Theorem of Asymptotic Stability

1 A sufficient precondition of asymptotic stability

1.1. Basis

1.1.1 Circle, spherical surface
A circle, a spherical surface, and its n-dimensional equivalent are the geometrical place of those points which in the plain, in space and in the n-dimensional space are at an equal distance from a given point, the so-called center. Because of this, traveling along these, the distance from the point does not change.

1.1.2 Tangent of a circle, spherical surface
The tangent of these curves is perpendicular to the so-called radius, the straight section connecting the center with the tangential point.

1.1.3 A property of a scalar product
The scalar product of two vectors can be written down as a product of the absolute value of the two vectors and the cosine of the angle between the vectors. It follows from this that the value of the scalar product of the two vectors perpendicular to each other is zero, as the cosine of the right angle is also zero.

1.1.4 Some consequences
On the basis of 1.1.2 and 1.1.3, the scalar product of the position vector pointing from the origo towards the tangential point and of the tangent as a vector is of zero value.

Also, the value of the angle enclosed by the position vector pointing from the origo towards the tangential point and the vector starting from the tangential point is higher than 90 degrees if, and only if, the latter vector points towards the inside of the circle, the sphere.

On the basis of this, in this case the value of the cosine of this angle is negative.

1.2 The theorem

In case the precondition

(r-r0).v<0					(1a)

is satisfied at every point of the space section containing the singular point, that is,

COS(Φ(t)) <0                                        	(1b)

or

SIG(COS(Φ(t))) = -1					(1c)

the singular point is asymptotically stable.

Here r is a position vector, r0 is the position vector of the singular point, that is, the difference vector is the position vector pointing from the singular point towards the examined point; v is the so-called phase- velocity vector; Φ is the angle enclosed by the difference vector and the phase-velocity vector; t is time; SIG is the sign function.

(The phase-velocity vector used in phase-plain methods is the
v = Σei.(dri/dt)
phrase, where ei is the “i”-direction unity vector, r is the “i”-direction component of the place vector.)

1.2.1 Proving the equivalence of the equations in (1a) and (1b)
As the scalar series of the two vectors can be written down in the following form:
ABS(rr0).ABS(v).COS(Φ(t)) <0
and the absolute values do not affect the sign of the series, they can be omitted.

1.2.2 Proving the equivalence of the equations in (1)
(1b) can be rewritten in the following way:
COS(Φ(t)) = (Σ(rr0);.(dr;/dt))/ (ABS(rr0).ABS(v)),
where the sign of the denominator can, for the same reason as above, be omitted. So, the precondition is:
SIG(COS(Φ(t))) = SIG(Σ(rr0)i.(dri/dt)) = -1

1.3 Proving the theorem

In case v points towards the inside of a circle of an r-r0 radius whose center is the singular point, the moving point moves towards the singular point.

The precondition of pointing towards the inside is that the angle enclosed between the components of the scalar product is an obtuse angle. And its cosine is negative.

As this holds true for every point of the space section, in the case of every motion moving forward in time the position vector pointing from the singular point towards the examined point becomes shorter, and so the motion is towards the singular point.

1.4 Not being necessary

The moving point can also get to the singular point if in the course of its way it does not always approaches it, but there are sections where it gets neither nearer, nor farther.

1.5 Being general

As in the proving there was no restriction whatsoever regarding the number of space dimensions, it is valid for the n-dimensional phase space. (Here, n represent an arbitrary number of dimensions.)

2 A more general sufficient precondition of asymptotic stability in the phase plain

2.1 The theorem

Taking point 1.3 into consideration, it follows that in case the preconditions in (1b) and (1c) are satisfied at every point of the space section containing the singular point, excepting a limited number of plain sections within which
COS(Φ(t)) = 0,
and there is no other stable singular point or stable limit cycle, the system is asymptotically stable.

2.2 Proving

Reaching a non-approaching section, the motion goes through it, as there is no singular formation there to stop the moving point, and it gets neither nearer, nor farther. After going through that section, it reaches an approaching section, through which it either reaches the singular point or gets nearer to it. Repeating these two kinds of motion, the moving point gets nearer and nearer to the singular point.

