Global positional synthesis and stabilization in finite time of MIMO generalized triangular systems by means of the controllability function method
We solve the problem of global stabilization in finite time for a general class of triangular multi-input and multi-output systems with singular input-output links. In order to obtain the main result of the work, we combine the controllability function method (which works locally, around the equilib...
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| Cite this: | Global positional synthesis and stabilization in finite time of MIMO generalized triangular systems by means of the controllability function method / V.I. Korobov, S.S. Pavlichkov, W.H. Schmidt // Нелінійні коливання. — 2012. — Т. 15, № 2. — С. 205-214. — Бібліогр.: 25 назв. — англ. |
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irk-123456789-1755962021-02-02T01:28:29Z Global positional synthesis and stabilization in finite time of MIMO generalized triangular systems by means of the controllability function method Korobov, V.I. Pavlichkov, S.S. Schmidt, W.H. We solve the problem of global stabilization in finite time for a general class of triangular multi-input and multi-output systems with singular input-output links. In order to obtain the main result of the work, we combine the controllability function method (which works locally, around the equilibrium only for our class of triangular forms) with a modification of the global construction developed for the generalized triangular form in the singular case in our previous works. Розв’язано задачу глобальної стабiлiзацiї в скiнченний час для загального класу трикутних систем з кiлькома входами i виходами та сингулярними зв’язками мiж входом i виходом. Для отримання основного результату поєднано метод функцiї керованостi (що працює в околi нерухомої точки тiльки для розглядуваного класу трикутних систем) з модифiкацiєю загальної побудови для узагальненої трикутної форми в сингулярному випадку, що розвинута в попереднiх роботах авторiв. 2012 Article Global positional synthesis and stabilization in finite time of MIMO generalized triangular systems by means of the controllability function method / V.I. Korobov, S.S. Pavlichkov, W.H. Schmidt // Нелінійні коливання. — 2012. — Т. 15, № 2. — С. 205-214. — Бібліогр.: 25 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/175596 517.9 en Нелінійні коливання Інститут математики НАН України |
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We solve the problem of global stabilization in finite time for a general class of triangular multi-input and multi-output systems with singular input-output links. In order to obtain the main result of the work, we combine the controllability function method (which works locally, around the equilibrium only for our class of triangular forms) with a modification of the global construction developed for the generalized triangular form in the singular case in our previous works. |
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| author |
Korobov, V.I. Pavlichkov, S.S. Schmidt, W.H. |
| spellingShingle |
Korobov, V.I. Pavlichkov, S.S. Schmidt, W.H. Global positional synthesis and stabilization in finite time of MIMO generalized triangular systems by means of the controllability function method Нелінійні коливання |
| author_facet |
Korobov, V.I. Pavlichkov, S.S. Schmidt, W.H. |
| author_sort |
Korobov, V.I. |
| title |
Global positional synthesis and stabilization in finite time of MIMO generalized triangular systems by means of the controllability function method |
| title_short |
Global positional synthesis and stabilization in finite time of MIMO generalized triangular systems by means of the controllability function method |
| title_full |
Global positional synthesis and stabilization in finite time of MIMO generalized triangular systems by means of the controllability function method |
| title_fullStr |
Global positional synthesis and stabilization in finite time of MIMO generalized triangular systems by means of the controllability function method |
| title_full_unstemmed |
Global positional synthesis and stabilization in finite time of MIMO generalized triangular systems by means of the controllability function method |
| title_sort |
global positional synthesis and stabilization in finite time of mimo generalized triangular systems by means of the controllability function method |
