517.9
We establish a compensator design via the separation principle in the practical sense for a class of semilinear evolution equations in Hilbert spaces. Under a restriction imposed on the perturbation, which is bounded by an integrable function, we propose a nonlinear time-varying practical Luenberger observer to estimate the states of the system and prove that the Luenberger observer based on the linear controller stabilizes the system. An illustrative example is given to demonstrate the applicability of our theoretical results. |
| doi_str_mv | 10.37863/umzh.v74i8.6152 |
| first_indexed | 2026-03-24T03:26:14Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v74i8.6152
UDC 517.9
H. Damak1 (Univ. Sfax, Tunisia)
COMPENSATOR DESIGN VIA THE SEPARATION PRINCIPLE
FOR A CLASS OF SEMILINEAR EVOLUTION EQUATIONS
РОЗРОБКА КОМПЕНСАТОРА ЗА ПРИНЦИПОМ ПОДIЛУ
ДЛЯ КЛАСУ НАПIВЛIНIЙНИХ ЕВОЛЮЦIЙНИХ РIВНЯНЬ
We establish a compensator design via the separation principle in the practical sense for a class of semilinear evolution
equations in Hilbert spaces. Under a restriction imposed on the perturbation, which is bounded by an integrable function,
we propose a nonlinear time-varying practical Luenberger observer to estimate the states of the system and prove that the
Luenberger observer based on the linear controller stabilizes the system. An illustrative example is given to demonstrate
the applicability of our theoretical results.
Встановлено схему компенсатора через принцип подiлу в практичному сенсi для класу напiвлiнiйних еволюцiйних
рiвнянь в гiльбертових просторах. За умов обмеження на збурення, яке обмежене iнтегровною функцiєю, запропоно-
вано застосовувати нелiнiйного змiнного у часi практичного спостерiгача Люенбергера для оцiнки станiв системи
та доведено, що спостерiгач Люенбергера на основi лiнiйного регулятора стабiлiзує систему. Наведено наочний
приклад, що демонструє можливiсть застосування наших теоретичних результатiв.
1. Introduction. Feedback compensator design of partial differential equations has been attracting
a lot of attention (see [4, 6, 7, 10, 11, 24]) over the past few decades, due to its wide potential
applications in heat exchanger [18], chemical engineering [5], and flexible mechanical systems [19].
The theory of compensator design is a straightforward extension of the finite dimensional theory
and has been used as a starting point in many control designs for distributed parameter systems,
see [1, 17, 20, 22, 25]. Alternative direct state-space finite-dimensional compensator designs can be
found in [4, 9]. For extensions to systems with unbounded input and output operators (see [7, 10]),
and for a comparison of various finite-dimensional control designs (see [8]). The authors in [24] give
an observer-based output feedback design for linear parabolic partial differential equation (PDE) with
local piecewise control and pointwise observation in space. Moreover, the problem of compensator
design for linear systems in Hilbert spaces can be solved (see [6, 11]), but if the system contains some
nonlinearities as a disturbances or perturbations, the problem remains a difficult task. However, the
problem of practical stabilization of the infinite-dimension time-varying control systems in Hilbert
spaces has been presented in [12]. In finite dimensions one simple way of designing a compensator
is to first construct a state feedback stabilizer and an observer for the system and then to combine
the two to design a compensator using a feedback of the observer instead of the state. This is the
so-called separation principle (see [2, 3, 13, 15, 16]). In the present paper, the result is obtained
in the Hilbert space setting and, hence, can be regarded as a contribution to this class of problems.
Give a compensator design based on analysis results for cascaded systems, as done for instance in
[15, 16, 21] used a nonlinear time-varying practical observer to estimate the system states for a class
of nonlinear systems.
1 e-mail: hanen.damak@yahoo.fr.
c\bigcirc H. DAMAK, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8 1073
1074 H. DAMAK
The main contribution of this paper is the study of the problem of practical feedback stabilization
for a class of semilinear evolution equations and the design of a Luenberger observer. We refer
the reader to [11, 15, 16, 24] and the references therein for more information on this direction.
A compensator design is given under a restriction about the perturbed term that the perturbation
is bounded by an integrable function where the nominal system is supposed to be exponentially
stabilizable by a linear feedback law. We show, how under the assumptions of stabilizability and
detectability of the pairs (A,B) and (A,C), we can construct a stabilizing feedback law and a
Luenberger observer. A practical approach is obtained.
This paper is organized as follows. The system description, notations and some preliminary
results are presented in Section 2. The required assumptions and the statement of the main results
are provided in Section 3. In Section 4, an example of application of the result is given. The paper
is concluded in Section 5.
