On Stabilization Problem for Nonlinear Systems with Power Principal Part
In the present paper, the stabilization problem for the uncontrollable with respect to the first approximation nonlinear system with power principal part is solved. A class of stabilizing controls for the nonlinear approximation of this system is constructed by using the Lyapunov function method. It...
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irk-123456789-1405502018-07-11T01:23:06Z On Stabilization Problem for Nonlinear Systems with Power Principal Part Bebiya, M.O. Korobov, V.I. In the present paper, the stabilization problem for the uncontrollable with respect to the first approximation nonlinear system with power principal part is solved. A class of stabilizing controls for the nonlinear approximation of this system is constructed by using the Lyapunov function method. It is proved that the same controls solve the stabilization problem for the original nonlinear system. An ellipsoidal approximation of the domain of attraction to the origin is given. 2016 Article On Stabilization Problem for Nonlinear Systems with Power Principal Part / M.O. Bebiya, V.I. Korobov // Журнал математической физики, анализа, геометрии. — 2016. — Т. 12, № 2. — С. 113-133. — Бібліогр.: 14 назв. — англ. 1812-9471 DOI: doi.org/10.15407/mag12.02.113 Mathematics Subject Classification 2000: 93C10, 93D15, 93D30 http://dspace.nbuv.gov.ua/handle/123456789/140550 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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In the present paper, the stabilization problem for the uncontrollable with respect to the first approximation nonlinear system with power principal part is solved. A class of stabilizing controls for the nonlinear approximation of this system is constructed by using the Lyapunov function method. It is proved that the same controls solve the stabilization problem for the original nonlinear system. An ellipsoidal approximation of the domain of attraction to the origin is given. |
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Article |
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Bebiya, M.O. Korobov, V.I. |
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Bebiya, M.O. Korobov, V.I. On Stabilization Problem for Nonlinear Systems with Power Principal Part Журнал математической физики, анализа, геометрии |
| author_facet |
Bebiya, M.O. Korobov, V.I. |
| author_sort |
Bebiya, M.O. |
| title |
On Stabilization Problem for Nonlinear Systems with Power Principal Part |
| title_short |
On Stabilization Problem for Nonlinear Systems with Power Principal Part |
| title_full |
On Stabilization Problem for Nonlinear Systems with Power Principal Part |
| title_fullStr |
On Stabilization Problem for Nonlinear Systems with Power Principal Part |
| title_full_unstemmed |
On Stabilization Problem for Nonlinear Systems with Power Principal Part |
| title_sort |
on stabilization problem for nonlinear systems with power principal part |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2016 |
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http://dspace.nbuv.gov.ua/handle/123456789/140550 |
| citation_txt |
On Stabilization Problem for Nonlinear Systems with Power Principal Part / M.O. Bebiya, V.I. Korobov // Журнал математической физики, анализа, геометрии. — 2016. — Т. 12, № 2. — С. 113-133. — Бібліогр.: 14 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT bebiyamo onstabilizationproblemfornonlinearsystemswithpowerprincipalpart AT korobovvi onstabilizationproblemfornonlinearsystemswithpowerprincipalpart |
| first_indexed |
2025-07-10T10:42:37Z |
| last_indexed |
2025-07-10T10:42:37Z |
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1837256313861570560 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2016, vol. 12, No. 2, pp. 113–133
On Stabilization Problem for Nonlinear Systems with
Power Principal Part
M.O. Bebiya and V.I. Korobov
Department of Mathematics and Informatics, V.N. Karazin Kharkiv National University
4 Svobody Sq., Kharkiv 61022, Ukraine
E-mail: m.bebiya@karazin.ua,
vkorobov@univer.kharkov.ua
Received June 5, 2015, revised October 21, 2015
In the present paper, the stabilization problem for the uncontrollable
with respect to the first approximation nonlinear system with power princi-
pal part is solved. A class of stabilizing controls for the nonlinear approxima-
tion of this system is constructed by using the Lyapunov function method.
It is proved that the same controls solve the stabilization problem for the
original nonlinear system. An ellipsoidal approximation of the domain of
attraction to the origin is given.
Key words: nonlinear stabilization, nonlinear systems, Lyapunov func-
tion method, nonlinear approximation.
Mathematics Subject Classification 2010: 93C10, 93D15, 93D30.
1. Introduction and Problem Statement
The stabilization problem for nonlinear systems is an interesting and difficult
problem. The most commonly used approach for solving this problem is the
method of stabilization with respect to the first approximation [1–4]. The present
work is devoted to the stabilization problem for nonlinear systems uncontrollable
with respect to the first approximation [5–12].
Consider the system
{
ẋ1 = u
ẋi = ϕi−1(t, x, u), i = 2, . . . , n,
(1)
where x ∈ Rn, Rn is n-dimensional Euclidian space with the norm ‖ · ‖, u ∈ R is
a control, ϕi(t, 0, 0) = 0 for all t ≥ 0, i = 1, . . . , n − 1. The functions ϕi(t, x, u)
c© M.O. Bebiya and V.I. Korobov, 2016
M.O. Bebiya and V.I. Korobov
are continuous with respect to all their arguments and are such that in some
neighborhood of the origin ‖x‖ < ρ they can be represented in the form
ϕi(t, x, u) = cix
2ki+1
i + fi(t, x, u), ki =
pi
qi
, i = 1, . . . , n− 1, (2)
where pi ≥ 0 are integers, qi > 0 are odd integers, ci are real numbers such
that
n−1∏
i=1
ci 6= 0. The functions fi(t, x, u) are continuous with respect to all their
arguments and Lipschitz continuous with respect to x and u, fi(t, 0, 0) = 0 for
all t ≥ 0, i = 1, . . . , n− 1.
Assume that there exists s such that 0 ≤ s ≤ n−2 and the following condition
holds:
ki = 0, i = 1, . . . , s and 0 < ks+1 < · · · < kn−1. (3)
We also assume that for some αi > 0 the functions fi(t, x, u) satisfy the estimates
|fi(t, x, u)| ≤ αi(x2k1+2
1 + x2k2+2
2 + · · ·+ x
2kn−1+2
n−1 ) (4)
for ‖x‖ < ρ, i = 1, . . . , n− 1. For s = n− 2, condition (3) means that
ki = 0, i = 1, . . . , n− 2, kn−1 =
pn−1
qn−1
> 0.