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Інститут математики НАН України |
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2012 |
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http://dspace.nbuv.gov.ua/handle/123456789/175596 |
| citation_txt |
Global positional synthesis and stabilization in finite time of MIMO generalized triangular systems by means of the controllability function method / V.I. Korobov, S.S. Pavlichkov, W.H. Schmidt // Нелінійні коливання. — 2012. — Т. 15, № 2. — С. 205-214. — Бібліогр.: 25 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT korobovvi globalpositionalsynthesisandstabilizationinfinitetimeofmimogeneralizedtriangularsystemsbymeansofthecontrollabilityfunctionmethod AT pavlichkovss globalpositionalsynthesisandstabilizationinfinitetimeofmimogeneralizedtriangularsystemsbymeansofthecontrollabilityfunctionmethod AT schmidtwh globalpositionalsynthesisandstabilizationinfinitetimeofmimogeneralizedtriangularsystemsbymeansofthecontrollabilityfunctionmethod |
| first_indexed |
2025-07-15T12:54:33Z |
| last_indexed |
2025-07-15T12:54:33Z |
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1837717598551146496 |
| fulltext |
UDC 517.9
GLOBAL POSITIONAL SYNTHESIS AND STABILIZATION IN FINITE
TIME OF MIMO GENERALIZED TRIANGULAR SYSTEMS BY MEANS
OF THE CONTROLLABILITY FUNCTION METHOD
ГЛОБАЛЬНИЙ ПОЗИЦIЙНИЙ СИНТЕЗ ТА СТАБIЛIЗАЦIЯ
В СКIНЧЕННИЙ ЧАС УЗАГАЛЬНЕНИХ ТРИКУТНИХ СИСТЕМ
З КIЛЬКОМА ВХОДАМИ ТА ВИХОДАМИ
ЗА ДОПОМОГОЮ МЕТОДУ ФУНКЦIЇ КЕРОВАНОСТI
V. I. Korobov*
Kharkov Nat. Univ.
Svobody sqr. 4, Kharkov 61077, Ukraine
Inst. Math., Szczecin Univ.
Wielkopolska str. 15, Szczecin 70451, Poland
e-mail: vkorobov@univer.kharkov.ua
korobow@sus.univ.szczecin.pl
S. S. Pavlichkov**
Taurida Nat. Univ.
Vernadsky Av. 4, Simferopol 95007, Ukraine
and Univ. Appl. Sci. Erfurt
Altonaer str. 25, Erfurt 99085, Germany
e-mail: s_s_pavlichkov@yahoo.com
W. H. Schmidt**
Inst. Math. and Inform., Greifswald Univ.
Walter-Rathenau str. 47, Greifswald 17487, Germany
e-mail: wschmidt@uni-greifswald.de
We solve the problem of global stabilization in finite time for a general class of triangular multi-input and
multi-output systems with singular input-output links. In order to obtain the main result of the work, we
combine the controllability function method (which works locally, around the equilibrium only for our
class of triangular forms) with a modification of the global construction developed for the generalized tri-
angular form in the singular case in our previous works.
Розв’язано задачу глобальної стабiлiзацiї в скiнченний час для загального класу трикутних сис-
тем з кiлькома входами i виходами та сингулярними зв’язками мiж входом i виходом. Для отри-
мання основного результату поєднано метод функцiї керованостi (що працює в околi нерухо-
мої точки тiльки для розглядуваного класу трикутних систем) з модифiкацiєю загальної по-
будови для узагальненої трикутної форми в сингулярному випадку, що розвинута в попереднiх
роботах авторiв.
∗ The work was partially supported by Polish Ministry of Science and High Education (Grant № N514238438).
∗∗ This work was partially supported by DAAD, Germany.
c© V. I. Korobov, S. S. Pavlichkov, W. H. Schmidt, 2012
ISSN 1562-3076. Нелiнiйнi коливання, 2012, т . 15, N◦ 2 205
206 V. I. KOROBOV, S. S. PAVLICHKOV, W. H. SCHMIDT
1. Introduction. The problem of feedback stabilization of control systems by means of the
Lyapunov second method is a wide area developing extensively and it is hardly possible to
give any detailed overview which covers all the aspects.
On the other hand, the problem of stabilization in finite time differs essentially from that of
classical Lyapunov stabilization (for instance, because the uniqueness property of the Cauchy
problem no longer holds true at the corresponding equilibrium point) which requires new
methods. One such constructive method is that of controllability function, which was proposed
in [10] and later developed in [1, 11, 12] and in many other works. In brief, the purport of
this approach is that, instead of looking for a Lyapunov function, one constructs the so-called
„controllability function” (whose properties are different from those of Lyapunov functions)
and finds a feedback to satisfy a certain inequality which can be regarded as an analog of the
Lyapunov inequality.
