Asymptotic behavior of a class of perturbed differential equations
UDC 517.9 This paper deals with the problem of stability of nonlinear differential equations with perturbations. Sufficient conditions for global uniform asymptotic stability in terms of Lyapunov-like functions and integral inequality are obtained. The asymptotic behavior is studied in the sense tha...
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| author | Dorgham, A. Hammi, M. Hammami , M. A. A. Dorgham, A. Hammi, M. Hammami , M. A. |
| author_facet | Dorgham, A. Hammi, M. Hammami , M. A. A. Dorgham, A. Hammi, M. Hammami , M. A. |
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| description | UDC 517.9
This paper deals with the problem of stability of nonlinear differential equations with perturbations. Sufficient conditions for global uniform asymptotic stability in terms of Lyapunov-like functions and integral inequality are obtained. The asymptotic behavior is studied in the sense that the trajectories converge to a small ball centered at the origin. Furthermore, an illustrative example in the plane is given to verify the effectiveness of the theoretical results.
 
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| doi_str_mv | 10.37863/umzh.v73i5.232 |
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DOI: 10.37863/umzh.v73i5.232
UDC 517.9
A. Dorgham, M. Hammi, M. A. Hammami (Univ. Sfax, Tunisia)
ASYMPTOTIC BEHAVIOR OF A CLASS
OF PERTURBED DIFFERENTIAL EQUATIONS
АСИМПТОТИЧНА ПОВЕДIНКА ОДНОГО КЛАСУ
ЗБУРЕНИХ ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ
This paper deals with the problem of stability of nonlinear differential equations with perturbations. Sufficient conditions
for global uniform asymptotic stability in terms of Lyapunov-like functions and integral inequality are obtained. The
asymptotic behavior is studied in the sense that the trajectories converge to a small ball centered at the origin. Furthermore,
an illustrative example in the plane is given to verify the effectiveness of the theoretical results.
Розглядається задача стiйкостi нелiнiйних диференцiальних рiвнянь iз збуреннями. Отримано достатнi умови гло-
бальної рiвномiрної асимптотичної стiйкостi у термiнах функцiй типу Ляпунова та iнтегральних нерiвностей. Асим-
птотична поведiнка вивчається в сенсi того, що траєкторiї збiгаються до малої кулi iз центром у початку координат.
Крiм того, з метою перевiрки ефективностi отриманих теоретичних результатiв наведено вiдповiдний приклад на
площинi.
1. Introduction. The problem of Lyapunov stability of time-varying differential equations has
attracted the attention of serval authors and has produced a large important results [1 – 6]. The two
major approaches for Lyapunov stability analysis [11, 15 – 19] are the linearization method and the
direct method. Stability of a system can be investigated via the first linearization method, but in
general and the most powerful technique is the second direct method. For this method one usually
assumes the existence of the so-called Lyapunov function which is a positive definite function with
negative derivative along the trajectories of the system. Therefore, because of the existence of
some perturbations, stability are defined in terms of the behavior of solutions under other restrictions
imposed on the term of perturbation. The natural assumption is to consider some stability property
for the unperturbed system with some information on the bound of the perturbed term. Many works
are given in this sense [4, 10, 13, 17]. The authors in [7, 8] introduce the concept of exponential
rate of convergence and for a specific classes of uncertain systems where they present controllers
which guarantee this behavior. In particular, when the origin is not necessarily an equilibrium point,
in this case one can study the asymptotic behavior of the solutions in a neighborhood of the origin.
This approach is used to study systems whose desired behavior is asymptotic stability about the
origin of the state space or a close approximation to this [9, 10, 12 – 14]. In [18], the authors
studied the robust stability of nonlinear systems, which admit simultaneously disturbances on the
structure of the system and the initial conditions. This kind stability requires that the disturbing
function is bounded. However, the system may oscillate close to the state, in such situation for
allowable uncertainties and nonlinearities, we can estimate the region of attraction from which all
solutions converge to a small ball containing the origin of the state space. The usual techniques
is to use the Lyapunov function associated to the nominal system as a Lyapunov candidate for the
perturbed system (see [16]). The idea used in [4] is to add in the Lyapunov function associated to the
nominal system a special function which is chosen such that the derivative along the trajectories of
c\bigcirc A. DORGHAM, M. HAMMI, M. A. HAMMAMI, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5 627
628 A. DORGHAM, M. HAMMI, M. A. HAMMAMI
the system in presence of perturbations is definite negative. Another approach is to use some integral
inequalities of Gronwall – Belmann type to deduce the asymptotic stability of systems in presence of
perturbations (see [14]). In this paper, under some restrictions on the perturbed term, we show that
all state trajectories are bounded and approach a sufficiently small neighborhood of the origin. One
also desires that the state approaches the origin (or some sufficiently small neighborhood of it) in a
sufficiently fast manner, in this sense, we prove that the solutions of the perturbed systems converge
to a small ball centered at the origin. Furthermore, we provide some sufficient conditions for the
exponential stability of a class of perturbed systems based on a new nonlinear inequality. Finally, a
numerical example is given to show the efficiency and accuracy of the method.
