doi: **10.7392/Mathematics.70081944**

This seminar basically consists of three chapters. The first chapter converses the background about Lyapunov’s second method of estimating region of asymptotical stability (RAS) and mathematical preliminaries which will be needed in the next two chapters. The second chapter deals with Lyapunov’s second method for estimating region of asymptotic stability of autonomous nonlinear differential system. Chapter three deals with estimating region of asymptotic stability of nonautonomous system. Finally I summarize Lyapunov’s second method of estimating RAS.

**Keywords: **Lyapunov’s second method, asymptotic stability, autonomous nonlinear differential system, nonautonomous system.

**Citation: **Yadeta, Z. (2013). Lyapunov’s Second Method for Estimating Region of Asymptotic Stability. *Open Science Repository Mathematics*, Online(open-access), e70081944. doi:10.7392/Mathematics.70081944

**Received:** February 21, 2013

**Published: **March 20, 2013

**Copyright:** © 2013 Yadeta, Z . Creative Commons Attribution 3.0 Unported License.

**Contact:** research@open-science-repository.com

The subject of this analysis is to estimate region of asymptotic stability using Lyapunov’s second method in differential equations (nonlinear dynamic system). This technique was discovered by Lyapunov’s in the 19

Lyapunov’s second method also determines the criteria for stability and asymptotical 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 stone areas of the qualitative theory of dynamical systems and is of fundamental 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 often 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.

Converse theorems are interesting because they show the universality of Lyapunov’s second method. If an invariant object is asymptotically stable then there exists a Lyapunov’s function. Thus there is always the possibility that we may actually find it, though this may be hard.

When we mention Lyapunov’s second method to estimate region of stability, particularly for asymptotical stability, we have to know first the properties of asymptotical stability of

dynamic system at equilibrium point = 0

1) x* = 0 is stable, and

2) x* = 0 is locally attractive; i.e., there exists such that

The main objective of this seminar is to estimate region of asymptotical stability of nonlinear differential equation for both autonomous and nonautonomous systems using Lyapunov’s second method. This directly influences the exact estimating region of the asymptotic stability.

Hence this
seminar is deliberately to explore the following specific objectives:

• To
provide the methods for determining region of asymptotical stability (RAS)

• To
analyze and demonstrate region of asymptotic stability basically using
Lyapunov’s second method.

• To
analyze some related theorem, application and examples of region of
asymptotical stability.

**1.3. Mathematical
preliminaries**

In
this subtopic we deal with some mathematical results which will be needed later
and we state some general principles related to Lyapunov’s second method to
estimating region of asymptotic stability

**1.3.1 Lyapunov’s
functions**

Lyapunov’s
functions are a powerful function for determining the stability or instability
of fixed points of nonlinear autonomous systems. To explain the method, we
begin with a few definitions. We study an autonomous system

* *(1.1)

We
assume that *x *and *f*(*x*) are in.
The domain of *f *may be an open subset of,
but we will only be interested in what happens near a particular point, so we
won't complicate the notation by explicitly describing the domain of *f*.

Let
*V *be a real valued function defined on an open subset *U *of and let *p *be a point in *U*.
We say that V is positive definite with respect to p (or just positive definite,
if *p *is understood) if for all *x ** **U
*and *V *(*x*) = 0 if and only
if *x *= *p*.

Suppose
that *V *in . We want to examine
the derivative of *V *along trajectories of the

system
Suppose
that is a solution of Then,
we have
* *
* *(1.2)

where
denotes the usual
dot product inand

,

is
the gradient vector of *V. * Since *x *is a solution of (1.1) we may
rewrite (1.2)

Thus,
if we define a function we have .

It
is trivial, but important, to observe that we can compute * without *having to solve the
system explicitly.
Where V(x) is Lyapunov’s functions

**1.3.2 Stability in the sense of Lyapunov’s second method**
** **

Let * *(1.3) be dynamic system that satisfies all
the standard condition existence and uniqueness of solutions. A point is equilibrium point of the
system, if .