Let us note that (by its construction) the controllability function method has been applied
to either linear systems or at least to linearizable ones. During the last decade the problem of
stabilization in finite time became a hot topic (see, for example, [2, 7, 8, 16]), especially in the
context of applicability of backstepping designs [7, 8].
Therefore, it is natural to raise the problem of (global) positional synthesis (i.e., that of
stabilization in finite time by means of the controllability function method) for some classes of
nonlinear control systems which are not feedback linearizable. The first candidate on this way
is the triangular form (next called TF) [4 – 6, 9, 13 – 15, 17 – 21, 23 – 25].
Although the TF was originally introduced in [9] as a class of nonlinear but feedback lineari-
zable systems (and later was treated by most authors in the context of feedback linearization,
see for instance [6, 15, 21, 23]), in fact, this class is much richer and the triangular systems of
ODE are not necessarily feedback linearizable (the so-called „singular case” [4, 5, 18, 19, 24]).
Unfortunately, even for the classical Lyapunov stabilization, most results were obtained for the
regular case; as to the singular case, this requires some specific restrictions and/or developing
very specific techniques see [5, 18, 24, 25].
The above-mentioned work [18] was devoted to the global Lyapunov stabilization of the so-
called „generalized” TF systems under the only assumption of the regularity of the equilibrium
point at which the system should be stabilized (see also related work [14] devoted to the global
robust controllability of this class without that assumption). Being motivated by this work as
well as by the controllability function method, we want to combine these two approaches and
to solve the problem of global positional synthesis for the class of TF considered in [18], i.e.,
to construct a feedback, which not only stabilizes this class globally, but also steers any initial
state into the equilibrium point in finite time (however, such a feedback cannot be of class C1
everywhere, because one cannot provide the uniqueness property for the Cauchy problem at
the equilibrium point with such a feedback).
2. Preliminaries. For any vectors ξ = [ξ1, . . . , ξN ] and η = [η1, . . . , ηN ] in RN (with arbitrary
N ∈ N), and any R > 0, we use the following notation:
〈ξ, η〉 :=
N∑
i=1
ξiηi, |ξ| :=
(
N∑
i=1
ξ2
i
) 1
2
, |η| :=
(
N∑
i=1
η2
i
) 1
2
;
BR(ξ) := {η ∈ RN ||η − ξ| < R}.
ISSN 1562-3076. Нелiнiйнi коливання, 2012, т . 15, N◦ 2
GLOBAL POSITIONAL SYNTHESIS AND STABILIZATION IN FINITE TIME OF MIMO GENERALIZED . . . 207
For any Ω ⊂ RN by Ω we denote its closure.
Our background is the following theorem (see [10] and [12, p. 19]).
Theorem 2.1 [10, 12]. Consider the control system
ẋ = f(x, u)
with states x ∈ Rn and controls u ∈ Ω ⊂ Rm, 0 ∈ int Ω and with continuous function f such
that f(0, 0) = 0 and in each set {[x, u]|0 < ρ1 ≤ |x| ≤ ρ2, u ∈ Ω} the following Lipschitz
condition holds:
∣∣f(x1, u1)− f(x2, u2)
∣∣ ≤ L1(ρ1, ρ2)
(
|x1 − x2|+ |u1 − u2|
)
.