2. Stability analysis. We consider the first unperturbed nonlinear system described by the
equation
\.x = f(t, x), t \geq 0, (1)
where x \in \BbbR n and f : \BbbR + \times \BbbR n \rightarrow \BbbR n is continuous in (t, x) and locally Lipschitz in x uniformly
in t. We suppose that f(t, 0) = 0, t \geq 0, in this case the origin is an equilibrium point of (1). The
second one, is a perturbed system given by
\.y = f(t, y) + g(t, y), t \geq 0, (2)
where f, g : \BbbR + \times \BbbR n \rightarrow \BbbR n is continuous in (t, y) and locally Lipschitz in y uniformly in t.
The function g(., .) is the perturbation term which could result from errors in modeling a non-
linear system, aging of parameters or uncertainties and disturbances. In practice, if we know some
information on the upper bound of this term and if the associated nominal system (1) has a uniformly
asymptotically stable equilibrium point at the origin, what we can say about the stability behavior of
the perturbed system (2). Unless otherwise stated, we assume throughout the paper that the function
f(., .) encountered is sufficiently smooth. We often omit arguments of function to simplify notation,
\BbbR n is the n-dimensional Euclidean vector space; \BbbR + is the set of all nonnegative real numbers;
\| x\| is the Euclidean norm of a vector x \in \BbbR n. Br = \{ x \in \BbbR n/\| x\| \leq r, r > 0\} denotes the
ball centered at the origin and of radius r > 0. For all x0 \in \BbbR n and t0 \in \BbbR +, we will denote by
x(t, t0, x0), or simply by x(t), the unique solution of (1) (respectively, by y(t, t0, y0), or simply by
y(t), the unique solution of (2)) at time t0 starting from the point x0 (respectively, starting from the
point y0 at time t0). We recall now some standard comparison functions which are used in stability
theory to characterize the stability properties and uniform asymptotic stability (see [11, 16]): \scrK is
the class of functions \BbbR + \rightarrow \BbbR + which are zero at the origin, strictly increasing and continuous.
\scrK \infty is the subset of \scrK functions that are unbounded. \scrL is the set of functions \BbbR + \rightarrow \BbbR + which are
continuous, decreasing and converging to zero as their argument tends to +\infty . \scrK \scrL is the class of
functions \BbbR + - \rightarrow \BbbR + which are class \scrK on the first argument and class \scrL on the second argument.
We will consider more general case when we do not know that g(t, 0) = 0 for all t \geq 0. The
origin may not be an equilibrium point of the perturbed system (2). We can non longer study the
stability of the origin as an equilibrium point, nor should we expect the solution of the perturbed
system to approach the origin as t goes to infinity. The best we can hope that for a small perturbation
term the solution approach to the a small set which contains the origin. The asymptotic behavior of
the solutions of (2) can be studied in a neighborhood of the origin, in this case the solutions converge
to a certain small ball. Let consider the system (2), in the case where g(t, 0) \not = 0 for a certain t \geq 0.
We introduce some basic definitions and preliminary facts which we shall need in the sequel. First,
we give the definition of global uniform asymptotic stability of a ball Br (see [3, 9]).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
ASYMPTOTIC BEHAVIOR OF A CLASS OF PERTURBED DIFFERENTIAL EQUATIONS 629
Definition 1. Let r > 0 be a positive number. The ball Br is said to be globally uniformly
asymptotically stable for (1), if there exists a class \scrK \scrL functions \beta such that the solution of (1) from
any initial state x0 \in \BbbR n and initial time t0 \in \BbbR + satisfies the estimation
\| x(t)\| \leq \beta (\| x0\| , t - t0) + r for all t \geq t0 \geq 0.
The next definition concerns the global uniform exponential stability.
Definition 2. Br is globally uniformly exponentially stable, if there exist \gamma > 0 and k > 0 such
that, for all t \geq t0 \geq 0 and x0 \in \BbbR n,
\| x(t)\| \leq k\| x0\| \mathrm{e}\mathrm{x}\mathrm{p}( - \gamma (t - t0)) + r.
Note that, in the above definition, if we take r = 0, then one deals with the standard concept of the
global exponential stability of the origin viewed as an equilibrium point. Moreover, if g(t, 0) \not = 0,
we shall study the asymptotic behavior of a small ball centered at the origin for 0 \leq \| x(t)\| - r
\forall t \geq t0 \geq 0, so that the initial conditions are taken outside the ball Br.