2. Preliminaries and system description. Throughout this paper we adopt the following
notations: \BbbR + denotes the set of all nonnegative real numbers, H denotes a Hilbert space with the
norm \| .\| and the inner product \langle ..\rangle . L(X) (respectively, L(X,Y )) denotes the space of all linear
bounded operators S mapping X into X (respectively, X into Y ) endowed with the norm
\| S\| = \mathrm{s}\mathrm{u}\mathrm{p}
\bigl\{
\| Sx\| : x \in X, \| x\| \leq 1
\bigr\}
.
The domain and the adjoint of an operator A are denoted by D(A) and A\ast , respectively. I denotes
the identity operator. C([0,\infty ), H) denotes the space of all continuous functions from [0,\infty ) to H.
In this paper, we consider the controlled system
\.x(t) = Ax(t) +Bu(t) +G(t, x), t \geq 0,
y = Cx,
(1)
where x \in H is the system state, u \in U is the control input, y \in Y is the measured output. H, U
and Y are assumed to be Hilbert spaces. Further, the operator A : D(A) \subset H \rightarrow H is the generator
of a C0-semigroup over H with a domain of definition D(A), B \in L(U,H) and C \in L(H,Y ).
Given an initial condition
x(t0) = x0 \in H.
Let x(t) = x(t, t0, x0, u) denote the state of a system (1) at moment t \geq t0 \geq 0 associated with an
initial condition x0 \in H at t = t0 and input u \in U.
We consider mild solutions of (1), i.e., solutions of the integral equation
x(t) = S(t - t0)x0 +
t\int
t0
S(t - s)[Bu(s) +G(s, x(s))]ds,
belonging to the class C([t0, t0+\delta ], H) for certain \delta > 0. Here \{ S(t), t \geq 0\} is a C0-semigroup on
a Hilbert space H with an infinitesimal generator A : D(A) \subset H \rightarrow H, Ax = \mathrm{l}\mathrm{i}\mathrm{m}t - \rightarrow 0+
S(t)x - x
t
,
whose domain of definition D(A) consists of those x \in H, for which this limits exists.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
COMPENSATOR DESIGN VIA THE SEPARATION PRINCIPLE FOR A CLASS . . . 1075
Definition 1. We call G : \BbbR + \times H - \rightarrow H locally Lipschitz continuous in x, uniformly in t on
bounded intervals if for every \~t \geq 0 and constant c \geq 0, there is a constant L(c, \~t ), such that\bigm\| \bigm\| G(t, u) - G(t, v)
\bigm\| \bigm\| \leq L(c, \~t )\| u - v\|
holds for all u, v \in H, with \| u\| \leq c, \| v\| \leq c, and t \in [0, \~t ].
We will use the following assumption concerning nonlinearity G throughout this paper.
(\scrH 1) We assume that G : \BbbR + \times H - \rightarrow H is continuous in t and locally Lipschitz continuous
in x, uniformly in t on bounded intervals and there exists a function \phi such that
\| G(t, x)\| \leq \phi (t) \forall t \geq 0, (2)
with
+\infty \int
0
\phi (s)ds \leq M\phi < +\infty .
The corresponding system without perturbations, called the nominal system, is described by
\.x(t) = Ax(t), x(0) = x0, t \geq 0. (3)
Next, we recall the definition of the generator of an exponentially stable semigroup as well as that of
the exponential stabilizability and detectability (see [11] for details).
Definition 2. The operator A generates an exponentially stable semigroup S(t) if the initial
value problem (3) has a unique solution x(t) = S(t)x0 and \| S(t)\| \leq Me - \alpha t for all t \geq 0 with
some positive numbers M and \alpha .
The \alpha is called the decay rate.
If S(t) is exponentially stable, then the solution to the abstract Cauchy problem (3) tends to zero
exponentially as t - \rightarrow \infty .
An important criterion for exponential stability is the following.
Lemma 1. The C0-semigroup S(t) on H is exponentially stable if and only if , for every x \in H,
there exists a positive constant \gamma x such that
\infty \int
0
\| S(t)x\| 2dt \leq \gamma x.
Definition 3. The pair \{ A,B\} is said to be exponentially stabilizable if there exists a feed-
back operator D \in L(H,U) such that the operator A + BD generates an exponentially stable
semigroup SBD.
Definition 4. The pair \{ A,C\} is said to be exponentially detectable if there exists an output
injection operator L \in L(Y, U) such that the operator A + LC generates an exponentially stable
semigroup SLC .
To study stability properties of (1) with respect to external inputs, we use the notion of practical
stabilizability.