Assume that there is a function θ(x) defined on G = {x| |x| ≤ R} with some R > 0 and such
that
1) θ(x) > 0, whenever x 6= 0 and θ(0) = 0;
2) θ(·) is of class C everywhere and of class C1 everywhere except the origin x = 0;
3) there exists C > 0 such that the setQ = {x| θ(x) ≤ C} is bounded andQ ⊂ {x| |x| < R};
4) there exists a feedback u(x) with u(x) ∈ Ω, whenever x ∈ Q, and
∂θ(x)
∂x
f(x, u(x)) =
n∑
i=1
∂θ(x)
∂xi
fi(x, u(x)) ≤ −βθ1− 1
α (x), x ∈ Q \ {0}, (1)
with some α > 0, β > 0, in addition, in each set K(ρ1, ρ2) = {x ∈ Rn|0 < ρ1 ≤ |x| ≤ ρ2} the
feedback u(x) satisfies the following Lipschitz condition:
|u(x1)− u(x2)| ≤ L2(ρ1, ρ2)
∣∣x1 − x2
∣∣ for all {x1, x2} ⊂ K(ρ1, ρ2).
Then, for each x0 ∈ Q and each t0 ∈ R, the trajectory, of the system ẋ = f(x, u(x)),
defined by x(t0) = x0 satisfies limt→t0+T x(t) = 0 and x(t) = 0 for all t ≥ t0 + T with some
T ≤
(
α
β
)
θ
1
α (x0); (note that if α = +∞ then x(t) → 0 as t → +∞).
3. Main result. We consider the following control system:
ẋk = fk(x1, . . . , xk+1), k = 1, . . . , n− 1,
(2)
ẋn = fn(x1, . . . , xn, u)
with controls u ∈ Rm = Rmn+1 , with states x = [x1, . . . , xn]T ∈ Rm1+...+mn , with xi ∈ Rmi ,
mi ≤ mi+1, and with fi(x1, . . . , xi+1) ∈ Rmi . We define
ISSN 1562-3076. Нелiнiйнi коливання, 2012, т . 15, N◦ 2
208 V. I. KOROBOV, S. S. PAVLICHKOV, W. H. SCHMIDT
f(x, u) =
f1(x1, x2)
f2(x1, x2, x3)
. . . . . . . . .
fn(x1, . . . , xn, u)
(3)
and assume that system (2) satisfies the following conditions:
(I) f ∈ Cn+1(Rm1+...+mn ×Rm;Rm1+...+mn),
(II) fi(x1, . . . , xi,R
mi+1) = Rmi for every [x1, . . . , xi] ∈ Rm1 × . . . × Rmi , and every i =
= 1, . . . , n,
(III) there exist x∗i ∈ Rmi , 1 ≤ i ≤ n, and u∗ = x∗n+1 in Rmn+1 such that
rank
∂fi
∂xi+1
(x∗1, . . . , x
∗
i+1) = mi for every i = 1, . . . , n,
and such that f(x∗, u∗) = 0, where x∗ = [x∗1, . . . , x
∗
n].
We want to study the problem of global finite-time stabilization of system (2) into [x∗, u∗].
This means that we are looking for a global feedback u = u(t, x) which is of classC1 everywhere
except R×{x∗} and uniformly bounded in R×Bσ(x∗) with some σ > 0 and which satisfies the
following conditions: (a) every solution of the closed-loop system ẋ = f(x, u(t, x)) given by any
initial condition x(t0) = x0 converges to x∗ in finite time, i.e., there exists a finite T (t0, x
0) > 0
such that x(t) → x∗ as t → t0 + T (t0, x
0) and x(t) = 0 for all t ≥ t0 + T (t0, x
0) and (b) x∗ is
a globally asymptotically stable equilibrium for the closed-loop system ẋ = f(x, u(t, x)). Our
main result is as follows.
Theorem 3.1. Suppose that system (2) satisfies conditions (I) – (III). Then there exists the
above-mentioned feedback u = u(t, x) (at least T -periodic in time with any T > 0) which
solves the above-mentioned problem. Furthermore, in some neighborhood of x∗ ∈ Rm1+...+mn ,
the feedback u(t, x) is time-invariant, i.e., u(t, x) = u(x) and it satisfies all the conditions 1) –
4) of Theorem 2.1 (along with (1)) everywhere in this neighborhood with some controllability
function θ(x).
Given a feedback u(t, x), any t0 ∈ R, and any x0 in Rm1+...+mn , let t 7→ x(t, t0, x
0, u(·, ·))
denote the trajectory, of system (2), defined by this feedback and by the initial condition x(t0) =
= x0 (in our argument, it will be clear from the context that the uniqueness of the solution of
the corresponding Cauchy problem is provided).