**Definition:
**The equilibrium point x*
= 0 (assume x* is isolated critical point) of
equation (1.3) is
stable (in the sense of Lyapunov’s*) *at if
for any * *there
existssuch
that * **<
δ* * *.* *Lyapunov’s
stability is a very gentle requirement on equilibrium points. In particular, it
does not require that trajectories starting close to the origin tend to the
origin asymptotically. Also, stability is defined at a time instant. Uniform stability* *is a
concept which assures that the equilibrium point is not losing stability. We
state that for a uniformly stable equilibrium point,
*δ *in the
definition of stability in the sense of Lyapunov’s not be a function of , so that equation

* **<
δ* * ** *(1.4)

may
hold for all.

Asymptotic
stability is made precise in the following definition. In many physical
situations, the origin may not be asymptotic stable for all possible initial
conditions but only for initial conditions contained in region around the
origin, such a region is called the region of asymptotic stability or finite
region of attraction. If such region exists, computation of such a region is of
great interest to the system designer.

**Definition:**
An equilibrium point
x* =
0 of (1) is
*asymptotically stable *at if

x*
= 0 is stable,
and

x*
= 0 is locally attractive; i.e., there exists such
that

* **<
δ ** ** *(1.5)

As
in the previous definition, asymptotic stability is defined at *t*_{0}.

Uniform
asymptotic stability* *requires:

x*
= 0 is uniformly stable,
and

x*
= 0 is uniformly locally
attractive; i.e., there exists *δ *independent
of *t*_{0}
for which equation (1.5) holds. Further, it is
required that the convergence in equation (1.5) is uniform. In other words, a
solution is called asymptotic stable if it
is stable and attractive. Finally, we say that an equilibrium point is unstable*
*if it is not stable.

In another way, the region of asymptotic stability of the origin is defined
as the set of all points X_{0 }that and
.

So, we can find an open invariant set Ω with boundary such that Ω is the region of
asymptotic stability (RAS) of system (1.1) defined by the property that every
trajectory starting from reaches the
equilibrium point of the corresponding system.**
**
** **

a) Stable in the sense of Lyapunov b) Asymptotically stable

To state Lyapunov’s second method of estimating region of asymptotic stability first define the two definitions concerned with the construction of certain scalar function. Let V(x) be scalar continuous function, i.e. real-valued function of the variables defined on the region containing the origin again Ω could be the whole space.

V(0) =0 and V(x)>0 for and x in.

Note that V*(x) can be computed directly from the differential equation without any knowledge of the solutions. Herein lies the power of Lyapunov’s method. We observe that if is any solution of (1.1), by rule the definition of solution, and (1.6) we have

In other words, along a solution the total derivative of V() with respect to t coincides with the derivatives of V with respect to the system evaluated at x=.

and the given function

We obtain, using the definition (1.6) with

With these definitions, we may state the basic theorem as follows.

Let U be an open neighborhood of p and let be a continuous function that is positive definite with respect to p and is C

a. If on, then p is a stable fixed point for

b. If on, then p is an asymptotically stable fixed point for

Geometrically, the condition says that the vector field

A function satisfying the conditions in the theorem and (

Since

of and

for all

for every

Consider a point mass with mass m attached to a spring, with the mass constrained to move along a line. Let y denote the position of the mass, with y = 0 the equilibrium position. Moving the mass in the positive y direction stretches the spring and moving the mass in the negative y direction compresses the spring. According to Hooke's law, the force exerted by the spring is

F = -ky,

for some constant k > 0 (this is an approximation that is good for displacements that are not to large).

Applying Newton's law F = ma, we see that the differential equation governing the motion of the mass is:

Appropriate initial conditions are the initial position and velocity of the mass. Of course, we can solve this equation explicitly to show that all of the motions are periodic and the equilibrium position (y = 0, velocity = 0) is stable. But, for illustration, let's prove that the equilibrium is stable using a Lyapunov function.

There is an obvious choice from physics, the total energy of the system.