Definition 5. System (1) is practically stabilizable if there exists a continuous feedback control
u : X \rightarrow U such that system (1) with u(t) = u(x(t)) satisfies the following properties:
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
1076 H. DAMAK
(i) for any initial condition x0 \in H, there exists a unique mild solution x(t) defined on \BbbR +;
(ii) there exist positive scalars \omega , k, and r such that the solution of the system (1) satisfies
\| x(t)\| \leq k\| x0\| e - \omega (t - t0) + r \forall t \geq t0 \geq 0.
When (i) and (ii) are satisfied for (1), we say that (1) with u(t) = u(x(t)) is globally uniformly
practically exponentially stable.
Remark 1. We deal with the practical stabilizability of (1) whose the origin is not necessarily an
equilibrium point. In the case of infinite-dimensional space, the practical stabilizability is studied by
[12] of a class of time-varying control systems having a time-varying linear part.
In what follows, we shall that V : X \rightarrow \BbbR + is a Lyapunov function.
Definition 6. The Lie derivative of V corresponding to the input u is defined by
\.V (x) = \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
t\rightarrow 0+
1
t
\bigl(
V (x(t, x, u) - V (x)
\bigr)
.
Recall that a self-adjoint operator \scrP \in L(H) is called positive if \langle \scrP x, x\rangle > 0 holds for all
x \in H\setminus \{ 0\} . A positive operator \scrP \in L(H) is called coercive if there exists k > 0 such that
\langle \scrP x, x\rangle \geq k\| x\| 2 \forall x \in D(\scrP ).
Proposition 1 (see [11, 14]). Suppose that A is the infinitesimal generator of the C0-semigroup
S(t) on the Hilbert space H. Then S(t) is exponentially stable if and only if there exists a coercive
positive self-adjoint operator \scrP \in L(H) such that
\langle Ax,\scrP x\rangle + \langle \scrP x,Ax\rangle = - \langle x, x\rangle \forall x \in D(A). (4)
Equation (4) is called a Lyapunov equation.
The following technical lemma will be needed in ours investigations.
Lemma 2 (generalized Gronwall – Bellman inequality [27]). Let \beta , \rho : \BbbR + \rightarrow \BbbR be continuous
functions and \varphi : \BbbR + \rightarrow \BbbR + a function such that
\.\varphi (t) \leq \beta (t)\varphi (t) + \rho (t) \forall t \geq t0.
Then, for any t \geq t0 \geq 0, we have the inequality
\varphi (t) \leq \varphi (t0) \mathrm{e}\mathrm{x}\mathrm{p}
\left( t\int
t0
\beta (v)dv
\right) +
t\int
t0
\mathrm{e}\mathrm{x}\mathrm{p}
\left( t\int
s
\beta (v)dv
\right) \rho (s)ds.
3. Main results. 3.1. Practical stabilization. We consider the nonlinear system (1) satisfying
the following assumptions.
(\scrH 2) The pair \{ A,B\} is exponentially stabilizable, there exists a constant operator D \in
\in L(X,U) such that a sufficient condition specially related to operator AD = A+BD is presented
in [11] as the following:
there exists a coercive positive self-adjoint operator \scrP 1
\mu I \leq \scrP 1 \leq \| \scrP 1\| I,
where \mu > 0, which satisfies
A\ast
D\scrP 1 + \scrP 1AD = - I.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
COMPENSATOR DESIGN VIA THE SEPARATION PRINCIPLE FOR A CLASS . . . 1077
We start with the following result which assures the global existence and uniqueness of mild
solutions of (1).
Lemma 3. Suppose that (\scrH 1) holds. Then the system (1) possesses a unique mild solution
x \in C([0,\infty ), H) for any x0 \in H.
Proof. The local Lipschitz of G suffices to assure that the mild solution of (1) exists and is
unique, according to a classical existence and uniqueness theorem (Theorem 1.4 in [23]).
Using the fact that u(t) = Dx(t), we have
\| x(t)\| \leq M(\| x0\| +M\phi ) +M
\left( t\int
t0
\| B\| \| D\| \| x\|
\right) ds. (5)
By applying Gronwall inequality (see [26, p. 42], Lemma 2.7) to inequality (5), any solution of this
equation is uniformly bounded
\| x(t)\| \leq M(\| x0\| +M\phi )e
M\| B\| \| D\| \delta ,
where M = \mathrm{s}\mathrm{u}\mathrm{p}
\bigl\{
\| S(t - s)\| : 0 \leq t0 \leq s \leq t \leq t + \delta
\bigr\}
on an arbitrary time interval [t0, t0 + \delta ].
Then, using Theorem 1.4 in [23], we have t0 + \delta = \infty , and so the corresponding x \in C([0,\infty ), H)
is a mild solution of (1).
The lemma is proved.