First of all, let us explain why we are looking for a feedback of the form u(t, x) although
the dynamics of (2) does not depend on time. This is caused by the global properties of the
“generalized triangular form” defined by conditions (I) – (III) and can be demonstrated by the
following example proposed in [14, 18] for the classical stabilization problem (this is applied
both to the problem of classical global asymptotic stabilization and to our synthesis problem).
Example 3.1. Consider the nonlinear system
ẋ1 = bx3
2 − (1− ax2
1)x2,
(4)
ẋ2 = u,
ISSN 1562-3076. Нелiнiйнi коливання, 2012, т . 15, N◦ 2
GLOBAL POSITIONAL SYNTHESIS AND STABILIZATION IN FINITE TIME OF MIMO GENERALIZED . . . 209
(a > 0, b > 0) which satisfies all the above-mentioned Conditions (I) – (III), and assume, for
instance, that there is a feedback law u = u(x1, x2) of class C1, which globally asymptotically
stabilizes (4) into [0, 0]. Define g(x) by g(x) := [bx3
2−(1−ax2
1)x2, u(x1, x2)]T , and C by C :=
:= {[x1, x2] ∈ R2|ax2
1 + bx2
2 = 1}. Since the feedback u = u(x) is continuous on C, and
globally stabilizes (4), we obtain u(x) 6= 0 for all x ∈ C. Then the map given by C3x 7→
7→ g(x)
|g(x)|
=
[
0,
u(x)
|u(x)|
]T
is well-defined. On the one hand its degree equals 0, but on the other,
there exists a homotopy between this map and the map given by C 3 x 7→ (−x) ∈ C (see the
proof of the well-known Brockett necessary condition for stabilization [3], which can be also
found in [22, p. 184]). This contradiction proves that there is no any feedback u = u(x) of class
C1 which globally stabilizes (4). The same argument can be applied for the global synthesis
problem considered in the current paper. (Note that (4) is even SISO (single-input and single-
output), i.e., with scalar xi and u).
Our next example is concerned with the MIMO (multi-input and multi-output) case and
with different mi.
Example 3.2. Consider the following MIMO nonlinear system
ẋ1 =
(
y2
3−y2
1−1
)
cos y2 − x2
2 − 1,
ẋ2 =
(
y2
3−y2
1−1
)
sin y2 − x2
1 − 1,
ẏ1 = ey1u2, (5)
ẏ2 = u3 cosu1 −
1
x2
1 + 1
,
ẏ3 = u3 sinu1 −
1
x2
2 + 1
,
with states [x1, x2, y1, y2, y3]T ∈ R5 and controls [u1, u2, u3]T ∈ R3. System (5) has form (2)
withm1 = 2, m2 = m3 = 3, and with n = 2. It is clear that system (5) has many singular points
(every point [x1, x2, y1, y2, y3] such that y2
3−y2
1−1 = 0 is singular). However, each equilibrium
point satisfies the following conditions:
(y2
3−y2
1−1)2 = (x2
2+1)2 + (x2
1+1)2 > 0, y2
1+y2
3>0,
and
u2
3 =
(
1
x2
1 + 1
)2
+
(
1
x2
2 + 1
)2
> 0,
and, therefore, it is regular, i.e., satisfies condition (III). Finally, (5) satisfies conditions (I), (II),
and, therefore, our Theorem 3.1 is applicable to system (5) for each equilibrium point of this
system.