The work done against the spring in moving from 0 to y is

This is the potential energy of the system at position y. The kinetic energy is , where v is velocity.

The total energy is

To apply Lyapunov's theorem, we need to convert the second order equation to a first order system. Using x

Looking

This is clearly positive definite with respect to the origin. The gradient of V is

Thus

This shows that the origin is stable.

The origin is a fixed point. Take , which is clearly positive definite with respect to the origin. Rather than computing , we can do the same computation by differentiating V (x, y), assuming that x and y are solutions of (i). Thus, we have

Thus, on, so 0 is an asymptotically stable fixed point for the system (i).

See figure below for a picture of the phase portrait of this system near the origin.

.

1. If V (x, t) is locally positive definite and locally in x and for all t, then the origin of the system is locally stable (in the sense of Lyapunov’s).

2. If V (x, t) is locally positive definite and decrescent, and locally in x and for all t, then the origin of the system is uniformly locally stable (in the sense of Lyapunov’s).

3. If V (x, t) is locally positive definite and decrescent, and is locally positive definite, then the origin of the system is uniformly locally asymptotically stable.

4. If V (x, t) is positive definite and decrescent, and is positive definite, then the origin of the system is globaly uniformly asymptotically stable.

Since the theorem 2 only gives sufficient conditions, the search for a Lyapunov’s function establishing stability of an equilibrium point could be arduous. However, it is a remarkable fact that the converse of this theorem also exists. If an equilibrium point is stable, then there exists a function

We present a converse Lyapunov’s result for nonlinear time-varying systems that are uniformly semi globally asymptotically stable. This stability property pertains to the case when the size of initial conditions may be arbitrarily enlarged and the solutions of the system converge, in a stable way, to a closed ball that may be arbitrarily diminished by change a design parameter of the system (typically but not exclusively, a control gain). This result is notably useful in cascaded-based control when uniform practical asymptotic stability is established without a Lyapunov’s function. We provide a tangible example by solving the stabilization problem of a hovercraft.

We have a non-increasing function V() of t which is bounded below. Therefore exists. Suppose that for some we could have

for

We will show that (**) is impossible. By continuity, for the , there exists a 0< such that whenever ||x||<

.

Therefore, the solution for which (**) holds must satisfy |||| for t Let S be the set of y lying between the spheres of radius and r, that is, S={x|}. Consider the function –V*(x) on the closed bounded set S. By hypothesis on f and V, -V*(y) define (1.6) is continuous and positive definite.

Let

Since 0 is not a point of S we have using also (1.7)

()

Integrating, we obtain V( for . But then clearly for t large enough V() is negative, which is an obvious contradiction. Thus (**) is impossible and we must have which implies that. Since this holds for every solution with ||x||<, this completes the proof.

This chapter deals with the application of Lyapunov’s second method on the estimating region of asymptotic stability. It is a technique of estimating region of asymptotic stability solution of differential equation (nonlinear autonomous) of

and

So, we can find an open invariant set Ω with boundary such that Ω is the region of asymptotic stability (RAS) of system (2.1) defined by the property that every trajectory starting from reaches the equilibrium point of the corresponding system.

Before discussing the Lyapunov’s second method for estimating region of asymptotic stability of autonomous System, let us state one theorem which is very important for estimating region of asymptotic stability.

and so is decreasing function of t. Therefore remains in for all. Since is closed and bounded, the positive limit set L(C

None of the results up to now, with the exception of theorem 2.1 has given any indication of the size region of asymptotic stability. Let be an open set in containing the origin. Let there exist a positive definite scalar function V(x) which, with respect to (2.1) as in. From corollary 1 and theorem 2.1 we might be lead to the guess that the set is contained in the region of asymptotic stability of the zero solution of (2.1).

