Freezing Method Approach to an Asymptotic Stability of the
Discrete-time Oscillator Equation
Artur Babiarz, Adam Czornik and Michał Niezabitowski
Institute of Automatic Control, Silesian University of Technology, 16 Akademicka St., 44-100, Gliwice, Poland
Keywords:
Freezing Method, Asymptotic Stability, Discrete-Time Oscillator Equation.
Abstract:
The presented research work considers stability criteria of second-order differential equation. The second-
order discrete-time oscillator equation is obtained from discretization of second order continuous-time equa-
tion using the forward difference operator. The stability criteria are drawn with freezing method and are pre-
sented in the terms of the equation coefficients. Finally, an illustrative example is shown.
1 INTRODUCTION
In the literature, there are known research work de-
scribing a model called the simple harmonic oscilla-
tor (Buonomo and di Bello, 1996), (Hyland and Bern-
stein, 1987). The simple harmonic oscillator is ex-
pressed by the formula:
m ¨y(t) + c ˙y(t) + ky(t) = 0, (1)
where:
• y(t) is a measure of the displacement from the
equilibrium point at a given time;
• m is the mass;
• k is the spring parameters;
• c is the friction parameter.
The generalization of the equation (1) is a damped
linear oscillator defined by the following formula:
¨y(t) + a(t)˙y(t) + ω
2
y(t) = 0, (2)
where:
• the spring constant ω is positive;
• the damping coefficient a(t) is continuous and
nonnegative for t ≥ 0.
The equation (2) is one of the most famous model
using to describe many physical phenomena (Sugie
and et al, 2012). Moreover, this equation is also
known (Smith, 1961) as:
¨y(t) + a(t)˙y(t) + y(t) = 0. (3)
In (Graef and Karsai, 1996), the damped impul-
sive equation is analysed and authors studied the sta-
bility of this model. Instead, authors of (Yan and
Zhao, 1998) researched the stability of oscillator
equation with delays. In this case, the mathematical
model was described by the first-order linear delay
impulsive differential equation.
In (Zhang and Cheng, 1995) researchers focuses
on several oscillation criteria for a related neutral
first-order difference equation with delay. Authors of
(Yu and Cheng, 1994), (Kulikov, 2010), (Yu, 1998)
consider stability criteria of the first-order difference
equation with various kind of delays. They use, inter
alia, the Lyapunov function method.
In our research, we concentrate on stability of
second-order difference equation which we obtain by
a certain discretization of the equation (3).
For the discretization of the equation (3), it has
been used the so-called forward difference operator
and denoted by ∆. Its formal definition is as follows
(Agarwal and et al, 2005):
Definition 1. The first forward difference operator is
expressed by following formula
∆x(k) = x(k + 1) −x(k) (4)
and the second forward difference operator ∆
2
is de-
fined as
∆
2
x(k) = ∆(∆x(k)) = ∆x(k + 1) −∆x(k). (5)
Using the Definition 1, the equation (3) can be dis-
cretized in the following way (see (Agarwal and et al,
2005), page 3):
∆(∆(y(n))) + a(n)∆(y(n)) + y(n) = 0. (6)
353
Babiarz A., Czornik A. and Niezabitowski M..
Freezing Method Approach to an Asymptotic Stability of the Discrete-time Oscillator Equation.
DOI: 10.5220/0005512903530357
In Proceedings of the 12th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2015), pages 353-357
ISBN: 978-989-758-122-9
Copyright
c
 2015 SCITEPRESS (Science and Technology Publications, Lda.)

Subsequently, the forward difference is used and we
get:
y(n + 2) + y(n + 1)(a(n) −2)−
y(n)(a(n) −2) = 0. (7)
y(n + 1) + y(n)(a(n −1) −2)−
y(n −1)(a(n −1) −2) = 0. (8)
Substituting
q(n) = a(n −1) −2, (9)
the equation (8) will obtained the following form:
y(n + 1) = −q(n)y(n) + q(n)y(n −1). (10)
The main objective of our further consideration is the
stability problem for the equation (10) with initial
conditions y(0) = y
0
, y(1) = y
1
, where (q(n))
n∈N
is
a sequence of real numbers. We will say that equation
(10) is asymptotically stable if for all initial condi-
tions (y
0
, y
1
) the corresponding solution tends to zero.
In the paper we will use the following notation:
• R
s×s
is the set of all s by s real matrices;
• I
s
is the identity matrix of size s;
•
k
·
k
is the Euclidean norm in R
s
and the induced
operator norm;
• A
T
means the transposition of the matrix A.
The idea, which we use to obtain stability criteria
for the equation (10) is called in the literature (Bylov
and et al, 1966), (Desoer, 1970), (Gil and Medina,
2001), (Medina, 2008), (Czornik and Nawrat, 2010)
the freezing method. For the discrete linear time-
varying system given by
x(n + 1) = D(n)x(n) (11)
the main conception of the freezing method is to
freeze the matrices D(n) and consider system
x(l + 1) = D(n)x(l). (12)
The stability of all equations (12) usually does not
imply the stability of the system (11) (see (Ge and
Sun, 2005), (Czornik and Nawrat, 2006)). The stabil-
ity criteria for (11) are usually a combination of the
assumption of the stability of the system (12) and con-
straints on variation of the parameter D (n) of (11).
The above-mentioned approach is presented in (Des-
oer, 1970), (Gil and Medina, 2001) and (Czornik and
Nawrat, 2010).
2 MAIN RESULT
Let us introduce the following notation
A(n) =