Lyapunov’s direct method allows us to determine the stability of a system without explicitly
integrating the differential equation.
Definition 3. Let V : \BbbR + \times \BbbR n \rightarrow \BbbR + a Lyapunov function
V (t, 0) = 0 \forall t \geq 0,
V (t, x) > 0 \forall (t, x) \in \BbbR + \times \BbbR n \setminus \{ 0\} .
(i) V (t, x) is positive definite, i.e., there exists a continuous, nondecreasing scalar function \alpha (.)
such that \alpha (0) = 0 and
0 < \alpha (\| x\| ) \leq V (t, x) \forall y \not = 0.
(ii) \.V (t, x) is negative definite, that is,
\.V (t, x) \leq - \gamma (\| x\| ) < 0,
where \gamma (.) is a continuous nondecreasing scalar function such that \gamma (0) = 0.
(iii) V (t, x) \leq \beta (\| x\| ), where \beta (.) is a continuous nondecreasing function and \beta (0) = 0, i.e.,
V is decrescent, i.e., the Lyapunov function is upper bounded.
(iv) V is radially unbounded, that is, \alpha (\| x\| ) \rightarrow \infty as \| x\| \rightarrow \infty .
The classic Lyapunov direct criterion establishes that, given a nonlinear system, the existence of
a smooth positive definite function with the property of having a nonpositive derivative along the
solutions implies that the origin of the system is stable. If such derivative happens to be negative,
the origin is asymptotically stable. If, in addition, the Lyapunov function of the system, is radially
unbounded, then the origin is globally asymptotically stable. Furthermore, converse Lyapunov the-
orems ensure that the existence of a Lyapunov function is also a necessary condition for stability.
The classic Lyapunov result for a nonlinear nonautonomous system can found in [11]. The following
theorems give sufficient conditions to ensure global stability of a ball. Their proofs can be deduced
from [7 – 9].
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
630 A. DORGHAM, M. HAMMI, M. A. HAMMAMI
Theorem 1. Consider system (1) and suppose that there exist a continuously differentiable func-
tion V : \BbbR + \times \BbbR n \rightarrow \BbbR , a class \scrK \infty functions \alpha 1(.), \alpha 2(.), a class \scrK function \alpha 3(.) and a small
positive real number \varrho such that the inequalities hold, for all t \geq t0 \geq 0 and x \in \BbbR n :
\alpha 1(\| x\| ) \leq V (t, x) \leq \alpha 2(\| x\| ),
\partial V
\partial t
+
\partial V
\partial x
f(t, x) \leq - \alpha 3(\| x\| ) + \varrho .
Then the ball
Br = \{ x \in \BbbR n/\| x\| \leq \alpha - 1
1 \circ \alpha 2 \circ \alpha - 1
3 (\varrho )\}
is globally uniformly asymptotically stable with
r = \alpha - 1
1 \circ \alpha 2 \circ \alpha - 1
3 (\varrho ).
Now we state sufficient conditions for global exponential stability (see [3]).
Theorem 2. Consider system (1) and let V : [0,+\infty [\times \BbbR n \rightarrow \BbbR be a continuously differentiable
Lyapunov function such that
c1\| x\| 2 \leq V (t, x) \leq c2\| x\| 2,
\partial V
\partial t
+
\partial V
\partial x
f(t, x) \leq - c3V (t, x) + \rho
for all t \geq t0 \geq 0 and x \in \BbbR n, where c1, c2 and c3 are positive constants. Then the ball
Br = \{ x \in \BbbR n/\| x\| \leq
\sqrt{}
\rho /c1c2\}
is globally uniformly exponentially stable with
r =
\sqrt{}
\rho /c1c2.
3. Stability of perturbed system. We study in this section the asymptotic behavior of perturbed
systems via different approaches for certain classes of perturbed differential equations.
3.1. Lyapunov approach. There exist equations of the form (1) such that the origin is globally
uniformly asymptotically stable, but there exists bounded perturbations such that (2) becomes unstab-
le. This motivates us to study the problem of uniform stability of perturbed systems by assuming that
the nominal associated system is globally uniformly asymptotically stable and some assumptions on
the size of perturbations.
We assume that the perturbation term g(., .) satisfies the uniform bound:
\| g(t, y)\| \leq \theta (t)\| y\| + \xi (t) \forall t \geq t0 \geq 0 \forall y \in \BbbR n, (3)
where \theta (.), \xi (.) \in C[\BbbR +,\BbbR +] such that
\mathrm{s}\mathrm{u}\mathrm{p}
t\geq t0
\theta (t) = \eta < +\infty
and the function \xi (t) satisfies
\mathrm{s}\mathrm{u}\mathrm{p}
t\geq t0
\xi (t) = \kappa < +\infty .