For the case ki ∈ N, for some special class of functions ϕi(t, x, u), the represen-
tation (2) can be obtained by using the Taylor series expansion. As an example,
we can take the system
ẋ1 = u,
ẋ2 = x1 − sinx1,
ẋ3 = x2 − x2 cosx2 − sinx2 + 1
2 sin 2x2,
which is studied below.
The stabilization problem for system (1) is to construct a control u(t, x) such
that the equilibrium point x = 0 of this system is asymptotically stable in the
sense of Lyapunov when u = u(t, x).
For the case ki = 0, i = 1, . . . n − 2, kn−1 ∈ N, the stabilization problem for
system (1) was solved in [6]. We present an approach for solving the stabilization
problem for nonlinear system (1) in more general case. The approach is based
on stabilization of the nonlinear approximation of system (1). For this nonlinear
approximation the stabilizing control is constructed in the form u = u(x) and it is
proved that this control solves the stabilization problem for the original nonlinear
system (1).
114 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 2
On Stabilization Problem for Nonlinear Systems with Power Principal Part
Consider the system
{
x1 = u,
xi = ci−1x
2ki−1+1
i−1 , i = 2, . . . , n,
(5)
as a nonlinear approximation of system (1) with right-hand side (2) satisfying
conditions (4). This system is uncontrollable with respect to the first approxi-
mation.
We will call system (5) with ci = 1, i = 1, . . . , n−1 the canonical system with
power nonlinearities. System (1) is called the system with power principal part
if its right-hand side satisfies conditions (4). For the case ki = 0, i = 1, . . . , n−2,
kn−1 ∈ N, the stabilization problem for system (5) was solved in [7].
We note that the terms cix
2ki+1
i are principal for representation (2). We mean
that, under conditions (4), system (1) with right-hand side (2) is stabilizable
independently of the functions fi(t, x, u), i = 1, . . . , n − 1. Namely, there exists
a control u = u(x), which solves the stabilization problem for system (1), such
that for all functions fi(t, x, u), i = 1, . . . , n − 1 that satisfy estimates (4) there
exists a common domain of attraction to the equilibrium point x = 0.
Similar systems were studied, for example, in [9–12]. The main assump-
tion was that the functions fi(t, x, u) are of bounded growth with respect to t,
x1, . . . , xi, i = 1, . . . , n− 1, and u, i.e.,
|fi(t, x, u)| ≤ M (|xi+1|pi+1 + · · ·+ |xn|pn) , i = 1, . . . , n− 1
with some additional restrictions imposed on the numbers pi. The last condition
is an analog of triangularity [13, 14]. In contrast, conditions (4) mean that the
functions fi(t, x, u) are of bounded growth with respect to t, xn, and u.
The approach for solving the stabilization problem, which was used in [9–
12], is based on a step-wise Lyapunov function and a stabilizing control design.
As a result, the Lyapunov function and the stabilizing control were obtained
recursively. That is why the practical construction of the stabilizing control is
difficult.
In the present work we show that it is possible to construct a Lyapunov
function as a quadratic form and find a stabilizing control in an explicit form.
The paper is organized as follows. In Sec. 2 the stabilizing control for system (5)
with ci = 1, i = 1, . . . , n − 1 is constructed in the form u = u(x). In Sec. 3 it
is shown that this control solves the stabilization problem for system (1) with
ci = 1, i = 1, . . . , n− 1. Finally, the case
n−1∏
i=1
ci 6= 0 is considered.
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 2 115
M.O. Bebiya and V.I. Korobov
2. Stabilization of Canonical System with Power Nonlinearities
In this section we solve the stabilization problem for system (5) for the case
ci = 1, i = 1, . . . , n− 1. We begin with considering the nonlinear system:
ẋ1 = u,
ẋ2 = x2k1+1
1 ,
· · · · ·
ẋn = x
2kn−1+1
n−1 ,
(6)
where ki = pi
qi
(pi ≥ 0 are integers, qi > 0 are odd integers), i = 1, . . . , n − 1
satisfy condition (3).
We introduce the following feedback control law:
u(x) = a1x1 + a2x2 + · · ·+ anxn +
n−1∑
i=s+1
an−s+ix
2ki+1
i , (7)
where {ai}2n−s−1
i=1 are negative real numbers, which will be defined below.
Our goal is to provide sufficient conditions under which this control solves the
stabilization problem for system (6). To this end, we use the Lyapunov function
method. The Lyapunov function V (x) will be found in the form
V (x) = (Fx, x), (8)
where F is some positive definite matrix.
We use the following notation:
A =
a1 a2 . . . as as+1 . . . an
1 0 . . . 0 0 . . . 0
· · · . . . . . . · · · · · · · · · · · ·
· · · · · · . . . . . . · · · · · · · · ·
0 0 0 1 0 . . . 0
0 0 0 0 0 · · · 0
· · · · · · · · · · · · · · · . . . · · ·
0 0 0 0 0 · · · 0
, (9)
hi = (an−s+i−1, 0, . . . , 0, 1︸ ︷︷ ︸
i
, 0, . . . , 0)T is an n-dimensional vector (1 is in the i-th
row), i = s + 2, . . . , n, A is an (n× n) matrix.
Suppose that the control u = u(x) is applied to system (6). Calculating the
derivative of V (x) along the trajectories of the closed-loop system (6), we obtain
V̇ (x)
∣∣∣
(6)
=
(
(AT F + FA)x, x
)
+ 2
n−1∑
i=s+1
(Fhi+1, x)x2ki+1
i . (10)
116 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 2
On Stabilization Problem for Nonlinear Systems with Power Principal Part
Note that because the matrix A is singular it is impossible to choose the
positive definite matrix F such that the matrix A∗F + FA is negative definite.