−q(n) q(n)
1 0

(13)
and
B(q) =

−q q
1 0

. (14)
Moreover for the solution (y(n))
n∈N
of (10) denote
x(n) =

y(n)
y(n −1)

(15)
for n = 1, 2, .... With this notation equation (10) may
be rewritten in the following form
x(n + 1) = A(n)x(n) (16)
with initial condition
x(0) =

y(1)
y(0)

. (17)
If for all initial conditions x (0) the solution of (16)
tends to zero, then we will called (16) asymptotically
stable. In the paper we will use the following facts.
Lemma 1. (Khalil, 1995). If for (16) there exists a
function V : N× R
2
→ [0, ∞) such that
1.
k
x
k
2
≤V (n, x) ≤C
1
k
x
k
2
2.
V (n, x (n)) −V (n, x (n + 1)) ≤ −C
2
k
x (n)
k
2
for all x ∈ R
2
, n ∈ N and certain positive C
1
, C
2
,
then (16) is asymptotically stable. The function V is
called the Lyapunov function.
Lemma 2. (Gajic and et al, 1995). For a matrix A ∈
R
s×s
the following conditions are equivalent:
1. matrix A ∈ R
s×s
has all eigenvalues in the open
unit circle;
2. for each positive definite matrix Q ∈ R
s×s
there
exists a positive definite matrix P ∈R
s×s
such that
the following Lyapunov equation is satisfied
A
T
PA −P = −Q; (18)
3. there are positive definite matrices P, Q ∈ R
s×s
such that (18) is satisfied.
Moreover if A ∈ R
s×s
has all eigenvalues in the
unit circle then the solution of (18) is given by
P =
∞
∑
k=0

A
T

k
QA
k
. (19)
ICINCO2015-12thInternationalConferenceonInformaticsinControl,AutomationandRobotics
354

Lemma 3. (Horn and Johnson, 1985). For any matrix
A ∈ R
s×s
, A = [a
i j
]
i, j=1,...,s
we have
k
A
k
≤ s max
i, j=1,...,s


a
i j


. (20)
Lemma 4. All the eigenvalues of B(q) lies in the open
unit circle if and only if q ∈

−1,
1
2

and then
k
B(q)
k
<
√
5 + 1
2
. (21)
Proof. Consider the Lyapunov equation (18) with
A = B(q) and Q = I
2
then the solution is given by
P =
1
2q
2
+ q −1

2q −2 2q
2
2q
2
2q
3
+ q −1

. (22)
It is easy to check that the solution is positive defi-
nite if and only if q ∈

−1,
1
2

. The first conclusion
follows now from point 3 of Lemma 2. We have
k
B(q)
k
=
r
q
2
+
1
2
p
4q
4
+ 1 +
1
2
(23)
and using the standard method of calculus we may
verify that the greatest value of the function f :

−1,
1
2

→ R,
f (x) =
r
x
2
+
1
2
p
4x
4
+ 1 +
1
2
(24)
is
f (−1) =
√
5 + 1
2
. (25)
Lemma 5. If
ε ∈

0,
3
4

, (26)
q ∈

−1 + ε,
1
2
−ε

(27)
and P (q) is the solution of the Lyapunov equation
(18) with A = B(q) and Q = I
2
, then
1 ≤
k
P(q)
k
≤
8 −4ε
−2ε
2
+ 3ε
. (28)
Proof. The most left inequality follows from the for-
mula (19). As we noticed in the proof of Lemma (4)
P(q) is given by (22).
Since for
q ∈

−1 + ε,
1
2
−ε

, (29)


2q
2
+ q −1


> −2ε
2
+ 3ε, (30)
therefore
k
P(q)
k
≤
1
−2ε
2
+ 3ε





2q −2 2q
2
2q
2
2q
3
+ q −1





. (31)
Using Lemma 3 we have
k
P(q)
k
≤
2
−2ε
2
+ 3ε
·
·
"
max
−1+ε<q <
1
2
−ε

|
2q −2
|
, 2q
2
,


2q
3
+ q −1



#
=
8 −4ε
−2ε
2
+ 3ε
. (32)
Theorem 6. If for certain ε ∈

0,
3
4

and η ∈(0, 1) the
sequence (q(n))
n∈N
satisfies the following two condi-
tions:
1.
q(n) ∈

−1 + ε,
1
2
−ε

(33)
2.
|
q(n) −q(n −1)
|
≤
(1 −η)