Consider the set the component of = {} containing the origin, for. For we get the origin. When n=2 for small we get closed bounded regions containing the origin and contained in. But this can fail to be true when becomes too large. In this case the sets can extend outside (and they may even be bounded); of course, the difficulty is that for such contains points at which needs not hold. However, from theorem 2.1 we can at least say that every closed bounded region contained in lies in the region of asymptotic stability, and is considerably better than anything we have been able to say up to now. We can compute the large, so that has the property, as the largest values of for which the component of = {} containing the origin actually meets the boundary of Ω. We can also say, even when Ω is unbounded, that all those sets that are completely contained in Ω are positively invariant sets with respect to (2.1) starting in at t=0 bounded , and thus tends to the origin.

This shows that such a set must be contained in the region of asymptotic stability.

The region of asymptotic stability is at least as large as the largest invariant set contained in.

In particular, the interior of is contained in region of asymptotic stability.

We shall now show these ideas by actual examples.

Consider the scalar equation

(*)

Let , then it is written as a system

There are only two critical points (-1,0) and (0,0) by using the linear approximation. The first is unstable and the second is asymptotically stable. However, we know nothing about the region of stability. To estimate region of asymptotic stability we have to use corollary (1) and theorem 2.1. We note that (*) is a Lienard equation with g(u)=u+u

implies

This function is positive definite on the set together with the origin and sketched in figure 2.2.

We have , so that on the whole plane, in particular, on to apply theorem 2.1 on above extent of asymptotic stability, we look at the subset E of given by

This is clearly y

We see that the points (-1,0), (0,0) are the only invariant subsets of E, because at all other points of E, . Clearly, that is with the point (-1, 0) deleted. Then, at least, the origin is the only invariant subset of that part of E which lies is. The boundary of consists of the and the point (-1, 0).We now look at the regions

those lie in. A little consideration shows that the curves V(y

However, it is almost certainly a larger set than. Notice that we cannot expect to enlarge our estimate by enlarging in such a way that the point (-1, 0) would be included.

Let us consider the region, which is together with the interior of the rectangle whose vertices are (0, 0) (-1, 0), as shown figure 2.3 below.

If we can show that no solution can leave across the left and top edge of the rectangle, we will have that is contained in the region of asymptotic stability. On the left edge of the figure . Then from (**) and. On the top edge, while and here the function assumes its maximum value at the point and from (**) we have.

Thus no solution starting in can leave through the edge y

Where g(0)=0, ug(u)>0 () for and for some

0<|u|<

the method already employed (corollary 1, theorem 2.1 above) that the zero solution is asymptotic stable. Again, the problem is to determine the region of asymptotic stability. Here we employ a different equivalent system, namely

where

Equation (*) and (**) are equivalent. To show the equivalence, let . Define and try the V function which is positive in the whole plane. Now) and since by hypothesis has the sign of, we have immediately that on the strip

Ω=

If we take the region separately as and, then. This implies origin is stable, and from we see that the set of theorem 2.1 is the y

But, since for () in E

We see that the origin is the only invariant subset of E and thus the origin is (corollary1 and theorem 2.1) asymptotically stable. We now wish to consider the curves for with increasing value of beginning with =0. These are closed curves symmetric about the axis. For decreases and for increases. Thus the curves, first makes contacts with the boundary of Ω at one of the point (-

Hence, directly from theorem 2.1 every solution starting in approaches the origin (see figure 2.5).

We present a converse Lyapunov’s result for nonlinear time-varying systems that are uniformly semi globally asymptotically stable. This stability property pertains to the case when the size of initial conditions may be arbitrarily enlarged and the solutions of the system converge, in a stable way, to a closed ball that may be arbitrarily diminished by tuning a design parameter of the system (typically but not exclusively, a control gain). This result is especially useful in cascaded based control when uniform practical asymptotic stability is established without a Lyapunov’s. function,

In many problems, in control theory or nuclear reactor dynamics, for examples, in which the zero solution of 2.1 is known to be asymptotically stable, it is important to determine whether all solutions, no matter what their initial values may be, approach the origin. In other words, we wish to determine whether the region of asymptotic stability is the whole space. If this is the case, we say the zero solution of 2.1 is globally asymptotically stable. As we will see, everything depends on finding a “good enough” V function. For, with enough hypotheses on the V function, it is very easy to give a criterion for global asymptotic stability.