So we try to choose the positive definite matrix F such that the matrix A∗F +FA
is negative semi-definite. To this end, we consider the Lyapunov matrix equation
AT F + FA = −W, (11)
where W is some given positive semi-definite matrix, the matrix A is given by (9),
F is an unknown matrix. We define the matrix F as a positive definite solution
of equation (11). Equation (11) is called a singular Lyapunov matrix equation.
In general, matrix equation (11) is not solvable in the class of all positive
definite matrices F . Therefore we consider equation (11) under the assumption
that the matrix W has the form
w11 · · · w1s+1 w1s+1
as+2
as+1
· · · w1s+1
an
as+1
· · · . . . · · · · · · · · · · · ·
w1s+1 · · · ws+1s+1 ws+1s+1
as+2
as+1
· · · ws+1s+1
an
as+1
w1s+1
as+2
as+1
· · · ws+1s+1
as+2
as+1
ws+1s+1
a2
s+2
a2
s+1
· · · ws+1s+1
as+2an
a2
s+1
· · · · · · · · · · · · . . . · · ·
w1s+1
an
as+1
· · · ws+1s+1
an
as+1
ws+1s+1
as+2an
a2
s+1
· · · ws+1s+1
a2
n
a2
s+1
. (12)
In this case, equation (11) has a positive definite solution F , but this solution
is not unique. In Theorem 1 we describe the class of positive definite solutions
of matrix equation (11). First we need the following lemma.
Lemma 1. If the matrix
Ws+1 =
w11 · · · w1s+1
· · · . . . · · ·
w1s+1 · · · ws+1s+1
(13)
is positive semi-definite, then the matrix W given by (12) is positive semi-definite.
P r o o f. To prove that matrix (12) is positive semi-definite, we show that
all principal minors of this matrix are non-negative. By M i1,...,ik
i1,...,ik
, we denote the
principal minors of the matrix W . These minors are formed by rows and columns
with the numbers 1 ≤ i1 ≤ . . . ≤ ik ≤ n.
From the positive semi-definiteness of the matrix Ws+1 we obtain that
M i1,...,ik
i1,...,ik
≥ 0, 1 ≤ i1 ≤ · · · ≤ ik ≤ s + 1. (14)
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 2 117
M.O. Bebiya and V.I. Korobov
Taking into account that the last (n − s) rows and columns of matrix (12) are
linearly dependent, from inequality (14) we obtain that
M
i1,...,ik,ik+1,...,im
i1,...,ik,ik+1,...,im
= 0, 1 ≤ i1 ≤ · · · ≤ ik < s + 1, s + 1 ≤ ik+1 < · · · < im ≤ n,
where m > k + 1 ≥ 1, and
M
i1,...,ik, ik+1
i1,...,ik, ik+1
=
a2
ik+1
a2
s+1
M i1,...,ik, s+1
i1,...,ik, s+1 ≥ 0, 1 ≤ i1 < · · · < ik ≤ s, s + 2 ≤ ik+1 ≤ n,
where k ≥ 0. Thus the lemma is proved.
Theorem 1. Let the matrices A and W be defined by (9) and (12), re-
spectively. Suppose that the matrix Ws+1 given by (13) is positive definite, the
eigenvalues of the matrix
As+1 =
a1 a2 · · · as as+1
1 0 · · · 0 0
...
. . . · · · ...
...
...
...
. . .
...
...
0 0 · · · 1 0
(15)
have negative real parts. Define the matrix Fs+1 = {fij}s+1
i,j=1 as a unique positive
definite solution of the equation
AT
s+1Fs+1 + Fs+1As+1 = −Ws+1. (16)
Then the Lyapunov matrix equation (11) is solvable, and the matrix
F =
f11 · · · f1s+1 f1s+1
as+2
as+1
· · · f1s+1
an
as+1
· · · . . . · · · · · · · · · · · ·
f1s+1 · · · fs+1s+1 fs+1s+1
as+2
as+1
· · · fs+1s+1
an
as+1
f1s+1
as+2
as+1
· · · fs+1s+1
as+2
as+1
fs+2s+2 · · · fs+2n
· · · · · · · · · · · · . . . · · ·
f1s+1
an
as+1
· · · fs+1s+1
an
as+1
fns+2 · · · fnn
, (17)
where the elements {fij}n
i,j=s+2 are arbitrary real numbers, is a solution of this
equation. Moreover, there exist numbers {fij}n
i,j=s+2 such that matrix (17) is
positive definite.
P r o o f. Due to the assumption that the matrix A has the form (9), the
matrix F = {fij}n
i,j=1 is a solution of equation (11) if and only if the matrix
118 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 2
On Stabilization Problem for Nonlinear Systems with Power Principal Part
Fs+1 = {fij}s+1
i,j=1 satisfies equation (16) and the elements fij , i = 1, . . . , s + 1,
j = s + 2, . . . , n satisfy the system
f1iaj + f1jai + fi+1j = −wij ,
i = 1, . . . , s, j = s + 2, . . . , n,
f1iaj + f1jai = −wij ,
i = s + 1, . . . , j, j = s + 2, . . . , n,
(18)
where wij are the entries of the matrix (12).
Note that the last statement is true for equation (11) with an arbitrary sym-
metric matrix W = {wij}n
i,j=1 on its right-hand side. In general, for an arbitrary
matrix A, equation (11) does not split into the Lyapunov matrix equation (16)
and system (18).
Since the eigenvalues of the matrix As+1 have negative real parts, matrix
equation (16) has a unique positive definite solution Fs+1 for an arbitrary positive
definite matrix Ws+1. We recall that this solution can be represented in the form
Fs+1 =
∞∫
0
eAT
s+1tWs+1e
As+1tdt.
Because the matrix Fs+1 = {fij}s+1
i,j=1 is a solution of equation (16), its ele-
ments satisfy the linear system
{
f1ias+1 + aif1s+1 + fi+1s+1 = −wis+1, i = 1, . . . , s + 1,
2as+1f1s+1 = −ws+1s+1.