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
ASYMPTOTIC BEHAVIOR OF A CLASS OF PERTURBED DIFFERENTIAL EQUATIONS 631
For the asymptotic behavior of equation (2), we shall suppose for system (1), that there exists a
continuously differentiable function V (., .) : \BbbR + \times \BbbR n \rightarrow \BbbR which satisfies the inequalities
\alpha 1(\| x\| ) \leq V (t, x) \leq \alpha 2(\| x\| ), (4)
\partial V
\partial t
+
\partial V
\partial x
f(t, x) \leq - \alpha 3(\| x\| ) + \varrho (t), (5)\bigm\| \bigm\| \bigm\| \bigm\| \partial V\partial x
\bigm\| \bigm\| \bigm\| \bigm\| \leq \alpha 4(\| x\| ) (6)
in [0,\infty )\times \BbbR n, where \alpha i(.), i = 1, 2, 3, 4, are class \scrK \infty functions and \varrho (.) \in C[\BbbR +,\BbbR +] such that
\mathrm{s}\mathrm{u}\mathrm{p}
t\geq t0
\varrho (t) = \~\varrho < +\infty .
Theorem 3. Suppose that the conditions (3), (4) – (6) hold and\bigl(
\kappa \alpha 4 + \eta \alpha 4.\alpha
- 1
1 \circ \alpha 2
\bigr)
< l\alpha 3, 0 < l < 1. (7)
Then the ball
Br = \{ x \in \BbbR n/\| x\| \leq \alpha - 1
1 \circ \alpha 2 \circ \alpha - 1
0 (\~\varrho )\}
is globally uniformly asymptotically stable respecting the system (2).
Proof. We use V (., .) as a Lyapunov function candidate for the perturbed system (2). The
derivative of V (t, y) along the trajectories of (2) gives
\.V (t, y) \leq - \alpha 3(\| y\| ) + \rho (t) +
\bigm\| \bigm\| \bigm\| \bigm\| \partial V\partial y
\bigm\| \bigm\| \bigm\| \bigm\| \| g(t, y)\| \leq
\leq - \alpha 3(\| y\| ) + \alpha 4(\| y\| )
\bigl(
\eta \| y\| + \kappa
\bigr)
+\varrho (t).
Thus,
\.V (t, y) \leq - \alpha 3(\| y\| ) +
\bigl(
\kappa \alpha 4 + \eta \alpha 4.\alpha
- 1
1 \circ \alpha 2
\bigr)
(\| y\| ) + \~\varrho .
The last expression in conjunction with the fact that the functions \alpha i, i = 1, . . . , 4, satisfy (7), yields
\.V (t, y) \leq - (1 - l)\alpha 3(\| y\| ) + \~\varrho .
Let \alpha 0(r) = (1 - l)\alpha 3(r), 0 < l < 1. One has \alpha 0 \in \scrK \infty , and by using Theorem 1, we deduce that
the ball Br with
r = \alpha - 1
1 \circ \alpha 2 \circ \alpha - 1
0 (\~\varrho )
is globally uniformly asymptotically stable.
Remark 1. If l approaches to zero in the inequality (7), then the radius r of the above ball
decreases.
For the exponential behavior of equation (2), we shall suppose for system (1) that there exists a
continuously differentiable function V : \BbbR + \times \BbbR n \rightarrow \BbbR which satisfies the inequalities
c1\| x\| 2 \leq V (t, x) \leq c2\| x\| 2, (8)
\partial V
\partial t
+
\partial V
\partial x
f(t, x) \leq - c3V (t, x), (9)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
632 A. DORGHAM, M. HAMMI, M. A. HAMMAMI\bigm\| \bigm\| \bigm\| \bigm\| \partial V\partial y
\bigm\| \bigm\| \bigm\| \bigm\| \leq \alpha 4(\| x\| ) (10)
for all t \geq 0 and x \in \BbbR n, where c1, c2, c3 and c4 are positive constants.
Theorem 4. Suppose that the conditions (3), (8) – (10) hold and
\eta <
c3c1
c4
.
Then the ball
Br = \{ x \in \BbbR n/\| x\| \leq \alpha - 1
1 \circ \alpha 2 \circ \alpha - 1
0 (\~\varrho )\}
is globally uniformly exponentially stable respecting the system (2).
Proof. We use V (., .) as a Lyapunov function candidate for the perturbed system (2). The
derivative of V (t, y) along the trajectories of (2) gives
\.V (t, y) \leq - c3V (t, y) +
\bigm\| \bigm\| \bigm\| \bigm\| \partial V\partial y
\bigm\| \bigm\| \bigm\| \bigm\| \| g(t, y)\| \leq
\leq - c3V (t, y) + c4(\| y\| )
\bigl(
\eta \| y\| + \kappa
\bigr)
.