Next, sufficient conditions are presented to guarantee the practical stabilizability of a perturbed
control system using the Gronwall – Bellman inequality and Lyapunov direct method.
Theorem 1. If assumptions (\scrH 1) and (\scrH 2) are fulfilled, then the system (1) in closed-loop with
the linear feedback u(t) = Dx is globally uniformly practically exponentially stable.
Proof. Let x(t) be the solution of system (1). We consider the Lyapunov function
W (x) = \langle \scrP 1x, x\rangle .
The Lie derivative of W in t along the solution of the system (1) in the closed-loop with the
controller u(t) = Dx leads to
\.W (x) = \langle \scrP 1 \.x, x\rangle + \langle \scrP 1x, \.x\rangle =
=
\bigl\langle
\scrP 1[(A+BD)x+G(t, x)], x
\bigr\rangle
+
\bigl\langle
\scrP 1x, [(A+BD)x+G(t, x)]
\bigr\rangle
.
By using (\scrH 2) with the help of Cauchy – Schwartz inequality, we obtain
\.W (x) \leq - \langle x, x\rangle + 2\| \scrP 1\| \| G(t, x)\| \| x\| .
It follows from (2) that
\.W (x) \leq - 1
\| \scrP 1\|
W (x) + 2
\| \scrP 1\| \surd
\mu
\phi (t)
\sqrt{}
W (x).
Let
\vargamma (t) =
\sqrt{}
W (x(t)).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
1078 H. DAMAK
The derivative of \vargamma is given by
\.\vargamma (t) =
\.W (x(t))
2
\sqrt{}
W (x)
,
which implies that
\.\vargamma (t) \leq - 1
2\| \scrP 1\|
\vargamma (t) +
\| \scrP 1\| \surd
\mu
\phi (t).
According to Lemma 2, we have
\vargamma (t) \leq \vargamma (t0)e
- 1
2\| \scrP 1\|
(t - t0) +
\| \scrP 1\| \surd
\mu
t\int
t0
\phi (s)e
- 1
2\| \scrP 1\|
(t - s)
ds.
Using (\scrH 1), we get
\vargamma (t) \leq \vargamma (t0)e
- 1
2\| \scrP 1\|
(t - t0) +
\| \scrP 1\| \surd
\mu
M\phi .
We deduce that
\| x(t)\| \leq
\sqrt{}
\| \scrP 1\|
\mu
\| x0\| e
- 1
2\| \scrP 1\|
(t - t0) +
\| \scrP 1\|
\mu
M\phi \cdot
Consequently, the system (1) in closed-loop with the linear feedback u(t) = Dx is globally uniformly
practically exponentially stable.
Theorem 1 is proved.
3.2. Practical Luenberger observer. In this subsection, we use the measurements to estimate the
full state (the construction of a Luenberger observer) and to apply state feedback on the estimated
state. The Luenberger observer is a dynamical system that is expected to reconstruct the states of
the system. Our objective is to design a state reconstructor for the system (1), such that the practical
global exponential stability of the resulting error system can be guaranteed.
We shall introduce the following assumptions:
(\scrH 3) The pair \{ A,C\} is exponentially detectable, there exists a constant operator L \in L(Y,H),
such that a sufficient condition specially related to operator AL = A+LC is presented in [11] as the
following:
there exists a coercive positive self-adjoint operator \scrP 2
\nu I \leq \scrP 2 \leq \| \scrP 2\| I,
where \nu > 0, which satisfies
A\ast
L\scrP 2 + \scrP 2AL = - I.
To design a Luenberger observer, we shall consider the system
\.\^x(t) = A\^x(t) +Bu(t) +G(t, \^x) + L(\^y(t) - y(t)), t \geq 0,
\^y(t) = C\^x(t),
(6)
where \^x is the Luenberger observer with output injection L \in L(Y,H).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
COMPENSATOR DESIGN VIA THE SEPARATION PRINCIPLE FOR A CLASS . . . 1079
Define estimation error e as e = \^x - x, which is governed by
\.e(t) = \.\^x(t) - \.x(t) = (A+ LC)e(t) +G(t, \^x(t)) - G(t, x(t)), (7)
where e0 = \^x0 - x0.
The following lemma provides sufficient conditions for the global solution of (7).
Lemma 4. Under assumption (\scrH 1), the system (7) possesses a unique mild solution e \in
\in C([0,\infty ), H) for any e0 \in H.
Proof. It is known from [23] that for every initial state e0 \in H, system (7) has a unique mild
solution given by
e(t) = SLC(t - t0)e0 +
t\int
t0
SLC(t - s)
\bigl[
G(s, \^x(s)) - G(s, x(s))
\bigr]
ds, t0 \leq t \leq t0 + \delta , \delta > 0,
where SLC is the C0-semigroup of AL.
