Mem. Differential Equations Math. Phys. 41 (2007), 157–162
N. A. Izobov, S. E. Karpovich, L. G. Krasnevsky,
and A. V. Lipnitsky
QUASI-INTEGRALS OF THREE-DIMENSIONAL LINEAR
DIFFERENTIAL SYSTEMS WITH SKEW-SYMMETRIC
COEFFICIENT MATRICES
(Reported on July 2, 2007)
We consider the linear system
ẋ = A(t)x, x ∈ R3 , t ≥ 0,
(1A )
with a continuous piecewise differentiable skew-symmetric matrix A(·) ≡
(aij )3i,j=1 for all t ≥ 0. Such systems coincide with kinematic equations of
the rigid body mechanics, in particular, they are applied in robotics [1] in
modelling automatized production based on automatic holonomic systems
for parametric construction of programmed motions of executive devices in
a three dimensional physical space. Four-dimensional systems with skewsymmetric coefficient matrix are also applied in the gyroscope theory [2].
Following [3,4], for the elements aij (t) of the skew-symmetric matrix A(t)
we define: the function vector
a(t) ≡ a23 (t), −a13 (t), a12 (t) ∈ R3 , t ≥ 0,
the scalar functions
C(η) ≡ cos
Zη
0
ka(τ )kdτ, S(η) ≡ sin
Zη
0
ka(τ )kdτ, t ≥ 0,
and the vector function of two-variables


−(a212 (t) + a213 (t))C(η)
v(t, η) ≡ −a13 (t)a23 (t)C(η) + a12 (t)ka(t)kS(η) , t, η ∈ [0, +∞).
a12 (t)a23 (t)C(η) + a13 (t)ka(t)kS(η)
In the above-mentioned works, for the quasi-integrals
L1 (x(t), t) ≡ (x(t), a(t)) − (x(0), a(0)),
L2 (x(t), t) ≡ (x(t), v(t, t)) − (x(0), v(0, 0)),
2000 Mathematics Subject Classification. 34A30, 34C41, 34D10.
Key words and phrases. Quasi-integrals, skew-symmetric matrix, linear differential
system.
158
of the non-stationary system (1A ) on its solutions x(·) : [0, +∞) → R3 ,
which are ordinary integrals in the stationary case and identically vanish,
the estimates
Zt
L1 (x(t), t) ≤ kx(0)k kȧ(τ )k dτ, t ≥ 0,
(21 )
0
L2 (x(t), t) ≤ c2 kx(0)k
Zt
0
ka(τ )k kȧ(τ )k dτ, t ≥ 0,
(22 )
√
are obtained with the constant c2 = 2 3. In those papers it is also proved
that the first estimate may turn into equality (be efficient), whereas for the
second one this was established for c2 = 1.
The authors of the present paper improved the estimate (22 ) up to the one
with c2 = 2 and proved that the latter is unimprovable. It should be noted
that the efficiency of both estimates (the estimate (21 ) and the estimate
(22 ) with the constant c2 = 2) is realized for different three-dimensional
systems (1A ). In this connection, we have the following two problems on
the simultaneous efficiency of the estimates (21 ) and (22 ) with c2 = 2: 1)
efficiency of these estimates for the common system (1A ) but, probably, for
its different its solutions; 2) simultaneous efficiency of both estimates for
one nontrivial solution x(t) of the same system (1A ).
The aim of this paper is to prove that the estimates (21 ) and (22 ) with
c2 = 2 cannot be efficient simultaneously for one nontrivial solution x(t) of
the system (1A ) at the same moment of time t = t0 > 0 such that ȧ(τ ) 6≡ 0
for τ ∈ [0, t0 ].
The following theorem establishes this.
Theorem. Let x(·) : R+ → R3 \ {0} be an arbitrary solution of any
three-dimensional system (1A ), and h be a fixed constant, h ∈ (0.9; 1]. If
for some t0 > 0 the estimate
Zt0
L1 (x(t0 ), t0 ) ≥ hkx(0)k kȧ(τ )k dτ
(3)
0
is fulfilled, then the inequality
L2 (x(t0 ), t0 ) ≤
t0
Z
√ (h − 0.8)(h − 0.9) i
≤ 2 1 − (2 − 2)
kx(0)k kȧ(τ )k ka(τ )k dτ
2+h
h
(4)
0
is valid.
Proof. The statement of the theorem is evident if ȧ(τ ) ≡ 0 for all τ ∈ [0, t0 ].
Thus let us consider the opposite case. We introduce the vectors
e1 := (1, 0, 0) ∈ R3 , w(t) := e1 × a(t),
f (t) := ka(t)ke1 , t ≥ 0.