Thus,
\.V (t, y) \leq -
\biggl(
c3 -
c4\eta
c1
\biggr)
V (t, y) +
c4\kappa \surd
c1
\sqrt{}
V (t, y).
By assumption, the term c3 -
c4\eta
c1
> 0. Let
v(t) =
\sqrt{}
V (t, y).
The derivative with respect time gives
\.v(t) \leq - 1
2
\biggl(
c3 -
c4\eta
c1
\biggr)
v(t) +
1
2
c4\kappa \surd
c1
.
It implies that
v(t) \leq v(t0) \mathrm{e}\mathrm{x}\mathrm{p} -
1
2
\biggl(
c3 -
c4\eta
c1
\biggr)
(t - t0) +
c4\kappa \surd
c1
c3 -
c4\eta
c1
.
The last expression in conjunction with (8) yields
\| y(t)\| \leq
\surd
c2\surd
c1
\| y(0)\| \mathrm{e}\mathrm{x}\mathrm{p} - 1
2
\biggl(
c3 -
c4\eta
c1
\biggr)
(t - t0) +
c4\kappa
c1c3 - c4\eta
.
Hence, the ball Br with
r =
c4\kappa
c1c3 - c4\eta
is globally uniformly exponentially stable.
Next, we will prove a new nonlinear generalized Gronwall – Bellman-type integral inequality, and,
as application, we give a new class of time-varying perturbed systems which is globally uniformly
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
ASYMPTOTIC BEHAVIOR OF A CLASS OF PERTURBED DIFFERENTIAL EQUATIONS 633
exponentially stable in the sense that the trajectories converge to a small ball centered at the origin.
Moreover, we give an example to illustrate the applicability of the result.
3.2. Integral inequality approach. We consider now, the nonlinear perturbed differential equa-
tion
\.y = A(t)y + g(t, y). (11)
Here, the nominal system is supposed linear, where A(.) is continuous bounded matrix defined on
\BbbR + and g : \BbbR + \times \BbbR n \rightarrow \BbbR n is continuous in (t, y) and locally Lipschitz with respect y uniformly
in t.
The natural assumption is to consider some stability property for the unperturbed system with
some information on the bound of the perturbed term where the nominal system is supposed to be
globally uniformly asmptotically stable.
Suppose that, for
\.x = A(t)x,
x = 0 is globally uniformly exponentially stable equilibrium point, this is equivalent to say that
\| \Phi (t, t0)\| \leq k \mathrm{e}\mathrm{x}\mathrm{p}( - \gamma (t - t0)), t \geq t0 \geq 0, (12)
where k > 0, \gamma > 0, \Phi (t, t0) is the state transition matrix of the matrix A(t).
We assume that the perturbation term g(., .) satisfies the uniform bound
\| g(t, y)\| \leq \theta (t)\| y\| + \xi (t) \forall t \geq t0 \geq 0 \forall y \in \BbbR n, (13)
where \theta (.), \xi (.) \in C[\BbbR +,\BbbR +].
The solution of the perturbed equation which starts at (t0, y0) is given by
y(t) = \Phi (t, t0)y(t0) +
t\int
t0
\Phi (t, s)g(s, y(s))ds, t \geq t0 \geq 0.
By using (12), we obtain
\| y(t)\| \leq ke - \gamma (t - t0)\| y(t0)\| +
t\int
t0
ke - \gamma (t - s)\| g(s, y(s))\| ds.
Thus,
e\gamma t\| y(t)\| \leq ke\gamma t0\| y(t0)\| +
t\int
t0
ke\gamma s\| g(s, y(s))\| ds.
By using inequality (13) imposed on the term of perturbation, we have
e\gamma t\| y(t)\| \leq ke\gamma t0\| y(t0)\| +
t\int
t0
ke\gamma s
\bigl(
\theta (s)\| y(s)\| + \xi (s)
\bigr)
ds.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
634 A. DORGHAM, M. HAMMI, M. A. HAMMAMI
It follows that
e\gamma t\| y(t)\| \leq ke\gamma t0\| y(t0)\| +
t\int
t0
k\theta (s)e\gamma s\| y(s)\| + (ke\gamma s\xi (s))
\bigr)
ds, t \geq t0 \geq 0. (14)
To obtain an estimation on the solutions of the perturbed system (11), we need the following lemma
which a generalization of Gronwall type integral inequality.