ISSN 1562-3076. Нелiнiйнi коливання, 2012, т . 15, N◦ 2
210 V. I. KOROBOV, S. S. PAVLICHKOV, W. H. SCHMIDT
4. Proof of Theorem 3.1. First, we rewrite system (2) in the following form:
ẋ
(1)
1 = f
(1)
1
(
x
(1)
1 , x
(1)
2 , x
(2)
2
)
,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ẋ
(1)
k = f
(1)
k
(
x
(1)
1 , . . . , x
(1)
k , . . . , x
(k)
k , x
(1)
k+1, . . . , x
(k+1)
k+1
)
,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ẋ
(k)
k = f
(k)
k
(
x
(1)
1 , . . . , x
(1)
k , . . . , x
(k)
k , x
(1)
k+1, . . . , x
(k+1)
k+1
)
, (6)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ẋ(1)
n = f (1)
n
(
x
(1)
1 , . . . , x(1)
n , . . . , x(n)
n , u(1), . . . , u(n+1)
)
,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ẋ(n)
n = f (n)
n
(
x
(1)
1 , . . . , x(1)
n , . . . , x(n)
n , u(1), . . . , u(n+1)
)
,
where
xk =
[
x
(1)
k , . . . , x
(k)
k
]T
∈ Rmk , fk =
[
f
(1)
k , . . . , f
(k)
k
]T
∈ Rmk , k = 1, . . . , n,
u =
[
u(1), . . . , u(n+1)
]T
∈ Rmn+1
with
x
(i)
i ∈ Rm1 , f
(i)
i ∈ Rm1 , i = 1, . . . , n, u(n+1) ∈ Rm1 , u(1) ∈ Rmn+1−mn
and
x
(j)
i ∈ Rmi−j+1−mi−j , f
(j)
i ∈ Rmi−j+1−mi−j , 1 ≤ j ≤ i− 1, i = 2, . . . , n,
and x(j)
i , f
(j)
i are defined by the following conditions:
rank
∂f
(j)
k
(
x∗1, . . . , x
∗
k+1
)
∂x
(k+1)
k+1
= m1, k = 1, . . . , n, rank
∂f
(n)
n (x∗, u∗)
∂u(n+1)
= m1,
(7)
rank
∂(f
(1)
k , . . . , f
(k)
k ) (x∗)
∂(x
(2)
k+1, . . . , x
(k+1)
k+1 )
= mk, k = 1, . . . , n, rank
∂f
(n)
n (x∗, u∗)
∂(u2, . . . , u(n+1))
= mn.
Let us note that if mi−j = mi−j+1, then x
(j)
i is omitted (its dimension equals zero); in
particular, for m1 = . . . = mn = mn+1, we have xk = x
(k)
k ∈ Rmk = Rmn+1 , k = 1, . . . , n,
u = u(n+1) ∈ Rmn+1 .
ISSN 1562-3076. Нелiнiйнi коливання, 2012, т . 15, N◦ 2
GLOBAL POSITIONAL SYNTHESIS AND STABILIZATION IN FINITE TIME OF MIMO GENERALIZED . . . 211
Then we consider the following transformation of states and controls (note that this is
a natural extension of the transformation considered in [9] for the special case m1 = . . .
. . . = mn+1 = 1):
ξ
(1)
1 = F
(1)
1 (x1) := x
(1)
1 ,
ξ
(1)
k+1 = F
(1)
k+1(x1, . . . , xk, xk+1) := x
(1)
k+1,
ξ
(l)
k+1 = F
(l)
k+1(x1, . . . , xk, xk+1) :=
:=
k∑
i=1
i∑
j=1
∂F
(l−1)
k
∂x
(j)
i
f
(j)
i (x1, . . . , xi+1), for 2 ≤ l ≤ k + 1, 1 ≤ k ≤ n− 1, (8)
v(1) = F
(1)
n+1(x, u) := u(1),
v(l) = F
(l)
n+1(x, u) :=
n−1∑
i=1
i∑
j=1
∂F
(l−1)
n
∂x
(j)
i
f
(j)
i (x, u) +
n∑
j=1
∂F
(l−1)
n
∂x
(j)
n
f (j)
n (x, u), l = 2, . . . , n+ 1.
It is easy to show that (8) is a local diffeomorphism around [x∗, u∗] and (in some nei-
ghborhood of [x∗, u∗]) maps diffeomorphically the trajectories of system (6) (i.e., system (2))
onto the trajectories of the following linear system:
ξ̇
(i)
k+i−1 = ξ
(i+1)
k+i , i = 1, . . . , n− k, ξ̇(n−k+1)
n = v(n−k+2), k = 1, . . . , n− 1,
(9)
ξ̇(1)
n = v(2).