(19)
From (12) and (17), we have
wij = wis+1
aj
as+1
, fij = fis+1
aj
as+1
, i = 1, . . . , s + 1, j = s + 2, . . . , n,
wij = ws+1s+1
aiaj
a2
s+1
, i = s + 2, . . . , n, j = s + 2, . . . , n. (20)
First we show that the elements of the matrix F defined by (17) satisfy sys-
tem (18). Using relations (20), we rewrite system (18) in the form
f1iaj + aiaj
as+1
f1s+1 + aj
as+1
fi+1s+1 = − aj
as+1
wis+1,
i = 1, . . . , s, j = s + 2, . . . , n,
2 aiaj
as+1
f1s+1 = − aiaj
a2
s+1
ws+1s+1,
i = s + 1, . . . j, j = s + 2, . . . , n.
Multiplying both sides of the first equation by as+1
aj
and both sides of the second
equation by
a2
s+1
aiaj
, we obtain that the last system is equivalent to system (19).
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 2 119
M.O. Bebiya and V.I. Korobov
Thus the elements of the matrix F defined by (17) satisfy system (18). Therefore
the matrix F is the solution of matrix equation (11).
Now we prove that there exist the numbers fij , i = s + 1, . . . , n, j = 1, . . . , n
such that the matrix F defined by (17) is positive definite. Suppose that the
elements {fij}n
i,j=s+2, for i 6= j, are chosen to be arbitrary fixed numbers such
that fij = fji. We denote by Mij the determinant of the (n− 1)× (n− 1) matrix
that results from deleting row i and column j of the matrix F . Then the leading
principal minors ∆(Fi) of the matrix F are defined by
∆(Fi) = fii∆(Fi−1) +
i−1∑
j=1
(−1)j+ifjiMji, j = s + 2, . . . , n.
We recall that ∆(Fs+1) > 0. Now choosing fii one-by-one so that
fii > max
0,
1
∆(Fi−1)
i−1∑
j=1
(−1)j+i+1fjiMji
, i = s + 2, . . . , n,
we obtain that ∆(Fi) > 0, i = s + 2, . . . , n. Taking into account that the matrix
Fs+1 is positive definite, we have that ∆(Fi) > 0, i = 1, . . . , n and the matrix F
is positive definite. This concludes the proof.
Now we find sufficient conditions under which the control u = u(x) defined
by (7) solves the stabilization problem for system (6).
Let the matrix Ws+1 = {wij}s+1
i,j=1 be positive definite. Suppose the matrix W
is defined by (12). We define the matrix F as a positive definite solution of the
Lyapunov matrix equation (11). Therefore, using Theorem 1, we obtain that (10)
takes the form
V̇ (x)
∣∣∣
(6)
= −(
Wx, x
)
+ 2
n−1∑
i=s+1
(Fhi+1, x)x2ki+1
i , (21)
where the matrix F is given by (17).
Let us introduce the following notation b i = −Fhi, i = s + 2, . . . , n. Then
b i
j = −
(
f1j an−s+i−1 + fjs+1
ai
as+1
)
, j = 1, . . . , s + 1, i = s + 2, . . . , n, (22)
b i
j = −
(
f1s+1
aj
as+1
an−s+i−1 + fji
)
, j = s + 2, . . . , n, i = s + 2, . . . , n. (23)
We chose the numbers an−s+i−1, s + 2 ≤ i ≤ n and fji, s + 2 < i < j ≤ n such
that
b i
j = 0, j = i, . . . , n, i = s + 2, . . . , n.
120 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 2
On Stabilization Problem for Nonlinear Systems with Power Principal Part
Solving the last system, we obtain
an−s+i−1 = −as+1
ai
fii
f1s+1
, i = s + 2, . . . , n, (24)
fji =
aj
ai
fii, j = i + 1, . . . , n, i = s + 2, . . . , n− 1. (25)
Thus the matrix {fij}n
i,j=s+2, which is formed by the last (n–s–1) rows and the
columns of the matrix F given by (17), takes the form
fs+2s+2 fs+2s+2
as+3
as+2
· · · fs+2s+2
an
as+2
fs+2s+2
as+3
as+2
fs+3s+3 · · · fs+3s+3
an
as+3
· · · · · · . . . · · ·
fs+2s+2
an
as+2
fs+3s+3
an
as+3
· · · fnn
.
It is easy to show that under conditions (25) the matrix F given by (17) is
positive definite if and only if
fii >
a2
i
a2
i−1
fi−1i−1, i = s + 2, . . . , n. (26)
From (22), (23), and (24) it follows that
b i
j =
as+1
ai
(
f1jfii
f1s+1
− fjs+1
a2
i
a2
s+1
)
, j = 1, . . . , s + 1, i = s + 2, . . . , n, (27)
b i
j =
aj
ai
(
fii − fjj
a2
i
a2
j
)
, j = s + 1, . . . , i− 1, i = s + 2, . . . , n. (28)
We choose fii > 0, i = s + 2, . . . , n such that condition (26) holds. Therefore,
using (28), we obtain
b i
i−1 =
ai−1
ai
(
fii − fi−1i−1
a2
i
a2
i−1
)
> 0, i = s + 2, . . . , n.
Finally, we obtain that (21) takes the form
V̇ (x)
∣∣∣
(6)
= −(
Wx, x
)− 2
n∑
i=s+2
i−1∑
j=1
b i
j xjx
2ki−1+1
i−1 , (29)
where b j
i are defined by (27) and (28).
We note that x2ki+2
i > 0 for x 6= 0 because 2ki + 2 = 2(pi+qi)
qi
, i = 1, . . . , n− 1
is a ratio of even and odd numbers.
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 2 121
M.O. Bebiya and V.I. Korobov
Consider the case n = 2. It follows from (28) and (29) that
V̇ (x)
∣∣∣
(6)
= −(
Wx, x
)− 2b 2
1x
2k1+2
1 < 0 for ‖x‖ 6= 0.