Lemma 1. Let u(.), \vargamma (.) and \zeta (.) be continuous functions on [0,+\infty ) such that \vargamma (.) \geq 0. We
suppose that the inequality
u(t) \leq c+
t\int
a
(\vargamma (s)u(s) + \zeta (s))ds for all t \geq a (15)
holds, where a and c are arbitrary positive constants. Then
u(t) \leq ce
\int t
a \vargamma (\tau )d\tau +
t\int
a
\zeta (s)e
\int t
s \vargamma (\tau )d\tau ds for all t \geq a.
Proof. For t \geq a, we put
F (t) =
t\int
a
(\vargamma (s)u(s) + \zeta (s))ds,
then F is differentiable on [a,+\infty ) and verifies
F \prime (t) = \vargamma (t)u(t) + \zeta (t).
By using the inequality (15), we obtain
F \prime (t) \geq c\vargamma (t) + \vargamma (t)F (t) + \zeta (t).
Let y the maximal solution of the linear equation
y\prime (t) = \vargamma (t)y(t) + c\vargamma (t) + \zeta (t)
with y(a) = 0. It is clear that
y(t) = ce
\int t
a \vargamma (s)ds - c+
t\int
a
\zeta (s)e
\int t
s \vargamma (\tau )d\tau .
By comparison, we deduce that
F (t) \leq y(t).
Finally, we obtain
u(t) \leq ce
\int t
a \vargamma (\tau )d\tau +
t\int
a
\zeta (s)e
\int t
s \vargamma (\tau )d\tau ds for all t \geq a.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
ASYMPTOTIC BEHAVIOR OF A CLASS OF PERTURBED DIFFERENTIAL EQUATIONS 635
Lemma 2. Let \theta \in Lp(\BbbR +,\BbbR +), where p \in [1,+\infty ) \cup \{ +\infty \} . We denote by \| \theta \| p the p-norm
of \theta . Then, for all t \geq t0 \geq 0,
t\int
t0
\theta (u)du \leq \| \theta \| p
p
+
\biggl[ \biggl(
1 - 1
p
\biggr)
\| \theta \| p
\biggr]
(t - t0).
Proof. We first consider the case p \in (1,+\infty ). By using Hölder inequality to the function \theta ,
we obtain, for all t \geq t0,
t\int
t0
\theta (\sigma )d\sigma \leq
\left( t\int
t0
\theta p(\sigma )d\sigma
\right)
1
p
\left( t\int
t0
d\sigma
\right)
p - 1
p
\leq
\leq (t - t0)
p - 1
p
\left( +\infty \int
t0
\theta p(\sigma )d\sigma
\right)
1
p
.
We put
f(x) =
1
p
+
p - 1
p
x - x
p - 1
p \forall x \geq 0,
then f is continuous on [0,+\infty ) and differentiable on (0,+\infty ) and verifying
f \prime (x) =
p - 1
p
\Bigl(
1 - x
- 1
p
\Bigr)
.
Hence, f is decreasing on [0, 1] and increasing on [1,+\infty ). Since f(1) = 0, we conclude that f is
positive on [0,+\infty ) which means that
x
p - 1
p \leq 1
p
+
p - 1
p
x \forall x \geq 0.
Consequently, we have
(t - t0)
p - 1
p \leq 1
p
+
p - 1
p
(t - t0) \forall t \geq t0.
Then
t\int
t0
\theta (u)du \leq \| \theta \| p
p
+
\biggl[ \biggl(
1 - 1
p
\biggr)
\| \theta \| p
\biggr]
(t - t0).
This inequality holds also for p \in \{ 1,+\infty \} .
Theorem 5. Suppose that the conditions (12), (13) hold and \xi \in L\infty (\BbbR +,\BbbR +), \theta \in Lp(\BbbR +,\BbbR +)
with
\| \theta \| p <
\biggl(
1 +
1
p - 1
\biggr)
\gamma
k
.
Then the ball
Br =
\left\{ x \in \BbbR n/\| x\| \leq k\| \xi \| \infty e
k\| \theta \| p
p
\gamma - k
\biggl(
1 - 1
p
\biggr)
\| \theta \| p
\right\}
is globally uniformly exponentially stable respecting the system (11).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
636 A. DORGHAM, M. HAMMI, M. A. HAMMAMI
Proof. Since the conditions (12) and (13) are satisfied, then one gets an estimation as in (14),
e\gamma t\| y(t)\| \leq ke\gamma t0\| y(t0)\| +
t\int
t0
(k\theta (s)e\gamma s\| y(s)\| + ke\gamma s\xi (s))ds.
Let
u(t) = e\gamma t\| y(t)\| .
Then, by applying Lemma 1, we obtain
u(t) \leq ku(t0)e
k
\int t
t0
\theta (s)ds
+
t\int
t0
ke\gamma s\xi (s)ek
\int t
s \theta (\tau )d\tau ds.