Rewrite system (9) in the following form:
ξ̇ = Aξ +Bv, (10)
where A and B are the corresponding matrices defined by system (9). (It is clear that (10) is
globally controllable.)
According to the result of [11] (see also [12, p. 121 – 139]), there exist some a0 > 0 and some
neighborhood of 0 such that the equation
2a0θ − 〈N−1(θ)ξ, ξ〉 = 0
with
N(θ) :=
+∞∫
0
e−
t
θ e−AtBB∗e−A
∗t dt
has a unique solution for each ξ ∈ Br(0) \ {0} (for some small r > 0) and defines a positive
definite function θ(ξ) of class C1 on Br(0) \ {0} and of class C on Br(0) (for ξ = 0, θ(0) = 0 by
the definition of θ(ξ)) and such that the control
w(ξ) :=
{
−B∗N−1(θ(ξ))ξ if ξ ∈ Br(0) \ {0},
0 if ξ = 0
ISSN 1562-3076. Нелiнiйнi коливання, 2012, т . 15, N◦ 2
212 V. I. KOROBOV, S. S. PAVLICHKOV, W. H. SCHMIDT
with this controllability function satisfies all the conditions 1) – 4) of Theorem 2.1 with some
α > 0, β > 0, and, therefore (locally, around the origin), resolves the synthesis problem in the
sense of Theorem 2.1. Furthermore ([12, p. 127] or [11]), for any d > 0, it is possible to find
some a0 > 0 such that |w(ξ)| ≤ d, whenever ξ ∈ Br(0).
Let x = Φ(ξ), u = Ψ(ξ, v) be a diffeomorphism which is inverse to the map given by (8)
and which (locally, around the origin) maps the trajectories of system (10) onto the trajectories
of system (2). Define
û(x) := Ψ(F (x), w(F (x))) and Θ(x) = θ(F (x)),
whereF (x) is the transformation of coordinates defined in (8) which (along with the transformati-
on of controls given by (8)) brings (2) (or (6)) to (10).
Then, in some open neighborhood of x∗, we obtain
∂Θ(x)
∂x
f(x, û(x)) ≤ −β
α
Θ1− 1
α (x), for all x 6= x∗ such that Θ(x) ≤ ρ0 (11)
with some ρ0 > 0. On the other hand, using the result of [18], we obtain that for any T > 0,
there exist R0 > 0 and some T -periodic feedback v(t, x) of class C1, which globally asymptoti-
cally stabilizes system (2), and there is a T -periodic Lyapunov function V (t, x) of class C1 defi-
ned in some neighborhood of R× {x∗} such that V is positive definite (V ≥ 0 and V (t, x) = 0
if and only if x = x∗); V has the form
V (t, x) = |x1 − x∗1|2 + |x2 − α1(t, x1)|2 + . . .+ |xn − αn−1(t, x1, . . . , xn−1)|2
with αi(t, x∗1, . . . , x
∗
i ) = x∗i+1 ∈ Rmi+1 and αi(t + T, x1, . . . , xi) = αi(t, x1, . . . , xi) for all t (αi
being of class C1) and such that
∂V
∂t
+
∂V
∂x
f(x, v(t, x)) ≤ −V (t, x) for all [t, x] such that V (t, x) ≤ R0.
Remark 1. Let us note that, since (2) is time-independent, the argument from [18] actually
yields that V (t, x) and v(t, x) are also time-independent locally in some neighborhood of R ×
×{x∗} (whereas v(t, x) is time-varying and T -periodic globally, in general), therefore, without
loss of generality we may assume that V (t, x) = V (x), v(t, x) = v(x) and αi(t, x1, . . . , xi) =
= αi(x1, . . . , xi) for all [t, x] such that V (t, x) ≤ R0.