−2ε
2
+ 3ε

2
√
2

√
5 + 1

(8 −4ε)
2
(34)
then the equation (10) is asymptotically stable.
Proof. Consider the solution P(n) of (18) with A =
A(n −1) and Q = I
2
. We will show that
V (n, x) = x
T
P(n)x (35)
is the Lyapunov function for (16). From Lemma 5 we
know that
k
x
k
2
≤V (n, x) ≤α
k
x
k
2
, (36)
where
α =
8 −4ε
−2ε
2
+ 3ε
. (37)
Let us estimate
R(n) = P(n + 1) −P(n). (38)
We have
A
T
(n)R(n)A(n) −R(n) = −Q(n), (39)
where
Q(n) =

A
T
(n) −A
T
(n −1)

P(n)A(n)+
A
T
(n −1)P(n)(A(n) −A(n −1)) (40)
and according to (19)
R(n) = Q(n) +
∞
∑
k=1

A
T
(n)

k
Q(n)A
k
(n). (41)
Because

A
T
(n)

k
Q(n)A
k
(n) ≤
k
Q(n)
k

A
T
(n)

k
A
k
(n)
(42)
FreezingMethodApproachtoanAsymptoticStabilityoftheDiscrete-timeOscillatorEquation
355

and by Lemmas 4 and 5
kQ(n)k ≤ 2kA(n) −A(n −1)kα
√
5 + 1
2
, (43)
then from (41) we have
R(n) ≤ Q(n) +
k
Q(n)
k
∞
∑
k=1

A
T
(n)

k
A
k
(n) ≤
k
Q(n)
k
I +
k
Q(n)
k
∞
∑
k=1

A
T
(n)

k
A
k
(n) =
k
Q(n)
k
"
I +
∞
∑
k=1

A
T
(n)

k
A
k
(n)
#
. (44)
By the definition of P(n) we know that
P(n) = I +
∞
∑
k=1

A
T
(n)

k
A
k
(n). (45)
From (44) and (45) we obtain
k
R(n)
k
≤
k
Q(n)
kk
P(n)
k
. (46)
Using (28) and (43) we may estimate R(n) as follows
k
R(n)
k
≤
k
A(n) −A(n −1)
k
α
2

√
5 + 1

=
√
2
|
q(n) −q(n −1)
|
α
2

√
5 + 1

≤ 1 −η. (47)
The last inequality implies that
V (n + 1, x (n + 1)) −V (n, x (n)) ≤
−η
k
x (n)
k
2
. (48)
Inequalities (36) and (48) show that V (n, x) is the
Lyapunov function for (16). By Lemma 1 we con-
clude that (16) is asymptotically stable what implies
that (10) is asymptotically stable.
Example 1. Consider equation (10) with
q(n) =
sin(ln (ln(n + 12)))
r
, (49)
where r > 0. Using Theorem 6 we will find the values
of r such that the conditions (33) and (34) are satisfied
with ε = 0.39 and certain η > 0, i.e.
q(n) ∈ (−0.61, 0.11) (50)
and
|
q(n) −q(n −1)
|
≤ (1 −η) ·3. 949 4 ×10
−3
(51)
with η > 0. According to Lagrange theorem
|
q(n) −q(n −1)
|
=


f
0
(c)


, (52)
where c ∈ (n −1, n) and
f (x) =
sin(ln (ln(x + 12)))
r
. (53)
Since
d
dx
sin(ln (ln(x + 12)))
r
=
1
ln(x + 12)
cos(ln (ln(x + 12)))
12r + rx
, (54)
then


f
0
(c)


≤
1
(12r + rc)ln (c + 12)
. (55)
It is easy to verify that for all c > 0 and r > 8.5 there
exists η > 0 such that
1
(12r + rc)ln (c + 12)
<
(1 −η) ·3. 949 4 ×10
−3
. (56)
It is also clear that for r > 10 the condition (50) is
satisfied. Finally we conclude from Theorem 6 that
equation (10) with q(n) given by (49) is asymptoti-
cally stable.
3 CONCLUSIONS
In this paper we have obtained the asymptotic stabil-
ity criteria for discrete version of damped linear os-
cillator. These criteria are obtained using the freezing
method. Typical situation in this method is that the
stability condition is a combination of requirements
about eigenvalues of certain matrices and variation of
the parameters. In our case we were able to present
the conditions in the terms of the original parameters
of the equation (10) only.
In further work, the authors intend to study the sta-
bility of various type of equation (10) using different
kind of exponents (Czornik et al., 2010a), (Czornik
et al., 2010b), (Czornik and Niezabitowski, 2013a),
(Czornik et al., 2013), (Czornik and Niezabitowski,
2013b), (Babiarz et al., 2015).
ACKNOWLEDGEMENTS
The research presented here were funded by
the Silesian University of Technology grant
BK-227/RAu1/2015/2 (A.B.), the National Sci-
ence Centre in Poland according to decisions
DEC-2012/05/B/ST7/00065 (A.C.) and DEC-
2012/07/B/ST7/01404 (M.N.).
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ICINCO2015-12thInternationalConferenceonInformaticsinControl,AutomationandRobotics
356

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