On the construction of regions of stability

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Pure and Applied Mathematics Journal
2014; 3(4): 87-91
Published online August 30, 2014 (http://www.sciencepublishinggroup.com/j/pamj)
doi: 10.11648/j.pamj.20140304.12
ISSN: 2326-9790 (Print); ISSN: 2326-9812 (Online)
On the construction of regions of stability
Luciano Miguel Lugo1, Juan Eduardo Nápoles Valdés1, 2, Samuel Iván Noya2
1
2
Facultad de Ciencias Exactas, UNNE, Av. Libertad 5540 (3400), Corrientes, ARGENTINA
Facultad Regional Resistencia, UTN, French 414 (3500), Resistencia, ARGENTINA
Email address:
[email protected] (L. M. Lugo), [email protected] (J. E. Nápoles Valdés), [email protected] (S. I. Noya)\
To cite this article:
Luciano Miguel Lugo, Juan Eduardo Nápoles Valdés, Samuel Iván Noya. On the Construction of Regions of Stability. Pure and Applied
Mathematics Journal. Vol. 3, No. 4, 2014, pp. 87-91. doi: 10.11648/j.pamj.20140304.12
Abstract: In this paper we built a stability region around the origin for the Liénard equation (4) to ensure stability and
boundedness of solutions of this equation, without making use of the classical Second Method of Lyapunov. We compare
our result with some others proposed by different authors.
Keywords: Lyapunov, Trajectories, Asymptotic Equilibrium
1. Introduction
The term “stability” originates in Mechanic to
characterize the equilibrium of a rigid body. So, the
equilibrium is called stable if the body returns to its original
position, having been “disturbed” by being moved slightly
from its position of rest. If the body after a slight
displacement tends toward a new position its equilibrium is
called unstable.
The Second Method of Lyapunov has been established as
the most general method to study the stability of
equilibrium positions of systems described by differential,
differences or functional equations (or systems). This
method was found in classical memory of Alexander
Mijaílovich Lyapunov 1 , published in Russian in 1892,
translated into French in 1907 (reprinted 40 years later 2 )
1
Born on June 6, 1857 in Yaroslavl, Russia and died on November 3, 1918 in
Odessa, Russia.
2
Lyapunov (1949).
and in English many years later3. In this work a key role is
played by the calls Lyapunov functions (functions of
energy from the physical point of view).
Lyapunov’s second method also determines the criteria
for asymptotic stability. In addition to giving us these
criteria, it gives us the way of estimating region of
asymptotic stability. Asymptotic stability is one of the
major areas of the qualitative theory of dynamical systems
and is of paramount importance in many applications of the
theory in almost all fields where dynamical effects play a
great role.
In the analysis of region of asymptotic stability
properties of invariant objects, it is very useful to employ
what is now called Lyapunov’s second method. It is an
important method to determine region of asymptotic
stability. This method relies on the observation that
asymptotic stability is very well linked to the existence of a
Lyapunov’s function, that is, a proper, nonnegative function,
vanishing only on an invariant region and decreasing along
those trajectories of the system not evolving in the invariant
region. Lyapunov proved that the existence of a Lyapunov’s
function guarantees asymptotic stability and, for linear
time-invariant systems, also showed the converse statement
that asymptotic stability implies the existence of a
Lyapunov’s function in the region of stability.
An excellent source for the study of this method (also
called direct method because this method allows us to
determine the stability and asymptotic stability of a system
without explicitly integrating the nonlinear differential
3
Lyapunov (1962) and Lyapunov (1992), in this last is included a biography by
Smirnov and an extensive bibliography of Lyapunov's work.
Pure and Applied Mathematics Journal 2014; 3(4): 87-91
equation) is the renamed text Yoshizawa of the sixty4. In
the qualitative study of a nonlinear system, whether
autonomous
x’=F(x); x∈Rn,
x
Let F(x)= f (u ) du and y=x’+F(x), then (3) can written as
∫
0
a system:
(1)
 x' = y − F ( x)