Since
\| y(t)\| = e - \gamma tu(t),
we obtain the estimation
\| y(t)\| \leq k\| y(t0)\| e
\int t
t0
k\theta (s)ds - \gamma (t - t0) +
t\int
t0
k\xi (s)e
\int t
s k\theta (\tau )d\tau - \gamma (t - s)ds.
Then
\| y(t)\| \leq k\| y(t0)\| e
k
\int t
t0
\theta (s)ds - \gamma (t - t0) + k\| \xi (t)\| \infty
t\int
t0
ek
\int t
s \theta (\tau )d\tau +\gamma (s - t)ds.
Now, by using Lemma 2, we get
t\int
t0
\theta (u)du \leq \| \theta \| p
p
+
\biggl[ \biggl(
1 - 1
p
\biggr)
\| \theta \| p
\biggr]
(t - t0).
Consequently,
\| y(t)\| \leq k\| y(t0)\| ek
\| \theta \| p
p e
-
\Bigl(
\gamma - k
\Bigl(
1 - 1
p
\Bigr)
\| \theta \| p
\Bigr)
(t - t0) + k\| \xi \| \infty e
k
\| \theta \| p
p
t\int
t0
e
\Bigl(
\gamma - k
\Bigl(
1 - 1
p
\Bigr)
\| \theta \| p
\Bigr)
(s - t)
ds.
Using the fact
\| \theta \| p <
\biggl(
1 +
1
p - 1
\biggr)
\gamma
k
,
which implies \gamma - k
\biggl(
1 - 1
p
\biggr)
\| \theta \| p > 0. Finally, we obtain the estimation
\| y(t)\| \leq ke
k
\| \theta \| p
p \| y(t0)\| e
-
\Bigl(
\gamma - k
\Bigl(
1 - 1
p
\Bigr)
\| \theta \| p
\Bigr)
(t - t0) +
k\| \xi \| \infty e
k
\| \theta \| p
p
\gamma - k
\biggl(
1 - 1
p
\biggr)
\| \theta \| p
.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
ASYMPTOTIC BEHAVIOR OF A CLASS OF PERTURBED DIFFERENTIAL EQUATIONS 637
Hence, the ball Br is globally uniformly exponentially stable with
r =
k\| \xi \| \infty e
k\| \theta \| p
p
\gamma - k
\biggl(
1 - 1
p
\biggr)
\| \theta \| p
.
3.3. Example. Consider the system
\.x1 = - x1 - tx2 +
1
(1 + t2)2
x21
1 +
\sqrt{}
x21 + x22
+
e - t
1 + t2)(1 + x21)
,
\.x2 = tx1 - x2 +
t
(1 + t2)2
x22
1 +
\sqrt{}
x21 + x22
(16)
which can be writing as
\.x = A(t)x+ h(t, x),
where
x =
\Biggl(
x1
x2
\Biggr)
,
A(t) =
\Biggl(
- 1 - t
t - 1
\Biggr)
and
h(t, x) =
\Biggl(
h1(t, x)
h2(t, x)
\Biggr)
with
h1(t, x) =
1
(1 + t2)2
x21
1 +
\sqrt{}
x21 + x22
+
e - t
(1 + t2)(1 + x21)
,
h2(t, x) =
t
(1 + t2)2
x22
1 +
\sqrt{}
x21 + x22
.
It is clear that the system
\.x = A(t)x
is globally uniformly asymptotically stable. Indeed, the transition matrix R(t, t0) satisfies
R(t, t0) = e - (t - t0)
\Biggl(
\mathrm{c}\mathrm{o}\mathrm{s} t - \mathrm{s}\mathrm{i}\mathrm{n} t
\mathrm{s}\mathrm{i}\mathrm{n} t \mathrm{c}\mathrm{o}\mathrm{s} t
\Biggr)
,
thus, we obtain
\| R(t, t0)\| = ke - \gamma (t - t0)
with \gamma = k = 1.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
638 A. DORGHAM, M. HAMMI, M. A. HAMMAMI
On the other hand,
\| h(t, x)\| 2 = h21(t, x) + h22(t, x) \leq
\leq 1
(1 + t2)2
(x21 + x22) +
2e - t
(1 + t2)2
.
By using the classic inequality \sqrt{}
a2 + b2 \leq a+ b \forall a, b \geq 0,
we get
\| h(t, x)\| \leq \phi (t)\| x(t)\| + \varepsilon (t) \forall t \geq 0,
where
\phi (t) =
1
1 + t2
and
\varepsilon (t) =
\surd
2e -
t
2
1 + t2
.