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Then the vector function v(t, η) satisfies the equality
v(t, η) = (a(t) × w(t))C(η) − ka(t)kw(t)S(η), t, η ≥ 0.
(5)
According to Lemma 2 in [1], the equality
L2 (x(t), t) =
Zt 0
x(τ ),
∂v(τ, t) dτ, t ≥ 0,
∂τ
(6)
is valid. We now estimate the absolute value of the scalar product under
the integral sign in (6) by using the inequality |ka(τ )k0 | ≤ kȧ(τ )k (see [3])
and the pairwise orthogonality of the vectors f (τ ) and w(τ ), as well as e 1
and e1 × ȧ(τ ):
∂v(τ, t) =
x(τ ),
∂τ
n
o0 = x(τ ), C(t) w(τ ) × a(τ ) − S(t) f (τ ) × a(τ )
=
τ
n
o
0
= x(τ ), C(t)w(τ ) − S(t)f (τ ) × a(τ )
≤
τ
0
≤ x(τ ), C(t)w(τ ) − S(t)f (τ ) τ × a(τ ) +
+ x(τ ), C(t)w(τ ) − S(t)f (τ ) × ȧ(τ ) ≤
≤ kx(0)k ka(τ )k C(t) e1 × ȧ(τ ) − S(t)ka(τ )k0 e1 +
o1/2
n
≤
+x(τ ) × ȧ(τ ) C 2 (t)kw(τ )k2 + S 2 (t)kf (τ )k2
(here the use is made of the equality kf (τ )k = ka(τ )k and the estimate
kw(τ )k ≤ ka(τ )k for all τ ≥ 0)
x(τ )
× ȧ(τ ) , 0 ≤ τ ≤ t.
≤ kx(0)k ka(τ )k kȧ(τ )k + kx(τ )k
Thus, by virtue of the above inequality, from the equality (6) we obtain the
following estimate for all t ≥ 0:
Zt
x(τ )
L2 (x(t), t) ≤ kx(0)k ka(τ )k ȧ(τ ) + × ȧ(τ ) dτ. (7)
kx(τ )k
0
Suppose now that the estimate (3) is fulfilled for some t = t0 > 0. Let
Zt0
p
s(τ ) := sin ∠ x(τ ), ȧ(τ ) , c(τ ) := 1 − s2 (τ ), I0 := ȧ(τ ) dτ.
0
√ Define also the set T0 ≡ τ ∈ [0, t0 ] : s(τ ) ≤ 1/ 2 and its complement
CT0 ≡ [0, t0 ] \ T0 in [0, t0 ]. Since every solution of the system (1A ) satisfies
for all t ≥ 0 the equality kx(τ )k ≡ kx(0)k, without loss of generality we can
160
assume that in the estimates (3) and (4) the equality kx(τ )k ≡ 1, τ ≥ 0, is
identically fulfilled.
Lemma 2 in [3] implies the estimates
Zt0
hI0 ≤ L1 (x(t0 ), t0 ) = x(τ ), ȧ(τ ) dτ ≤
0
≤
≤
Zt0
Z0
T0
x(τ ), ȧ(τ ) dτ ≤
ȧ(τ ) dτ +
Z
CT0
Zt0
0
ȧ(τ ) cos ∠ x(τ ), ȧ(τ ) dτ ≤
ȧ(τ ) |c(τ )| dτ ≤
(we now use the evident equality | cos α| ≤ 1 − 2−1 sin2 α)
Z
Z
ȧ(τ ) dτ + ȧ(τ ) dτ.
≤ max 1 − 2−1 s2 (τ )
τ ∈CT0
CT0
T0
√
Since the estimate s(τ ) ≥ 1/ 2 holds for all τ ∈ CT0 , we have
Z
1
ȧ(τ ) dτ,
hI0 ≤ I0 −
4
CT0
whence
R
CT0
kȧ(τ )k dτ ≤ 4(1 − h)I0 . The last estimate yields
Z
T0
ȧ(τ ) dt ≥ (4h − 3)I0 .
(8)
Consider now the case ka(t0 )k ≥ ka(0)k (the opposite case can be treated
analogously). Under this assumption, the estimate (3) implies that
n
o
ka(t0 )k = max ka(t0 )k, ka(0)k ≥ max (x(t0 ), a(t0 )), (x(0), a(0)) ≥
≥ 2−1 (x(t0 ), a(t0 )) − (x(0), a(0)) ≥ hI0 /2.