 y ' = − g ( x)
or non-autonomous
x’= F(x,t); x∈Rn,
(2)
suppose that F(t,x) is continuous in (t,x) on IxD, where D is
a connected open set in Rn, I denote the interval 0≤t<∞ and
Rn denote the Euclidean n-space with the norm x . In the
qualitative theory some of the most studied qualitative
properties are stability, asymptotic stability and the
boundedness (also called continuability)5:
The solution x=x*(t) of (2) is stable in the Lyapunov
sense, if for any ε>0 and any t0∈I, there exists δ(t0,∈)>0
such that if
x * (t 0 ) − x (t 0 ) < δ then x * (t ) − x(t ) <ε, ∀t≥t0.
The solution x=x*(t) of (2) is asymptotically stable in the
Lyapunov sense, if x*(t) and if there exists a δ0(t0)>0 such
that if
x * (t 0 ) − x(t 0 ) < δ 0 (t 0 ) then x * (t ) − x(t ) → 0 as t→∞.
Remark1. In many physical situations, the origin may not
be asymptotic stable for all possible initial value (t0,x0) but
only for initial value contained in some region around the
origin, such a region is called the region of asymptotic
stability and the value of δ allows to define a neighborhood
of x*(t0), commonly called base attraction. If this
neighborhood coincides with the whole space, then it is
said to be asymptotically stable in the global sense.
A solution x=x*(t) of (2) is bounded, if there exists a β>0
such that x * (t ) < β for all t≥t0, where β may depend on
each solution. In other words if and only if for all T>t0 we
have lim x(t ) < ∞ .
t →T
−
One way to ensure that these properties are fulfilled for
all solutions of the system, is to propose a bounded region
Ω around equilibrium point in which remain all those
solutions beginning on Ω, ie, if x(t0)∈Ω, then x(t)∈Ω, for
all t>t0. Throughout the work, and for convenience, we take
t0=0.
In this paper we consider the Liénard equation:
x’’+f(x)x’+g(x)=0; x∈R, t≥0
(3)
where f is a continuous function, g is a derivable function
and the following assumptions are fulfilled:
a) f(x)>0, if x≠0,
b) xg(x)>0, if x≠0.
Yoshizawa (1966).
5
Yoshizawa (1966), p.27, p.28 and p.36.
(4)
This system has the origin as a single equilibrium point,
so the properties will be referred to the trivial solution
x(t)=y(t)=0.
In our work we will need the following basic results6.
Theorem A. Let Ω be a bounded neighborhood of the
origin and let Ωc be its complement. Assume that W(x) is a
scalar function with continuous first partials in Ωc and
satisfying:
1. W(x)>0, ∀x∈Ωc,
2. W´(x)≤0, ∀x∈Ωc,
3. W(x)→∞ as x→∞.
Then each solution of (1) is bounded for all t≥0.
Theorem B. Let V(x) be a scalar function with
continuous first parcials satisfying:
1. V(x)>0, ∀x≠0,
2. V´(x)≤0, ∀x,
3. V(x)→∞ as x→∞.
If V´ is not identically zero along any solution other than
the origin, then the system (1) is completely stable7.
The main difficulty in using Theorem B often is that one
can construct a Lyapunov function satisfying the three
requirements. Hence it is much easier to study the
boundedness of the solutions as a separate problem, from
which arises the need to build appropriate regions where
we can ensure the boundedness.
Building the stability region of a given equation is
another way to study the problem of convergence, as t
tends to infinity, of all solutions of this equation. This
problem is of a paramount relevance in the qualitative
theory.
The purpose of this note is to construct a new stability
region for equation (3), using a different approach of earlier
results and without making uses of common conditions.
First we summarize know results, we present by illustration
the proof of first, and later we present our results.
2. Preliminary Results
While there are some previous results in the fifties, the
first result of this nature was obtained by LaSalle in 1960,
when he showed that all solutions of (4) are stable and
bounded using an appropriate bounded region.
2.1. Region 18
Theorem 1. Under assumptions a) and b) if we have
6
4
88
Cf. Theorems 4 and 5 of LaSalle (1960).
Continuable for us.
8
LaSalle (1960).
7
89
Luciano Miguel Lugo et al.: On the Construction of Regions of Stability
x
F (x) =
∫ f (u) du →+∞ as
x →∞,
0
then, all solutions of (2) are stable and bounded.
Proof. Let (x(t),y(t)) be a solution of (4), and let l and a
positive real numbers such that (x(0), y(0)) Ω, where Ω is
the region:
1