To prove that the last inequality holds, we consider the following two cases. If
x1 6= 0, then the last inequality is true because b 2
1 = a1
a2
(
f22 − f11
a2
2
a2
1
)
> 0 and
the matrix W given by (12) is positive semi-definite. If x1 = 0, then −(
Wx, x
)
=
−w11
a2
2
a2
1
x2
2 < 0 for x2 6= 0 and the inequality holds.
Let us introduce the following notation In,n−s = diag
(
1, . . . , 1, 0, . . . , 0︸ ︷︷ ︸
n−s
)
is an
n × n diagonal matrix, Is+1,1 = diag (1, . . . , 1, 0) is an
(
s + 1
)×(
s + 1
)
diagonal
matrix, Is+1 is the identity matrix of dimension
(
s + 1
)× (
s + 1
)
.
Now we consider the case n ≥ 3. Since the matrix Ws+1 is positive semi-
definite, then the following estimate holds
(Ws+1y, y) ≥ λmin(y, y), y ∈ Rs+1, (30)
where λmin > 0 is the smallest eigenvalue of the matrix Ws+1. Using esti-
mate (30), we obtain that
(
(Ws+1 − λminIs+1,1)x, x
)
=
(
(Ws+1 − λminIs+1)y, y
)
+ λminx2
s+1 ≥ 0,
where y = (x1, . . . , xs+1)T , i.e., the matrix Ws+1 − λminIs+1,1 is positive semi-
definite.
Let us show that the matrix W −λminIn,n−s is positive semi-definite. Indeed,
the matrix Ws+1−λminIs+1,1 is formed by the first (s+1) rows and the columns
of the matrix W − λminIn,n−s. Since Ws+1 − λminIs+1,1 ≥ 0, using Lemma 1, we
obtain that the matrix W − λminIn,n−s is positive semi-definite, that is,
(
(W − λminIn,n−s)x, x
) ≥ 0 (31)
for all x ∈ Rn.
Let us rewrite equality (29) in the form
V̇ (x)
∣∣∣
(6)
= −(
(W − λminIn,n−s)x, x
)− λmin
s∑
i=1
x2
i−
−2
n−1∑
i=s+1
i∑
j=1
b i+1
j xjx
2ki+1
i .
(32)
We recall that according to Young’s inequality for any a, b, r > 0 we have
ab ≤ 1
1 + r
a1+r +
r
1 + r
b1+ 1
r . (33)
122 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 2
On Stabilization Problem for Nonlinear Systems with Power Principal Part
Below we will show that V̇ (x)
∣∣∣
(6)
is negative in some deleted neighborhood
of the origin. Let us choose the real numbers rj > 0, j = 1, . . . , n − 1 such that
the following conditions hold:
0 < rj < 2ks+1, j = 1, . . . , s, (34)
2kj + 1 < rj < 2kj+1 + 1, j = s + 1, . . . , n− 1. (35)
Using inequality (33), we obtain the following estimates:
xjx
2ki+1
i ≤ |xj | |xi|2ki+1−rj |xi|rj ≤ 1
2
(|xj |2 + |xi|4ki+2−2rj
)|xi|rj , (36)
where j = 1, . . . , s, i = s + 1, . . . , n− 1, and
xjx
2ki+1
i ≤ 1
1 + rj
|xj |rj+1 +
rj
1 + rj
|xi|2ki+1+
2ki+1
rj , (37)
where j = 1, . . . , i− 1, i = s + 1, . . . , n− 1.
From (32), using (36) and (37), we obtain
V̇ (x)
∣∣∣
(6)
≤ −(
(W − λminIn,n−s)x, x
)− λmin
s∑
i=1
x2
i − 2
n−1∑
i=s+1
b i+1
i x2ki+2
i
+
n−1∑
i=s+1
s∑
j=1
|b i+1
j |
(
|xj |2 |xi|rj + |xi|4ki+2−rj
)
+2
n−1∑
i=s+2
i−1∑
j=s+1
|b i+1
j |
(
1
1+rj
|xj |rj+1 + rj
1+rj
|xi|2ki+1+
2ki+1
rj
)
.
(38)
It is easy to show that
n−1∑
i=s+2
i−1∑
j=s+1
|b i+1
j |
1 + rj
|xj |rj+1 =
n−2∑
i=s+1
n∑
j=i+2
|b j
i |
1 + ri
|xi|ri+1.
Therefore inequality (38) takes the form
V̇ (x)
∣∣∣
(6)
≤ −(
(W − λminIn,n−s)x, x
)−
s∑
i=1
(
λmin −
n−1∑
j=s+1
|b j+1
i | |xj |ri
)
x2
i
−2
n−1∑
i=s+1
(
b i+1
i − 1
2
s∑
j=1
|b i+1
j | |xi|2ki−rj −
n∑
j=i+2
| b j
i |
1+ri
|xi|ri−2ki−1 (39)
−
i−1∑
j=s+1
rj
1+rj
|b i+1
j | |xi|
2ki+1−rj
rj
)
x2ki+2
i .
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 2 123
M.O. Bebiya and V.I. Korobov
Using the notation
βj = min
1≤i≤s
(
λmin
(n− s− 1) |bj+1
i |
) 1
ri
, j = s + 1, . . . , n− 1,
we have
λmin −
n−1∑
j=s+1
|b j+1
i | |xj |ri > 0, i = 1, . . . , s (40)
for xj such that |xj | < βj , j = s + 1, . . . , n− 1.
Consider a family of functions gi : R −→ R of the form
gi(x) = 2b i+1
i −
s∑
j=1
|b i+1
j | |x|2ki−rj − 2
n∑
j=i+2
| b j
i |
1+ri
|x|ri−2ki−1
−2
i−1∑
j=s+1
rj
1+rj
|b i+1
j | |x|
2ki+1−rj
rj , i = s + 1, . . . , n− 1.
From (34) and (35) it follows that
2ki − rj > 0, j = 1, . . . , s, i = s + 1, . . . , n− 1,
ri − 2ki − 1 > 0,
2ki + 1− rj
rj
> 0, j < i, i = s + 1, . . . , n− 1.