If V is positive definite with respect to 2.1, and if in the set E= {x|V*(x)=0} the origin is the only invariant subset, then the zero solution of 2.1 is asymptotically stable (corollary 1) and all bounded solution of 2.1 approach zero as (corollary 2). Thus we only need to prove a result that insures that all solution of 2.1 bounded. The bounded of the solution is often called langrage stability. Let above V function is the additional property

Therefore (a constant) and the hypothesis

i. V(x) is positive definite on

ii. With respect to ,

iii. The origin is the only invariant subset of the set

Then the zero solution of is globally asymptotic stable.

From above theorem we have the following consequence.

where g(u) is continuously differential for all u and

as

we assert that the zero solution is globally asymptotically stable.

If we choose the familiar V function

then we have already shown, using it and corollary 1, theorem 2.1 that the origin is asymptotic stable. Since hypothesis (i) of theorem 2.2 is clearly fulfilled because of the requirement as , the result follows from theorem 2.

Occasionally it happens that, there is difficulty to find a V function that satisfies all the three condition of theorem 2.2. In that case it can sometimes be useful to prove the bound of all solution first, and separately establish the proposition that all bounded solutions approach zero. To illustrate this, consider the Lienard equation again, but under different assumptions. We now remove the requirement that

and replace it by stronger requirement on the damping.

In this chapter we deals with the extended of chapter two which discuss about autonomous system to non autonomous system. We consider the system

where

In which f depends explicitly on t. We will discuss with region of asymptotic stability by means of V function that may depends on t and y. In this we have to modify the definition as follows. Let V(t,y) be a scalar continuous function having continuous first order partial derivatives with respect to t and the component of y in a region in (t, y) space. We will also assume that contains the set H= {(t, 0): t}.

V(t,0)=0 and there exists a scalar function W(y), independent of t, with

for (t,y) in and such that W(y) is positive definite in the definition under 1.3.2.

1. If n=2, let for and W(y) is positive definite on the whole y-space. Thus V(t,y) is positive on ={(t,y):t}.

2. If n=2, let. Consider. Since this approaches zero as, we cannot hope to find a suitable function W. This V function is not positive definite in the sense of definition 1 above, even though V(t, y)>0 for .

If is any solution of (1), we have .

Where b

We wish to show the solution is asymptotic stable. Consider the function

=

Clearly V is positive definite and satisfies an infinitesimal upper bound; moreover, V*(t, y

Where are as example 1 above and where are real continuous functions defined on the region {(t,y

=0 (i=1, 2)

This condition implies that g

To establish asymptotic stability consider the same V function as before. However, with respect to the system (*) we now obtain V*(t,y

for

|y

Since | (i=1, 2) we obtain V*(t,y

Choose (any number less than will do) and determining the corresponding we obtain

V*(t,y

Thus V* is negative definite on the set {{(t, y

Consider the scalar equation

where , where is a constant. We may think of as a damping term of a linear oscillator. It can be shown that if is also bounded above, and then every solution of the equation (i) together with its derivative approaches zero. This, however, does not follow from theorem 1. For consider the equivalent system

and the positive definite scalar function having an infinitesimal upper bound. However .

Thus is positive, but not negative definite.

For the problem such as the one consider on equation (i) above, it is natural to inquire whether the results and techniques discussed in chapter 2 for autonomous systems can be carried over to the nonautonomous cases. One difficulty in doing this is that the notion of invariant set, natural for autonomous systems, cannot be defined directly for non autonomous systems. However, for a class of problems that may be called asymptotically autonomous systems, analogous results do hold and we state one such theorem below.