Ω =  ( x, y ) ∈ R 2 / V ( x, y ) = y 2 + G ( x ) < l ∧
2

Where
V(x,y)
is
the
x
y2
V(x, y) =
+ G ( x) with G ( x) = g ( s )ds
2
∫
( y + F ( x) )2 <

a

Lyapunov
Function
. However, since it may
Because g ( x) ( y + F ( x) ) >0. Hence, (x(t),y(t)) cannot
leave Ω, and every solution is bounded for t≥0. Thus, under
somewhat different conditions we have again shown that
Liénard’s equation is completely stable.
Remark 2. If we consider (3) under an external force
x´´+f(x)x´+g(x)=p(t) with p a continuous function and of
class L1[0,+∞), this result still valid and the proof is
practically the same10. It's funny, comparing the similarity
of the paper of LaSalle and Hasan and Zhu, and dates of
both, how in the second job is not mentioned first,
suggesting a lack thereof.
2.2. Region 211
0
not be true that G(x)→∞ as x→∞, we can conclude only
that every solution bounded for all t≥0 approaches the
origin as t→∞. Thus, to establish complete stability, we
need to show that all solutions are bounded for t≥0. To do
this, we consider the region Ω (see figure of the Region 1).
Theorem 2. Under condition b) we suppose that
∃a>0/ 0<x<a⇒xF(x)>0.
Then all solution of (4) are stable and bounded.
Remark 3. It is clear that the above condition is more
general than a).
In the proof we consider the same Lyapunov’s function
1
V ( x, y ) = y 2 + G ( x) . And we take Ω as the infinite band:
2
{
}
Ω = ( x, y ) ∈ R 2 / − a ≤ x ≤ a .
In this region we have:
V’(4)(x,y)=yy’+g(x)x’=–yg(x)+g(x)(y-F(x))=–g (x)F(x)≤0.
So we have V’(4)(x,y)≤0 in any subset of Ω. Let Cλ be the
region definite by Cλ = {( x, y ) ∈ R 2 / V ( x, y ) ≤ λ } , with λ is a
Region 1
For any l and a, Ω is a bounded region9. Let (x(t),y(t) be
any solution, and select l and a so large that the solution
starts in Ω. Then the solution cannot leave Ω without
crossing the boundary of Ω. It must cross either V=l or
y+F(x)=a, or y+F(x)=-a. We can select a suficiently large
that the part of y+F(x)=a which is the boundary of Ω
correspond to x>0 and the part of y+F(x)=-a corresponds to
x<0.
The derivative of V(x,y) along the system (4) is:
positive real number such that Cλ⊂Ω. It is clear that the
better value of λ is the lowest value between G(a) and G(–a).
Again we have the no positivity of V’(4)(x,y) we
guarantee the boundedness and stability of solutions
starting in Cλ (Region 2 in the figure bellow).
V’(4)=yy’+g(x)x’=
–yg(x)+g(x)(y-F(x))=–g(x)F(x)≤0, ∀(x, y)∈R2,
(5)
From (5) we have that a solution starting inside Ω cannot
cross V=l.
Now in the rest of boundary of Ω we have
d
2
( y + F ( x) ) = 2 ( y + F ( x) )( y ' + f ( x) x ')
dt
Region 2
2.3. Our Region12
= 2 ( y + F ( x) ) ( − g ( x) + f ( x) ( y − F ( x) ) )
While the construction of the stability region can be at
= 2 ( − g ( x) ( y + F ( x) ) + f ( x) ( y − F ( x) )( y + F ( x) ) )
= −2 a g ( x) < 0
10
See Hasan and Zhu (2007).
Cf. Yadeta (2013).
12
A preliminary version of this result was presented at the Annual Meeting of
the UMA last year. See Lugo, Nápoles and Noya (2013).
11
9
Cf. Miller and Michel (1982).
Pure and Applied Mathematics Journal 2014; 3(4): 87-91
times by the Second Method of Lyapunov, we shall
disregard it in the results presented below.
We next give a stability region allowing us to obtain
sufficient conditions for the boundedness and stability of
solutions of the system (4), and consequently for the
equation (3).
Theorem 3. If in addition to conditions a) and b) we have
value of a is fixed in advance, the proposed region
Cλ can not cover all solutions of the equation.
Region 3 has the advantage that it was not needed
to define a Lyapunov function (so, in certain sense
is a converse theorem), and also includes any
solution to the equation, since this region is
constructed when the initial condition. The
disadvantage is that it has 2 points of intersection
between the given edges where no one can speak of
derived.
As an attempt to overcome the above mentioned,
Guidorizzi constructed a family Lyapunov
functions 13 which we extended 14 to nonautonomous
Liénard
equation
x´´+f(x)x´+a(t)g(x)=0 taking as a Lyapunov
function for the family
f∈Fλ,g(R)={f∈C(R)/f(x)–λg(x)>0 if x>0; f(x)+λg(x)>0 if
x<0},
then all solutions of (4) are stable and bounded.
Proof. Let (x(t),y(t)) be a solution of (4) with initial
value(x(0),y(0))=(x0,y0), and we take the region:


1
k
Ω kG =  ( x , y )∈ R 2 / α ≤ x ≤ β ; y ≤ − k G( x ) +  .
Vα (t , x, y ) =
Where k is a positive real number satisfying the equation
1
y 0 = − k G ( x 0 ) + ; and α and β are the solutions of equation
k
1 (note that proposed equation to find the value of
G( x ) = 2
k
k is equivalent to asking that the solution “start” on the
boundary of the region).
Calculating the slopes of the boundary we have:
− kg ( x) =
dy
g ( x) ,
=−
dx
x'
kg ( x) =
90
1
Wα ( x, y ) + G ( x) .
a (t )
y − F ( x)
x
∫
∫
0
0
With G( x) = g (s)ds and Wα ( x, y ) =
sds
αs + 1
which
allowed us to define Ωα as the following sets:
Ωα≡R2 if α≡0,
from here we have that x'= 1 if y>0,
Ωα={(x,y): y>F(x)-α-1} if α>0,
k
Ωα={(x,y): y<F(x)-α-1} if α<0.
dy
g ( x)
, where x' = − 1 if y<0.
=−
dx
x'
k
•
The result obtained in the Theorem 3 is consistent
with some previous results of the second author 15.
References
[1]
Acosta, J., L. M. Lugo, J. E. Nápoles V. and S. I. Noya
(2013)-“On some qualitative properties of a nonautonomous
Liénard equation”, submitted.
[2]
Guidorizzi, H. L. (1996)-“The family of functions Sα,k and
the Liénard equation”, Tamkang J. of Math. 27, 37-54.
[3]
Hasan, Y. Q. and L. M. Zhu (2007)-“The bounded solutions
of Liénard equation”, J. Applied Sciences 7(8), 1239-1240.
[4]
LaSalle, J. P. (1960)-“Some Extensions of Lyapunov’s
Second Method”, IRE Transactions on Circuit Theory, Dec,
520-527.
[5]
Lugo L. M., J. E. Nápoles V. and S. I. Noya (2013)-“About a
region of boundedness for some nonautonomous Lienard’s
Equation”, Annual Meeting of the UMA, Universidad
Nacional de Rosario, September 17-20 (Spanish).
[6]
Lyapunov, A. M. (1949)-“Problème general de la stabilité du
movement”, Annals of Math. Studies, No.17, Princeton
University Press, Princeton, N.J.
Our Region
So, if we have
x' =
1
k
if y>0 and
x' = −
1
k
if y<0, the
trajectories that begin at the boundary of this region, “fall”
into the same, which ensures the stability and continuability
of the solution considered (see Our Region in the previous
figure).
3. Final Remarks
Region 1 has 4 points of intersection in the given
boundary, and those points can not speak of derived.
The Region 2 does not have that problem, but as the
13
Guidorizzi (1996).
Cf. Acosta, Lugo, Nápoles and Noya (2013).
15
Nápoles (2000) and Nápoles and Ruiz (1997).
14
91
Luciano Miguel Lugo et al.: On the Construction of Regions of Stability
[7]
Lyapunov, A. M. (1966)-“Stability of Motion”, Academic
Press, New-York & London, 1966.
[8]
Lyapunov, A. M. (1992)-“The General Problem of the
Stability of Motion”, (A. T. Fuller trans.) Taylor & Francis,
London 1992.
[9]
Miller, R. K. and A. N. Michel (1982)-„Ordinary
Differential Equation“, New York, Lawa Stat University.
[10] Nápoles Valdes, J. E. (2000)-“On the boundedness and
global asymptotic stability of Liénard equation with
restoring term”, Revista de la Unión Matemática Argentina
41(4), 47-59.
[11] Nápoles, J. E. and A. I. Ruiz (1997)-“Convergence in
nonlinear systems with a forcing term”, Revista de
Matemática: Teoría y Aplicaciones 4(1): 1-4.
[12] Yadeta, Z. (2013)-“Lyapunov´s Second Method for
Estimating Region of Asymptotic Stability”, Open Science
Repository
Mathematics,
online,
doi:
dx.doi.org/10.7392/Mathematics.70081944, available in
http://www.open-science-repository.com/lyapunovs-secondmethod-for-estimating-region-of-asymptotic-stability.html
[13] Yoshizawa, T. (1966)-“Stability theory by Liapunov´s
Second Method”, The Math. Soc. of Japan.

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