Therefore the functions gi(x), i = s+1, . . . , n−1 are continuous, symmetric, and
gi(x) have their global maximum at the point x = 0. Moreover,
gi(0) = 2b i+1
i > 0, i = s + 1, . . . , n− 1. (41)
We denote by x∗i the smallest positive root of the equation gi(x) = 0. Then,
using (41), we obtain
gi(x) > 0 for |x| ≤ x∗i , i = s + 1, . . . , n− 1. (42)
From (39) we obtain
V̇ (x)
∣∣∣
(6)
≤ −(
(W − λminIn,n−s)x, x
)−
s∑
i=1
(
λmin −
n−1∑
j=s+1
|b j+1
i | |xj |ri
)
x2
i
−
n−1∑
i=s+1
gi(xi)x2ki+2
i < 0 (43)
for x ∈ Rn such that |xi| < min {βi, x
∗
i }, i = s + 1, . . . , n− 1, and ‖x‖ 6= 0.
124 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 2
On Stabilization Problem for Nonlinear Systems with Power Principal Part
Let us show that the last inequality holds. Indeed, consider the following two
cases. If xi 6= 0 for some i such that 1 ≤ i ≤ n − 1, then it follows from (31),
(40), and (42) that inequality (43) is true. If x1 = x2 = · · · = xn−1 = 0, then
−(
(W − λminIn,n−s)x, x
)
= −ws+1s+1
a2
n
a2
s+1
x2
n < 0 for xn 6= 0
and inequality (43) is true.
Inequality (43) implies that the control u = u(x) defined by (7) solves the
stabilization problem for system (6).
Let us construct an ellipsoidal approximation of the domain of attraction to
the equilibrium point x = 0 for the case n ≥ 3. To this end, we find the largest
c > 0 such that the ellipsoid (Fx, x) ≤ c is contained in the set
Ω = {x ∈ Rn : |xi| ≤ γi, i = s + 1, . . . , n− 1} ,
where γi = min {βi, x
∗
i }, i = s + 1, . . . , n− 1.
In this connection we consider the extremal problem of finding a minimum of
the function (Fx, x) under the restriction (x, ei) = γi, where i is a fixed number
such that s + 1 ≤ i ≤ n − 1, ei is the i-th column of the n × n identity matrix.
Let us introduce the Lagrange function
L(x, λ) = (Fx, x)− λ
(
(x, ei)− γi
)
.
Let x∗ be a point of global minimum. The necessary condition of the extremum
gives that Lx(x∗, λ) = 2Fx∗−λei = 0. Thus we have x∗ = 1
2λF−1ei. Substituting
x∗ in the restrictions, we get 1
2λ(F−1ei, ei) = γi. Finding λ from the last equation,
we have x∗ = γi
(F−1ei,ei)
F−1ei. Finally, we obtain (Fx∗, x∗) = γ2
i
(F−1ei,ei)
.
So we have proved that, for n ≥ 3, the domain of attraction to the equilibrium
point x = 0 of the closed-loop system (6) contains the ellipsoid
Φ =
{
x ∈ Rn : (Fx, x) < c, c = min
s+1≤i≤n−1
γ2
i
(F−1ei, ei)
}
. (44)
Now we can summarize the previous discussion and formulate the main re-
sult of this section. In the following theorem we describe the solution of the
stabilization problem for system (6).
Theorem 2. Suppose ai < 0, i = 1, . . . , s + 1 are real numbers such that
the eigenvalues of the matrix As+1 given by (15) have negative real parts, and
ai < 0, i = s + 2, . . . , n are arbitrary real numbers. Suppose Ws+1 given by (13)
is an arbitrary positive definite matrix. Let the matrix Fs+1 = {fij}s+1
i,j=1 be a
unique positive definite solution of the equation (16). Choose the numbers fij,
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 2 125
M.O. Bebiya and V.I. Korobov
i = j, . . . , n, j = s + 2, . . . , n such that the conditions (25) and (26) are satisfied.
Define the matrix F by (17); and define the numbers an−s+i−1, i = s + 2, . . . , n
by (24). Then the control u = u(x) defined by (7) solves the stabilization problem
for system (6). Moreover, the domain of attraction to the equilibrium point x=0 of
the closed-loop system (6) contains ellipsoid (44) in the case n ≥ 3 and coincides
with the whole space in the case n = 2.
3. Stabilization of Systems with Power Principal Part
In this section we solve the stabilization problem for system (1). First consider
the case ci = 1, i = 1, . . . , n− 1. Then system (1) takes the form
ẋ1 = u,
ẋ2 = x2k1+1
1 + f1(t, x, u),
ẋ3 = x2k2+1
2 + f2(t, x, u),
· · · · · · · · ·
ẋn = x
2kn−1+1
n−1 + fn−1(t, x, u),
(45)
where ki = pi
qi
, pi ≥ 0 are integers, qi > 0 are odd integers, i = 1, . . . , n− 1.
As before, we assume that the numbers ki, i = 1, . . . , n − 1 satisfy condi-
tion (3). Moreover, we assume that the functions fi(t, x, u) satisfy conditions (4),
i.e., for some αi > 0 we have
|fi(t, x, u)| ≤ αi(x2k1+2
1 + x2k2+2
2 + · · ·+ x
2kn−1+2
n−1 ), i = 1, . . . , n− 1
when ‖x‖ < ρ, ρ > 0.
We take system (6) as a nonlinear approximation of system (45). Below we
will prove that the control u = u(x), constructed in the previous section, solves
the stabilization problem for system (45).
Let the control u = u(x) defined by (7) solve the stabilization problem for
system (6), and let the conditions of Theorem 2 hold. Let us show that this
control solves the stabilization problem for system (45).
Assume that the control u = u(x) is applied to system (45). Calculating the
derivative of the Lyapunov function V (x) given by (8) along the trajectories of
closed-loop system (45), we obtain
V̇ (x)
∣∣∣
(45)
=−(
Wx, x
)
+2
n−1∑
i=s+1
(Fhi+1, x)x2ki+1
i +2
n−1∑
i=1
(
Fei+1, x
)
fi(t, x, u), (46)
where ei is the i-th column of the n× n identity matrix.