It is easy to verify that \phi and \varepsilon are continuous, positive and bounded on [0,+\infty ). In particular,
\| \varepsilon \| \infty =
\surd
2, \phi \in Lp(\BbbR +,\BbbR +)
and
\| \phi \| p =
(2(p - 1))!
22(p - 1)((p - 1)!)2
\pi
2
\forall p \in [1,+\infty ).
Then
\| \phi \| p < 1 +
1
p - 1
\forall p \geq 1,
and we can apply Theorem 5 to prove that the ball Br, where
r =
\surd
2e
(2(p - 1))!
22(p - 1)((p - 1)!)2
\pi
2
p
1 -
\biggl(
1 - 1
p
\biggr)
(2(p - 1))!
22(p - 1)((p - 1)!)2
\pi
2
,
is globally uniformly exponentially stable respecting the system (16).
Remark 2. The above example show that the trajectories of the system converge exponentially
to a small ball centered at the origin under some sufficient conditions on the perturbed term. This
fact motivated to study systems whose desired behavior is asymptotic stability about the origin of the
state space or a close approximation to this, e.g., all state trajectories are bounded and approach a
sufficiently small neighborhood of the origin. Quite often, one also desires that the state approaches
the origin (or some sufficiently small neighborhood of it) in a sufficiently fast manner. So, one can
introduce a small parameter \epsilon > 0 on the perturbations so that when \epsilon \rightarrow 0 the solutions of the
system tend to zero when t goes to infinity.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
ASYMPTOTIC BEHAVIOR OF A CLASS OF PERTURBED DIFFERENTIAL EQUATIONS 639
4. Conclusion. Sufficient conditions for global uniform exponential stability of perturbed sys-
tems are obtained using Lyapunov techniques or integral inequalities approach. It is shown that
the trajectories converge to a small ball centered at the origin where a new nonlinear generalized
Gronwall – Bellman-type integral inequality is proved. As an application, we give a new class of
time-varying perturbed systems which is globally uniformly exponentially stable. Moreover, an ex-
ample is given to illustrate the applicability of the main result.
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Received 18.07.18,
after revision — 24.01.19
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 5
|
| id | umjimathkievua-article-232 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:18Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/01/5592af6f4dd4f7a1f77698d875995401.pdf |
| spelling | umjimathkievua-article-2322025-03-31T08:48:07Z Asymptotic behavior of a class of perturbed differential equations Asymptotic behavior of a class of perturbed differential equations Dorgham, A. Hammi, M. Hammami , M. A. A. Dorgham, A. Hammi, M. Hammami , M. A. Differential equations perturbations Lyapunov function stability Differential equations perturbations Lyapunov function stability UDC 517.9 This paper deals with the problem of stability of nonlinear differential equations with perturbations. Sufficient conditions for global uniform asymptotic stability in terms of Lyapunov-like functions and integral inequality are obtained. The asymptotic behavior is studied in the sense that the trajectories converge to a small ball centered at the origin. Furthermore, an illustrative example in the plane is given to verify the effectiveness of the theoretical results. &nbsp; &nbsp; УДК 517.9 Асимптотична поведінка одного класу збурених диференціальних рівнянь Розглядається задача стійкості нелінійних диференціальних рівнянь із збуреннями. Отримано достатні умови глобальної рівномірної асимптотичної стійкості у термінах функцій типу Ляпунова та інтегральних нерівностей. Асимптотична поведінка вивчається в сенсі того, що траєкторії збігаються до малої кулі із центром у початку координат. Крім того, з метою перевірки ефективності отриманих теоретичних результатів наведено відповідний приклад на площині. Institute of Mathematics, NAS of Ukraine 2021-05-24 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/232 10.37863/umzh.v73i5.232 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 5 (2021); 627 - 639 Український математичний журнал; Том 73 № 5 (2021); 627 - 639 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/232/9015 Copyright (c) 2021 Mohamed Ali Hammami |
| spellingShingle | Dorgham, A. Hammi, M. Hammami , M. A. A. Dorgham, A. Hammi, M. Hammami , M. A. Asymptotic behavior of a class of perturbed differential equations |
| title | Asymptotic behavior of a class of perturbed differential equations |
| title_alt | Asymptotic behavior of a class of perturbed differential equations |
| title_full | Asymptotic behavior of a class of perturbed differential equations |
| title_fullStr | Asymptotic behavior of a class of perturbed differential equations |
| title_full_unstemmed | Asymptotic behavior of a class of perturbed differential equations |
| title_short | Asymptotic behavior of a class of perturbed differential equations |
| title_sort | asymptotic behavior of a class of perturbed differential equations |
| topic_facet | Differential equations perturbations Lyapunov function stability Differential equations perturbations Lyapunov function stability |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/232 |
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