From the above equation, we get
\| e(t)\| \leq N\| e0\| +N
\left( 2
t\int
t0
\phi (s)ds
\right) ,
where N = \mathrm{s}\mathrm{u}\mathrm{p}
\bigl\{
\| S(t - s)\| : 0 \leq t0 \leq s \leq t \leq t + \delta
\bigr\}
on an arbitrary time interval [t0, t0 + \delta ].
Then, using Theorem 1.4 in [23], we have t0 + \delta = \infty , and so the system (7) has a unique mild
solution e which exists for all t \geq t0.
By arguing in exactly the same way as in Theorem 1, we prove that the output injection L can be
chosen in such a way that system (6) is a practical exponential Luenberger observer for system (1).
Theorem 2. Under assumptions (\scrH 1) and (\scrH 3), the system (6) is a practical exponential Lu-
enberger observer for the system (1).
Proof. Let e(t) be the solution of system (7). We consider the Lyapunov function candidate
Z(e) = \langle \scrP 2e, e\rangle .
The Lie derivative of Z along the trajectories of system (7) is given by
\.Z(e) = \langle \scrP 2 \.e, e\rangle + \langle \scrP 2e, \.e\rangle =
= \langle \scrP 2[(A+ LC)e+G(t, \^x) - G(t, x)], e\rangle +
+\langle \scrP 2e, [(A+ LC)e+G(t, \^x) - G(t, x)]\rangle .
By using (\scrH 3) with the help of Cauchy – Schwartz inequality, we have
\.Z(e) \leq - \langle e, e\rangle + 2\| \scrP 2\| \| G(t, \^x) - G(t, x)\| \| e\| .
It follows from (2) that
\.Z(e) \leq - \| e\| 2 + 4
\| \scrP 2\| \surd
\nu
\phi (t)
\sqrt{}
Z(e).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
1080 H. DAMAK
Let
\theta (t) =
\sqrt{}
Z(e(t)).
The derivative of \theta is given by
\.\theta (t) =
\.Z(e(t))
2
\sqrt{}
Z(e)
,
which implies that
\.\theta (t) \leq - 1
2\| \scrP 2\|
\theta (t) +
2\| \scrP 2\| \phi (t)\surd
\nu
\cdot
Applying Lemma 2 on the above inequality, we get
\theta (t) \leq \theta (t0)e
- 1
2\| \scrP 2\|
(t - t0) +
2\| \scrP 2\| M\phi \surd
\nu
\cdot
Hence,
\| e(t)\| \leq
\sqrt{}
\| \scrP 2\|
\nu
\| e0\| e
- 1
2\| \scrP 2\|
(t - t0) +
2\| \scrP 2\| M\phi
\nu
\cdot
We deduce that the system (7) is globally uniformly practically exponentially stable. Consequently,
the system (6) is a global uniform practical exponential Luenberger observer for the system (1).
Theorem 2 is proved.
3.3. Compensator design. To obtain a compensator design to (1) just consider (1) controlled by
the linear feedback control u(t) = D\^x(t) estimated by the Luenberger observer (6).
Theorem 3. Consider the nonlinear system (1) and assume that assumptions (\scrH 1), (\scrH 2) and
(\scrH 3) hold. If D \in L(H,U) and L \in L(Y,H) are such that A + BD and A + LC generate
exponentially stable semigroups, then the controller u = D\^x, where \^x is the Luenberger observer
with output injection L, stabilizes the closed-loop system. The stabilizing compensator is given by
\.\^x = (A+ LC)\^x+Bu(t) +G(t, \^x) - Ly(t),
u(t) = D\^x(t).
Proof. Under assumptions (\scrH 2) and (\scrH 3), there exist operators D and L such that SBD(t)
and SLC(t) are exponentially stable. Combining the abstract differential equations, we see that the
closed-loop system is given by the dynamics of the extended state xe =
\biggl(
\^x
e
\biggr)
,
\Biggl(
\.\^x
\.e
\Biggr)
(t) =
\Biggl(
A+BD LC
0 A+ LC
\Biggr)
\times
\Biggl(
\^x
e
\Biggr)
(t) +
\Biggl(
G(t, \^x)
G(t, \^x) - G(t, x)
\Biggr)
, t \geq 0. (8)
Let us define the Lyapunov function
Y (xe) = \alpha W (\^x) + Z(e),
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
COMPENSATOR DESIGN VIA THE SEPARATION PRINCIPLE FOR A CLASS . . . 1081
where W (\^x) = \langle \scrP 1\^x, \^x\rangle , Z(e) = \langle \scrP 2e, e\rangle and \alpha > 0 is a Lyapunov parameter to be determined.