2.3 Not being necessary

Following the reasoning in point 1.3, the singular point can be stable even in case the moving point gets farther from it at times.

2.4 Restriction

In a non-approaching section of the phase space there can be motion which meets this theorem but which does not leave that section. For example, such motion is every closed curve staying within one single spherical surface.

Thus, this theorem holds true for phase spaces only.

3 An even more general sufficient precondition of asymptotic stability

3.1 The theorem

If in the course of motion the precondition

Φ0 + 2Φ

 	∫ (ABS(r-r0).ABS(v).COS(Φ(t))dΦ(t)) < 0		(3a)

Φ0
or the SIG function form corresponding with it is always satisfied in the section containing the singular point, the singular point is asymptotically stable.

3.1.1 Márton Simponyi’s interpretation
(3a) means that the absolute value of the sum of one rotation approaches is higher than that of getting-farther motions.

3.2 Proving

For stability, it is enough for the moving point to get nearer to the singular point after each rotation it makes, for it moves towards it anyway.

(In case no rotation is made, this precondition cannot be satisfied , as the integral does not exist.)

3.3 Not being necessary

Rotations can be made in the course of which this precondition is not satisfied, yet the moving point moves towards the singular point.

3.4 Being general

The same holds true here as in point 1.4.

3.5 A simple example

All through making a circle, the motion is non-damped oscillation, the equation of which is:
x = x0.COS(t),
but after making the rotation, from point (x1,0) it changes to damped oscillation, that is, it will be

x = x1.EXP(-t) 			(Fig. 1).

Fig. 1

In the phase plain, it is
dy/dx = -x/y
then
dy/dx = -1
(Fig. 2).

Fig. 2

4 A necessary precondition of asymptotic stability

4.1 The theorem

If in the course of motion the precondition

Φ∞

∫ (ABS(r-r0).ABS(v).COS(Φ(t))dΦ(t)) < 0		(4a)

Φ0
is satisfied (where 0 – a t = 0, a ∞ – at ∞ ) and there is no other stable singular point or stable limit cycle in the examined range, the examined point is asymptotically stable.

In case

vr(t).dt = - r0					(4b)

0
is also satisfied, the moving point gets to the singular point.
(4b) is a necessary and sufficient precondition of getting to the singular point.

4.2 Proving

As in the case of (4a) being satisfied full motion takes nearer to the singular point, the precondition of stability is satisfied. In the case of (4b) it is even more the case, as it does get to the singular point. However, it cannot get there any other way than through the complete “waning” of the distance.

4.3 Being general

As in this case, either, there was nothing depending on the number of dimensions, both holds true for any number of dimensions.

4.4 A simple example

In section one
dy/dx = -1
xa = ya
x = xa – t
y = xa – t
In section two
dy/dx = -1
xb = yb
x = xb + t
y = xb – t

In section three
x = xa.EXP(-t)
y = 0

That is what Fig. 3 shows.

Fig. 3

5 Why is it necessary to have so many versions?

The reason for this is the simplest and at the same time the strictest as well. As we move towards the more general, so its application becomes more complicated.

6 An interesting example

In the case shown in Fig. 4, singular point 1 can be reached from section 1 a over the green line bordering area 2 from above only by first getting farther from it. From the upper boundary of this area it can also be reached by first getting neither nearer nor farther from it. From the other areas, however, there is a way that constantly nears it.

Fig. 4

Thus, this system realizes all the possibilities at the same time.

7 Necessary and sufficient preconditions and stability types

7.1 Depending on direction

If precondition (4a) holds true only in the case of certain Φ0, stability depends on the direction

7.2 Not depending on direction

If precondition (4a) holds true in the case of every Φ0, stability does not depend on the direction.

7.3 Local

If precondition (4b) is satisfied only in the case of certain, limited r0, stability is local.

7.4 Global

If precondition (4b) is not satisfied only in the case of certain, limited r0, stability is global.

7.5 Combinations

All four combinations (depending on the direction, local; depending on the direction, global; not depending on the direction, local; not depending on the direction, global) are possible.

Dr. Endre Simonyi