(9)
Next we define the set
Zt0
T ≡ t ∈ [0, t0 ] :
ȧ(τ ) dτ ≤ 0.4I0
t
for which the equality
R
T
kȧ(τ )k dτ = 0.4I0 is obviously fulfilled. Using the
estimate (8), we obtain the inequalities
Z
Z
Z
Z
ȧ(τ ) dτ = . . . dτ + . . . dτ −
I0 ≥
T0 ∪T
T0
T
T0 ∩T
. . . dτ ≥
161
≥ (4h − 2.6)I0 −
Z
T0 ∩T
ȧ(τ ) dτ.
R
These inequalities result in the estimate
T0 ∩T
Moreover, the inequality
min ka(t)k ≥ ka(t0 )k − max
t∈T
t∈T
implies the estimates
Z
T0
≥ min ka(τ )k
τ ∈T
Z
T0 ∩T
Zt0
t
kȧ(τ )k dτ ≥ (4h − 3.6)I0 .
ȧ(τ ) dτ = ka(t0 )k − 0.4I0
ka(τ )k ȧ(τ ) dτ ≥
ȧ(τ ) dτ ≥ (4h − 3.6) ka(t0 )k − 0.4I0 I0 .
Thus, by virtue of the inequality (11), the following estimates are valid:
J0 ≡
Zt0
≤
Z
0
s(τ )ka(τ )k ȧ(τ ) dτ ≤
ka(τ )k ȧ(τ ) dτ + max s(τ )
τ ∈T0
CT0
Z
T0
ka(τ )kȧ(τ ) dτ ≤
≤
Zt0
√
Z
2−1
ka(τ )k ȧ(τ ) dτ − √
ka(τ )k ȧ(τ ) dτ ≤
2
≤
Zt0
√
ka(τ )k ȧ(τ ) dτ − 2(2 − 2)(h − 0.9) ka(t0 )k − 0.4I0 I0 .
0
0
T0
(10)
Moreover, the inequalities
Zt0
J1 ≡ ka(τ )k ȧ(τ ) dτ ≤
0
≤ max ka(τ )k
τ ∈[0,t0 ]
Zt0
t
ȧ(τ ) dτ ≤ ka(t0 )k + I0 I0
(11)
are also fulfilled. Obviously, the inequality (9) is equivalent to the estimate
ka(t0 )k − 0.4I0 ≥ b(ka(t0 )k + I0 ,
where b ≡ (h − 0.8)/(2 + h).
Using (7), (10) and (11), we get the relations
L2 (x(t0 ), t0 ) ≤ (J1 + J0 ) ≤
162
√
2)(h − 0.9) ka(t0 )k − 0.4I0 I0 ≤
√
≤ 2J1 − 2b(2 − 2)(h − 0.9) ka(t0 )k + I0 I0 ≤
h
√ (h − 0.8)(h − 0.9) i
J1 .
≤ 2 1 − (2 − 2)
2+h
By virtue of Lemma 2 in [3], the latter inequalities imply the desired inequality (4).
Thus the theorem is proved.
≤ 2J1 − 2(2 −
Acknowledgement
The research was carried out by the Financial support of the National
Academy of Sciences of Belarus, the State complex scientific research program “Mechanics”.
References
1. N. B. Ignat’ev, Holonomic automatic systems. (Russian) Izd. Akad. Nauk SSSR,
Moscow–Leningrad, 1963.
2. I. V. Gaishun, Introduction into the linear non-stationary systems theory. (Russian)
Nauka, Minsk, 1999.
3. S. E. Karpovich, L. G. Krasnevskiı̆, N. A. Izobov, and E. A. Barabanov, On
the integrals of three-dimensional linear differential systems with skew-symmetric
matrices of coefficients. (Russian) Differ. Uravn. 42(2006), No. 8, 1027–1034.
4. S. E. Karpovich, L. G. Krasnevskiy, N. A. Izobov, E. A. Barabanov, On quasiintegrals of non-stationary three-dimensional linear differential systems with antisymmetric coefficient matrix. Mem. Differential Equations Math. Phys. 39(2006),
149–153.
(Received 9.07.2007)
Authors’ addresses:
N. A. Izobov and A. V. Lipnitsky
Institute of Mathematics
National Academy of Sciences of Belarus
Surganov 11, 220072 Minsk, Republic of Belarus
E-mail: izobov@im.bas-net.by
S. E. Karpovich
Byelorussian state university of informatics and radioelectronics
P. Brovki 6, 220013 Minsk, Republic of Belarus
E-mail: mmts@bsuir.by
L. G. Krasnevsky
Joint Institute of Machinery
National Academy of Sciences of Belarus
Academicheskaja 12, 220072 Minsk, Republic of Belarus
E-mail: adashek@mail.ru