Consider the case n = 2. Then s = 0. Thus we have
V̇ (x)
∣∣∣
(45)
= −(Wx, x)− 2b 2
1x
2k1+2
1 + 2(Fe2, x)f1(t, x, u).
126 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 2
On Stabilization Problem for Nonlinear Systems with Power Principal Part
Using estimates (4), we deduce that
V̇ (x)
∣∣∣
(45)
≤ −(
Wx, x
)− 2
(
b 2
1 − α1‖Fe2‖ ‖x‖
)
x2k1+2
1 < 0
for 0 < ‖x‖ < L1, L1 = min
{
ρ,
b21
α1 ‖Fe2‖
}
.
Let us show that the last inequality holds. Indeed, if x1 6= 0, then the inequal-
ity holds because the matrix W is positive semi-definite and b 2
1−α1‖Fe2‖ ‖x‖ > 0
for ‖x‖ <
b21
α1 ‖Fe2‖ . If x1 = 0, then we have −(Wx, x) = −w22
a2
2
a2
1
x2
2 < 0 for x2 6= 0
and the inequality holds.
Now we consider the case n ≥ 3. Using (43), from (46) we obtain
V̇ (x)
∣∣∣
(45)
≤ −(
(W − λminIn,n−s)x, x
)−
s∑
i=1
(
λmin −
n−1∑
j=s+1
|b j+1
i | |xj |ri
)
x2
i
−
n−1∑
i=s+1
gi(xi)x2ki+2
i + 2
n−1∑
i=1
(
Fei+1, x
)
fi(t, x, u) (47)
for x ∈ Rn such that |xi| < min {βi, x
∗
i }, i = s + 1, . . . , n− 1, ‖x‖ 6= 0.
Let us choose the numbers εi, i = 1. . . . n− 1 such that
0 < εi < λmin, i = 1, . . . , s, and 0 < εi < b i+1
i , i = s + 1, . . . , n− 1.
Let x̂i be the smallest positive root of the equation gi(xi) = εi, i = s+1, . . . , n−1.
It is obvious that x̂i < x∗i , i = s + 1, . . . , n− 1.
Then the following inequalities hold:
λmin −
n−1∑
j=s+1
|b j+1
i | |xj |ri ≥ εi, i = 1, . . . , s, (48)
for |xj | ≤ m̂j , where
m̂j = min
1≤i≤s
(
λmin − εi
(n− s− 1) |b j+1
i |
) 1
ri
, j = s + 1, . . . , n− 1,
and
gi(xi) ≥ εi, i = s + 1, . . . , n− 1 (49)
for |xi| ≤ x̂i, i = s + 1, . . . , n− 1.
Thus, from (48) and (49), it follows that inequality (47) takes the form
V̇ (x)
∣∣∣
(45)
≤ −(
(W − λminIn,n−s)x, x
)−
s∑
i=1
εix
2
i −
n−1∑
i=s+1
εix
2ki+2
i
+2
n−1∑
i=1
(
Fei+1, x
)
fi(t, x, u) (50)
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 2 127
M.O. Bebiya and V.I. Korobov
for x ∈ Rn such that |xi| < mi, where mi = min {x̂i, m̂i}, i = s + 1, . . . n − 1.
Using estimates (4), we have
n−1∑
i=1
(
Fei+1, x
)
fi(t, x, u) ≤
n−1∑
i=1
αi‖Fei+1‖ ‖x‖ ‖y‖2, (51)
where y =
(
xk1+1
1 , xk2+1
2 , . . . , x
kn−1+1
n−1
)T
, ‖y‖2 =
n−1∑
i=1
x2ki+2
i .
Using the notation
ε = min
1≤i≤n−1
εi, m = min
s+1≤i≤n−1
mi,
from inequalities (50) and (51) we deduce that
V̇ (x)
∣∣∣
(45)
≤ −(
(W − λminIn,n−s)x, x
)−
(
ε−
n−1∑
i=1
αi‖Fei+1‖ ‖x‖
)
‖y‖2 < 0 (52)
for 0 < ‖x‖ < L2, where L2 = min
ρ,
ε
n−1∑
i=1
αi‖Fei+1‖
,m
.
Let us show that inequality (52) is true. To this end, we consider the following
two cases. If ‖y‖ 6= 0, then ε −
n−1∑
i=1
αi‖Fei+1‖ ‖x‖ > 0 for ‖x‖ < L2. Hence,
combining the last inequality with inequality (31), we obtain that inequality (52)
is true. If ‖y‖ = 0, then
−(
(W − λminIn,n−s)x, x
)
= −ws+1s+1
a2
n
a2
s+1
x2
n < 0 for xn 6= 0
and inequality (52) is true.
We have shown that the equilibrium point x = 0 of the closed-loop system (45)
is asymptotically stable, and thus we have proved that the control u = u(x)
defined by (7) solves the stabilization problem for system (45).
Let us construct an ellipsoidal approximation of the domain of attraction to
the equilibrium point x = 0 of the closed-loop system (45). To this end, we find
c > 0 such that the ellipsoid (Fx, x) < c is inscribed in the ball ‖x‖ ≤ L, where
L =
{
L1, n = 2
L2, n ≥ 3
. It is easy to verify that this ellipsoid has the form
Φ =
{
x ∈ Rn : (Fx, x) < c, c = λmin(F )L2
}
, (53)
where λmin(F ) > 0 is the smallest eigenvalue of the matrix F .
128 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 2
On Stabilization Problem for Nonlinear Systems with Power Principal Part
Now, basing on results obtained in this section, we state the following theorem.
Theorem 3. Suppose that the conditions of Theorem 2 hold. Then the control
u = u(x) defined by (7) solves the stabilization problem for system (45); and the
domain of attraction to the equilibrium point x = 0 of the closed-loop system
contains ellipsoid (53).