The Lie derivative of Y along the trajectories of system (8) is given as follows:
\.Y (xe) = \alpha \.W (\^x) + \.Z(e) =
= \alpha (\langle \scrP 1
\.\^x, \^x\rangle + \langle \scrP 1\^x, \.\^x\rangle ) + \langle \scrP 2 \.e, e\rangle + \langle \scrP 2e, \.e\rangle =
= \alpha (\langle \scrP 1[A\^x+BD\^x+G(t, \^x) + LCe], \^x\rangle + \langle \scrP 1\^x,A\^x+BD\^x+G(t, \^x) + LCe\rangle +
+\langle \scrP 2[(A+ LC)e+G(t, \^x) - G(t, x)], e\rangle + \langle \scrP 2e, (A+ LC)e+G(t, \^x) - G(t, x)\rangle .
By using (\scrH 2) and (\scrH 3), with the help of Cauchy – Schwartz inequality, we obtain
\.Y (xe) \leq \alpha ( - \langle \^x, \^x\rangle + 2\| \scrP 1\| \| G(t, \^x)\| \| \^x\| +
+2\| \scrP 1\| \| LCe\| \| \^x\| ) - \langle e, e\rangle + 2\| \scrP 2\| \| G(t, \^x) - G(t, x)\| \| e\| .
It follows that
\.Y (xe) \leq \alpha
\biggl(
- 1
\| \scrP 1\|
W (\^x) + 2\| \scrP 1\| \phi (t)\| \^x\| + 2\| \scrP 1\| \| LCe\| \| \^x\|
\biggr)
-
- 1
\| \scrP 2\|
Z(e) + 4\| \scrP 2\| \phi (t)\| e(t)\| .
Let \varepsilon > 0. Using Young’s inequality
2\| \^x\| \| e\| \leq 1
\varepsilon
\| \^x\| 2 + \varepsilon \| e\| 2,
we can continue the above estimates as
\.Y (xe) \leq
\biggl(
- 1
\| \scrP 1\|
+
\| \scrP 1\| \| LC\|
\mu \varepsilon
\biggr)
\alpha W (\^x)+
+
\biggl(
- 1
\| \scrP 2\|
+
\alpha \varepsilon \| \scrP 1\| \| LC\|
\nu
\biggr)
Z(e) +
2\alpha \| \scrP 1\| \phi (t)\surd
\mu
\sqrt{}
W (\^x) +
4\| \scrP 2\| \phi (t)\surd
\nu
\sqrt{}
Z(e).
Choose \varepsilon > 0 such that
1
\| \scrP 1\|
- \| \scrP 1\| \| LC\|
\mu \varepsilon
> 0.
Let
\varepsilon =
2\| \scrP 1\| 2\| LC\|
\mu
\cdot
Also, choose for this value of \varepsilon the scalar \alpha such that
1
\| \scrP 2\|
- \alpha \varepsilon \| \scrP 1\| \| LC\|
\nu
> 0. Then let
\alpha =
\mu \nu
4\| \scrP 1\| 3\| \scrP 2\| \| LC\| 2
\cdot
We get
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
1082 H. DAMAK
\.Y (xe) \leq - \alpha
2\| \scrP 1\|
W (\^x) - 1
2\| \scrP 2\|
Z(e) +
2\alpha \| \scrP 1\| \phi (t)\surd
\mu
\sqrt{}
W (\^x) +
4\| \scrP 2\| \phi (t)\surd
\nu
\sqrt{}
Z(e).
Thus,
\.Y (xe) \leq - \lambda 1Y (xe) + \lambda 2\phi (t)
\Bigl( \sqrt{}
\alpha W (\^x) +
\sqrt{}
Z(e)
\Bigr)
with
\lambda 1 = \mathrm{m}\mathrm{i}\mathrm{n}
\biggl(
\alpha
2\| \scrP 1\|
,
1
2\| \scrP 2\|
\biggr)
and
\lambda 2 = \mathrm{m}\mathrm{a}\mathrm{x}
\biggl(
2
\surd
\alpha \| \scrP 1\| \surd
\mu
,
4\| \scrP 2\| \surd
\nu
\biggr)
.
Since
\surd
a+
\surd
b \leq 2
\surd
a+ b, for all a, b \geq 0, one can get, for all t \geq t0,\Bigl( \sqrt{}
\alpha W (\^x) +
\sqrt{}
Z(e)
\Bigr)
\leq 2
\sqrt{}
\alpha W (\^x) + Z(e),
which implies that
\.Y (xe) \leq - \lambda 1Y (xe) + 2\lambda 2\phi (t)
\sqrt{}
Y (xe).