Fix any R in ]0, R0[. For any c > 0, denote:
Γc := R×
{
x ∈ Rm1+...+mn |Θ(x) ≤ c
}
,
Λc :=
{
[t, x] ∈ R×Rm1+...+mn |V (t, x) < c
}
.
Remark 2. It is straightforward that both Γc (see [12, p. 122] as well as [1, 10, 11]) and Λc (see
the form of V (t, x)) have the following properties: Γρ̂ and ΛR̂ are bounded for all ρ̂ ∈ [0, ρ0] and
R̂ ∈ [0, R0], respectively, and sup[t,x]∈Γc |x− x
∗| → 0 and sup[t,x]∈Λc |x− x
∗| → 0 as c → +0.
Take any small ε0 > 0 and any small ρ ∈]0, ρ0[ such that Γρ+ε0 ⊂ ΛR.
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GLOBAL POSITIONAL SYNTHESIS AND STABILIZATION IN FINITE TIME OF MIMO GENERALIZED . . . 213
Given any ε ∈]0, ε0[, take some T -periodic function λ(t, x) of class C∞(R × Rm1+...+mn ;
[0, 1]) such that λ(t, x) = 1 if [t, x] ∈ Γρ; λ(t, x) = 0 if [t, x] /∈ Γρ+ε. Let uε(t, x) of class
C(R×Rm1+...+mn ;Rmn+1)
⋂
C1(R× (Rm1+...+mn \ {x∗});Rmn+1) be given by
uε(t, x) = λ(t, x)û(x) + (1− λ(t, x))v(t, x).
(Note that, although û(x) was defined locally around x∗ only, uε(t, x) is well-defined everywhere
because, if [t, x] /∈ Γρ+ε0 , then λ(t, x) = 0, i. e., the first term is vanishing.) It is clear that for
every [t0, x
0] ∈ R×Rm1+...+mn we obtain:
1) if [t0, x0] ∈ Γρ, then [t, x(t, t0, x
0, uε(·, ·))] ∈ Γρ for all t ≥ t0 and there is T (t0, x
0) > t0
such that x(t, t0, x
0, uε(·, ·)) → x∗ as t → T (t0, x
0) (furthermore, along such a trajectory, the
inequality (11) holds);
2) if [t0, x0] ∈ ΛR, then [t, x(t, t0, x
0, uε(·, ·))] ∈ ΛR for all t ≥ t0;
3) for every [t0, x
0] ∈ R×Rm1+...+mn there exists t1 ≥ t0 such that [t, x(t, t0, x
0, uε(·, ·))] ∈
∈ ΛR for all t ≥ t1.
Take any σ > 0 such that Λσ+δ ⊂ Γρ for some small δ > 0. Then, there are some n ∈ N
and some m ∈ N such that for any [t0, x0] ∈ ΛR we have [t, x(t, t0, x
0, v(·, ·))] ∈ Λσ for all
t ≥ t0 + nT and for any [t0, x0] ∈ Γρ we have x(t, t0, x
0, û(·)) = x∗ for all t ≥ t0 +mT.
Given h ∈
]
0,
T
2
[
, find any (m+n+2)T -periodic function µh(t) of class C∞(R; [0, 1]) such
that
µh(t) =
0 if k(m+n+2)T+h ≤ t ≤ (k+1)(n+1)T+k(m+1)T−h,
1 if (k+1)(n+1)T+k(m+1)T+h ≤ t ≤ (k+1)(m+n+2)T−h
for all k ∈ Z, and define
ûh(t, x) := µh(t)uε(t, x) + (1− µh(t))v(t, x).
Then it is straightforward to check that if h ∈
]
0,
T
2
[
is small enough, then the feedback ûh(t, x)
resolves our problem of global stabilization in finite time and satisfies Theorem 3.1.
Acknowledgement. This work was performed during April – May 2008 while the second
author was visiting Institute of Mathematics and Informatics, Ernst-Moritz-Arndt University
of Greifswald. Svyatoslav Pavlichkov is grateful for the warm hospitality provided by Professor
Werner H. Schmidt.
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Received 18.05.10,
after revision — 15.11.11
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