Let f(t,y) and h(y) be continuous together with their first derivatives with respect to the components of y in a set {(t,y)|0} where is some set in y-space.

is said to be asymptotically autonomous on the set Ω if

a) and this convergence is uniform for y in closed bounded subsets of Ω.

b) For every and every there exists such that

This equation is asymptotically autonomous on any closed bounded se is a constant. For:

Uniformly with respect to y in Ω

whenever

Then every bonded solution of approaches M as In particular, if all solutions of (3.2) are bounded, then every solution of approaches

We will now establish the asymptotic stability (global) of the zero solution of the system considered in Example 2 above. In the notation of previous theorem 2 we have

So that and Ω = {(y

(*)

is asymptotically autonomous and the corresponding limiting system is

which is obtained from (*) by putting y

for but also for where k>0 is constant. This insures that |f(t,y)| is bounded whenever |y| is bounded.

We observe next that every solution of

exists on and is then for as long as it exists we have from example

Therefore,

Thus by a familiar argument exists on and is bounded.

To apply theorem 2 we need to obtain M, the largest invariant subset of Ω with respect to the system (**) since every solution of (**) has the from where c

A region of asymptotic stability is a set of points surrounding a stable equilibrium point for which every system trajectory starting at a point in the set asymptotically takes to the equilibrium point. This seminar paper is to develop and validate computationally good methods of estimating regions of asymptotic stability of nonlinear systems and apply Lyapunov’s second method.

In general we can summarize the main points of this seminar paper as follows:

• Lyapunov’s second method used to determine stability of differential equation without explicitly integrating the nonlinear differential equation.

• It also determines asymptotic stability and estimate the region of asymptotic stability (RAS) of both autonomous and nonautonomous nonlinear systems.

• For this Lyapunov’s direct method is very useful method for estimating the region of asymptotic stability for non linear system.

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2. Fred Brauer and John A.Nohel, 1869. The** Qualitative Theory of Ordinary Differential Equations. **University of Wisconsin, over publication, Inc., New York. FIND ONLINE

3. Garrett Birkhoff Gion Carlorot, **Ordinary Differential Equations**, 4th edition, John Wiley & Sons.FIND ONLINE

4. R. M. Murray, Z. Li and S. S. Sastri. **Lyapunov stability theory. **Caltech. FIND ONLINE

5. Mao, J. (1999) **Differential Equation Stochastic. LaSalle theorems for SDEs, limits sets of SDEs**.

6. Richard E.Williamson, 1997. **Introduction to Differential Equations and Dynamical systems.** the McGraw-Gill companies, Higher Education 20INB001, Dartmouth College. FIND ONLINE

7. G.P. Szegö, **A contribution to Liapunov's second method: Nonlinear autonomous systems**. Trans. ASME Ser. D.J.Basic Engrg., 84 (1962), 571-578.

**APA**

Yadeta, Z. (2013). Lyapunov’s Second Method for Estimating Region of Asymptotic Stability. *Open Science Repository Mathematics*, Online(open-access), e70081944. doi:10.7392/Mathematics.70081944

**MLA**

Yadeta, Zelalem. “Lyapunov’s Second Method for Estimating Region of Asymptotic Stability.” *Open Science Repository Mathematics *Online.open-access (2013): e70081944.

**Chicago**

Yadeta, Zelalem. “Lyapunov’s Second Method for Estimating Region of Asymptotic Stability.” *Open Science Repository Mathematics* Online, no. open-access (March 20, 2013): e70081944. http://www.open-science-repository.com/lyapunovs-second-method-for-estimating-region-of-asymptotic-stability.html.

**Harvard**

Yadeta, Z., 2013. Lyapunov’s Second Method for Estimating Region of Asymptotic Stability. *Open Science Repository Mathematics*, Online(open-access), p.e70081944. Available at: http://www.open-science-repository.com/lyapunovs-second-method-for-estimating-region-of-asymptotic-stability.html.

**Science**

1. Z. Yadeta, Lyapunov’s Second Method for Estimating Region of Asymptotic Stability, *Open Science Repository Mathematics* **Online**, e70081944 (2013).

**Nature**

1. Yadeta, Z. Lyapunov’s Second Method for Estimating Region of Asymptotic Stability. *Open Science Repository Mathematics* **Online**, e70081944 (2013).

Research registered in the DOI resolution system as:** ****10.7392/Mathematics.70081944**.

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