Let
\omega (t) =
\sqrt{}
Y (xe(t)).
The derivative of \omega is given by
\.\omega (t) =
\.Y (xe(t))
2
\sqrt{}
Y (xe(t))
,
which implies that
\.\omega (t) \leq - \lambda 1
2
\omega (t) + \lambda 2\phi (t).
It follows from Lemma 2 that
\omega (t) \leq \omega (t0)e
- \lambda 1
2
(t - t0) + \lambda 2M\phi .
By using the inequality, (b1 + b2)
2 \leq 2b21 + 2b22 for all b1, b2 \geq 0, we obtain
Y (xe(t)) \leq 2Y (xe0)e
- \lambda 1(t - t0) + 2(\lambda 2M\phi )
2,
where xe0 = (\^x0, e0).
Hence,
\| \^x(t)\| \leq
\sqrt{}
2
\alpha
\Bigl[
\mathrm{m}\mathrm{a}\mathrm{x}(
\sqrt{}
\alpha \| \scrP 1\| ,
\sqrt{}
\| \scrP 2\| )(\| \^x0\| + \| e0\| )e -
\lambda 1
2
(t - t0) + \lambda 2M\phi
\Bigr]
.
Consequently, the cascade system (8) is globally uniformly practically exponentially stable.
Theorem 3 is proved.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
COMPENSATOR DESIGN VIA THE SEPARATION PRINCIPLE FOR A CLASS . . . 1083
4. Illustrative example.
Example. We consider the controlled metal bar equation
\partial x(\zeta , t)
\partial t
=
\partial 2x(\zeta , t)
\partial \zeta 2
+ b(\zeta )u(t) +
(t+ 1)e - t
1 + \| x(\zeta , t)\|
,
\partial x
\partial \zeta
(0, t) = 0 =
\partial x
\partial \zeta
(1, t), x(\zeta , 0) = x0(\zeta ), t \geq 0,
y(t) =
1\int
0
c(\zeta )x(\zeta , t)d\zeta ,
where x(\zeta , t) represents the temperature at position \zeta at time t \geq 0 and x0 represents the initial
temperature profile, u(t) the addition of heat along the bar and b, c represents the shaping functions
around the control \zeta 0 and the sensing point \zeta 1, respectively,
b(\zeta ) =
1
2\delta
\bfone [\zeta 0 - \delta ,\zeta 0+\delta ]
and
c(\zeta ) =
1
2\kappa
\bfone [\zeta 1 - \kappa ,\zeta 1+\kappa ],
with [\zeta 0 - \delta , \zeta 0 + \delta ] \cap [\zeta 1 - \kappa , \zeta 1 + \kappa ] = \varnothing and
\bfone [\vargamma ,\upsilon ](x) =
\left\{ 1, if \vargamma \leq x \leq \upsilon ,
0, otherwise.
Notice that b and c in this example are both elements in L2(0, 1) for a fixed small, nonnegative
constants \delta and \kappa .
The partial differential equation is equivalent to system (1) where H = L2(0, 1), U = \BbbC , Y = \BbbC ,
A =
\partial 2
\partial 2\zeta
, with D(A) =
\biggl\{
h \in L2(0, 1), h,
\partial h
\partial \zeta
are absolutely continuous,
\partial 2h
\partial \zeta 2
\in L2(0, 1) and
dh
d\zeta
(0) = 0 =
dh
d\zeta
(1)
\biggr\}
, the input operator
Bu = b(\zeta )u,
B \in L(\BbbC , H), and has norm
1\surd
2\delta
\cdot
Furthermore, the measured output operator
Cx =
1\int
0
c(\zeta )x(\zeta , t)d\zeta ,
where C \in L(H,\BbbC ), and has norm
1\surd
2\kappa
, and
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
1084 H. DAMAK
G(t, x) =
(t+ 1)e - t
1 + \| x(\zeta , t)\|
\cdot
A has the eigenvalues 0, - n2\pi 2, n \geq 1, and the corresponding orthogonal eigenvectors are
\upsilon n =
\left\{ 1, if n = 0,
\surd
2 \mathrm{c}\mathrm{o}\mathrm{s}(n\pi \zeta ), if n \geq 1.
It follows that A is the infinitesimal generator of the C0-semigroup (see [11] for details).
We can take as a stabilizing feedback u(t) = Dx with
Dx = - 3\langle x, \upsilon 0\rangle = - 3\langle x, 1\rangle ,
where \langle , \rangle is the inner product on L2(0, 1).
It is easy to verify that A+BD has the eigenvalues - 3, - (n\pi )2, n \geq 1. Then the pair \{ A,B\}
is exponentially stabilizable.