Finally, we solve the stabilization problem for system (1) for the case where
ci, i = 1, . . . , n − 1 are some real numbers such that
n−1∏
i=1
ci 6= 0. So system (1)
has the form
{
ẋ1 = u,
ẋi = ci−1x
2ki−1+1
i−1 + fi−1(t, x, u), i = 2, . . . , n.
(54)
We recall that the numbers ki satisfy condition (3) and the functions fi(t, x, u),
i = 1, . . . , n− 1 satisfy conditions (4).
The following theorem provides a way to solve the stabilization problem for
nonlinear system (54), which is uncontrollable with respect to the first approxi-
mation.
Theorem 4. Suppose that the conditions of Theorem 2 hold. Put
L̂1 = min
ρ
max
1≤i≤n
ĉi
,
b 2
1
α̂1‖Fe2‖
, L̂2 = min
ρ
max
1≤i≤n
ĉi
,
ε
n−1∑
i=1
α̂i‖Fei+1‖
,m
,
where α̂i =
αi
|ĉi+1| max
1≤j≤n−1
ĉ
2kj+2
j and ĉi are defined by the following relations
ĉ1 = 1, ĉ2 = c1, ĉi = ci−1(ĉi−1)2ki−1+1, i = 3, . . . , n.
Then the control
u(x) = a1x1 +
a2
ĉ2
x2 + · · ·+ an
ĉn
xn +
n−1∑
i=s+1
an−s+i
(
xi
ĉi
)2ki+1
solves the stabilization problem for system (54). Moreover, the domain of attrac-
tion to the equilibrium point x = 0 of the closed-loop system (54) contains the
ellipsoid
Φ =
{
x ∈ Rn : (Ĉ−1FĈ−1x, x) < λmin(F )L̂2
}
,
where Ĉ =diag
(
ĉ1, ĉ2, . . . , ĉn
)
is a diagonal matrix of dimension n×n, λmin(F )>0
is the smallest eigenvalue of the matrix F and L̂ =
{
L̂1, n = 2
L̂2, n ≥ 3
.
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 2 129
M.O. Bebiya and V.I. Korobov
P r o o f. Suppose that the control u = u(x) is applied to system (54). The
change of variables x = Ĉy (xi = ĉiyi, i = 1, . . . , n) maps system (54) to the
system
ẏ1 = v(y),
ẏi = y
2ki−1+1
i−1 +
1
ĉi
fi−1(t, Ĉy, u), i = 2, . . . , n,
(55)
where v(y) = u(Ĉy), y = (y1, . . . , yn)T . From estimates (4) it follows that
| 1
ĉi+1
fi(t, Ĉy, u)| ≤ αi
|ĉi+1|
n−1∑
i=1
(ĉiyi)2ki+2 ≤ α̂i
n−1∑
i=1
y2ki+2
i , i = 1, . . . , n− 1
for ‖y‖ ≤ ρ
max
1≤i≤n
ĉi
.
Thus, using Theorem 3, we obtain that the control
v(y) = a1y1 + a2y2 + · · ·+ anyn +
n−1∑
i=s+1
an−s+iy
2ki+1
i
solves the stabilization problem for the open-loop system (55). Moreover, the
domain of attraction to the equilibrium point x = 0 of the closed-loop system
contains the ellipsoid
{
y ∈ Rn : (Fy, y) < c, c = λmin(F )L̂2
}
.
Making the reverse change of variables y = Ĉ−1x, we obtain the stabilizing control
for system (54) and the ellipsoidal approximation of the domain of attraction to
the equilibrium point x = 0 of the closed-loop system.
E x a m p l e 1. Consider the system
ẋ1 = u,
ẋ2 = x1 − 1
2 sin 2x1,
ẋ3 = x2 − x2 cosx2 − sinx2 + 1
2 sin 2x2.
(56)
In this case
ϕ1(t, x, u) = x1 − 1
2
sin 2x1, ϕ2(t, x, u) = x2 − x2 cosx2 − sinx2 +
1
2
sin 2x2.
Using the Taylor expansion formula, we obtain
ϕ1(t, x, u) =
2
3
x3
1 + f1(x1), ϕ2(t, x, u) =
1
12
x5
2 + f2(x2),
130 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 2
On Stabilization Problem for Nonlinear Systems with Power Principal Part
where |f1(x1)| ≤ 2
15
|x1|5 and |f2(x2)| ≤ 1
90
|x2|7.
Thus system (56) takes the form of (54) with n = 3, k1 = 1, k2 = 2, c1 =
2
3
,
c2 =
1
12
, α1 =
2
15
, α2 =
1
90
, ρ = 1, s = 0. Then ĉ1 = 1, ĉ2 =
2
3
, ĉ3 =
8
729
.
Put a1 = −1, a2 = −2, a3 = −1
2
, w11 = 2. Then, according to (9) and (12),
we have
A =
−1 −2 −1
2
0 0 0
0 0 0
, W =
2 4 1
4 8 2
1 2
1
2
.
From (17) and (25) we obtain that the solution of matrix equation (11) when
f22 = 5, f33 = 1 has the form
F =
1 2
1
2
2 5
5
4
1
2
5
4
1
.
Thus, according to Theorem 4, we obtain that the control
x3
x2
x1
-1
-0.5
0.5
-0.6
-0.4
-0.2
0.2
0.1
0.05
Fig. 1. Phase trajectory x(t) in R3.
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 2 131
M.O. Bebiya and V.I. Korobov
u(x) = −x1 − 3x2 − 729
16
x3 − 5
2
x3
1 −
243
16
x5
2
solves the stabilization problem for system (56); and the domain of attraction to
the equilibrium point x = 0 of the closed-loop system contains the ellipsoid
Φ =
{
x ∈ R3 : (Fy, y) ≤ 0.0891 . . .
}
, where y =
(
x1
ĉ1
,
x2
ĉ2
,
x3
ĉ3
)T
.
Numerical analysis shows that we can take a point out of this ellipsoid. For
example, let x0 = (−0.1,−0.3, 0.15)T be the initial point. This point does not
belong to the ellipsoid Φ, but it belongs to the domain of attraction to the origin.
Graphic of the phase trajectory is shown in Fig. 1.
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