In addition, the stabilizing output injection is given by
Ly = - 3y\upsilon 0 = - 3y.1.
The system A+LC has the eigenvalues - 3, - (n\pi )2, n \geq 1. Then the pair \{ A,C\} is exponentially
detectable.
Moreover, the assumption (\scrH 1) is satisfied with \phi (t) = (t + 1)e - t is a continuous nonnegative
function with
+\infty \int
0
(t+ 1)e - t = M\phi = 2 < \infty .
From Theorem 3, we conclude that a stabilizing compensator is given by
\partial \^x(\zeta , t)
\partial t
=
\partial 2\^x(\zeta , t)
\partial \zeta 2
- 3
2\kappa
\zeta 1+\kappa \int
\zeta 1 - \kappa
\^x(\zeta , t)d\zeta +
1
2\delta
\bfone [\zeta 0 - \delta ,\zeta 0+\delta ](\zeta )u(t) +
(t+ 1)e - t
1 + \| \^x(\zeta , t)\|
+ 3y(t),
\partial \^x
\partial \zeta
(0, t) = 0 =
\partial \^x
\partial \zeta
(1, t), \^x(\zeta , 0) = \^x0(\zeta ), t \geq 0,
u(t) = - 3
1\int
0
\^x(\zeta , t)d\zeta .
5. Conclusion. We have presented the problems of state observation and state trajectory control
via output feedback for a class of nonlinear system. It is shown that the system can be practically
stabilizable by means of an estimated state feedback given by a designated Luenberger observer.
Furthermore, a compensator design of a class of nonlinear control systems has been considered. An
example has been introduced to validate the developed methods.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
COMPENSATOR DESIGN VIA THE SEPARATION PRINCIPLE FOR A CLASS . . . 1085
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Received 05.06.20
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| id | umjimathkievua-article-6152 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:26:14Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/2d/378e69d254ebce337f820f3590c1392d.pdf |
| spelling | umjimathkievua-article-61522022-10-24T09:23:18Z Compensator design via the separation principle for a class of semilinear evolution equations Compensator design via the separation principle for a class of semilinear evolution equations Damak, H. Damak, H. Practical stabilization, Practical observer, Compensator design, Semilinear evolution equations. 93C10, 93C25, 93B15 UDC 517.9 We establish a compensator design via the separation principle in the practical sense for a class of semilinear evolution equations in Hilbert spaces. Under a restriction imposed on the perturbation, which is bounded by an integrable function, we propose a nonlinear time-varying practical Luenberger observer to estimate the states of the system and prove that the Luenberger observer based on the linear controller stabilizes the system. An illustrative example is given to demonstrate the applicability of our theoretical results. UDC 517.9Розробка компенсатора за принципом подiлудля класу напiвлiнiйних еволюцiйних рiвняньВстановлено схему компенсатора через принцип подiлу в практичному сенсi для класу напiвлiнiйних еволюцiйних рiвнянь в гiльбертових просторах. За умов обмеження на збурення, яке обмежене iнтегровною функцiєю, запропоновано застосовувати нелiнiйного змiнного у часi практичного спостерiгача Люенбергера для оцiнки станiв системи та доведено, що спостерiгач Люенбергера на основi лiнiйного регулятора стабiлiзує систему. Наведено наочний приклад, що демонструє можливiсть застосування наших теоретичних результатiв. Institute of Mathematics, NAS of Ukraine 2022-10-04 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6152 10.37863/umzh.v74i8.6152 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 8 (2022); 1073 - 1085 Український математичний журнал; Том 74 № 8 (2022); 1073 - 1085 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6152/9287 Copyright (c) 2022 Damak Hanen |
| spellingShingle | Damak, H. Damak, H. Compensator design via the separation principle for a class of semilinear evolution equations |
| title | Compensator design via the separation principle for a class of semilinear evolution equations |
| title_alt | Compensator design via the separation principle for a class of semilinear evolution equations |
| title_full | Compensator design via the separation principle for a class of semilinear evolution equations |
| title_fullStr | Compensator design via the separation principle for a class of semilinear evolution equations |
| title_full_unstemmed | Compensator design via the separation principle for a class of semilinear evolution equations |
| title_short | Compensator design via the separation principle for a class of semilinear evolution equations |
| title_sort | compensator design via the separation principle for a class of semilinear evolution equations |
| topic_facet | Practical stabilization Practical observer Compensator design Semilinear evolution equations. 93C10 93C25 93B15 |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6152 |
| work_keys_str_mv | AT damakh compensatordesignviatheseparationprincipleforaclassofsemilinearevolutionequations AT damakh compensatordesignviatheseparationprincipleforaclassofsemilinearevolutionequations |