Investigations of the properties of solutions of impulsive differential systems in the linear approximation
Проведено якiсний аналiз властивостей розв’язкiв нелiнiйних iмпульсних систем диференцiальних рiвнянь iз збуреннями на деяких гiперповерхнях {ti(x)}. Отримано новi умови обмеженостi, практичної стiйкостi за Четаєвим (рiвномiрної, стискуючої), стiйкостi за Ляпуновим (рiвномiрної, асимптотичної), прит...
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| Cite this: | Investigations of the properties of solutions of impulsive differential systems in the linear approximation / Yu.A. Mitropolsky, S.D. Borysenko, S. Toscano // Доп. НАН України. — 2008. — № 7. — С. 36-42. — Бібліогр.: 15 назв. — англ. |
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| author | Mitropolsky, Yu.A. Borysenko, S.D. Toscano, S. |
| author_facet | Mitropolsky, Yu.A. Borysenko, S.D. Toscano, S. |
| citation_txt | Investigations of the properties of solutions of impulsive differential systems in the linear approximation / Yu.A. Mitropolsky, S.D. Borysenko, S. Toscano // Доп. НАН України. — 2008. — № 7. — С. 36-42. — Бібліогр.: 15 назв. — англ. |
| collection | DSpace DC |
| description | Проведено якiсний аналiз властивостей розв’язкiв нелiнiйних iмпульсних систем диференцiальних рiвнянь iз збуреннями на деяких гiперповерхнях {ti(x)}. Отримано новi умови обмеженостi, практичної стiйкостi за Четаєвим (рiвномiрної, стискуючої), стiйкостi за Ляпуновим (рiвномiрної, асимптотичної), притягування розв’язкiв збурених систем за лiнiйним наближенням.
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| first_indexed | 2025-11-27T15:17:56Z |
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UDC 517.911
© 2008
Academician of the NAS of Ukraine Yu. A. Mitropolsky ,
S. D. Borysenko, S. Toscano
Investigations of the properties of solutions of impulsive
differential systems in the linear approximation
Проведено якiсний аналiз властивостей розв’язкiв нелiнiйних iмпульсних систем ди-
ференцiальних рiвнянь iз збуреннями на деяких гiперповерхнях {ti(x)}. Отримано но-
вi умови обмеженостi, практичної стiйкостi за Четаєвим (рiвномiрної, стискуючої),
стiйкостi за Ляпуновим (рiвномiрної, асимптотичної), притягування розв’язкiв збуре-
них систем за лiнiйним наближенням.
In work [1], two-parametric scale increasing functions were first considered to investigate the
problem of stability of solutions of the nonlinear system dx/dt = A(t)x + F (t, x) in the linear
approximation dy/dt = A(t)y (shorted system). The Cauchy matrix of the shorted system sati-
sfies such an estimate: ‖C(t, τ)‖ 6 c exp[α(t − τ)][t/τ ]β , for τ > 1, where α > α∗ + ε, ε > 0,
α∗ = max
k
αk(k = 1, 2, . . . , n), αk = lim
t→∞
t−1 ln ‖yk(t)‖, k = 1, n, (αk is the characteristic index
of a Lyapunov nontrivial solution yk(t) of the shorted system), β > β = max
k
βk, k = 1, n, where
the characteristic degree by Lyapunov βk = lim
t→∞
[(ln t)−1 ln{‖yk(t)‖ exp[−αkt]}], k = 1, n.
In work [2], the author investigated the problem of stability by Lyapunov of solutions of a
nonlinear system in the linear approximation by using an estimate of the Cauchy matrix of the
shorted system of such a type: ‖C(t, τ)‖ 6 η(t)l(τ), η(t) : R
+ → R
+, η(t) ∈ C(R+), l(τ) ∈ C(R+).
Later on, Borysenko and Martynyuk (Mat. Fizika, 1980, N 2) used this estimate to investigate,
in the linear approximation, the problem of practical stability (by Chetaev; uniform, attractive)
of solutions of a nonlinear regular system with nonlinearities on the right-hand side of the system
either the Lipschitz or Hölder type (the evolution of processes which describe the system can
be either finite or infinite). The estimate from [2] was also used to estimate the Cauchy matrix
of a system of variations and to investigate the problem of stability of solutions of a nonlinear
system in the nonlinear approximation [see Mat. Fizika, 1981, N 1 (Borysenko)]. These results
are generally based on the method of integral inequalities for continuous functions [3] and its
applications. In 1983 (Ukr. Math. Journ., N 2), Borysenko considered a generalization of the
idea of Demidovich [1] by using the two-parametric scale of increasing functions to investigate
the properties of solutions of nonlinear impulsive differential systems in the linear approximation
(linear impulsive differential systems) and used the following estimate of the Cauchy matrix of
an impulsive shorted system: ‖C(t, t0)‖ 6 ceα(t−t0)[t/t0]
β, where c, α, and β are some constants,
and t > t0 > 1.
In the monograph by Lakshmikantham, Bainov, Simeonov [4], the problem of stability by
Lyapunov of solutions of the impulsive nonlinear differential system under a pulse influence at
fixed time moments,
dx
dt
= a(t)x+ g(t, x), t 6= ti, ∆x|t=ti = Bx+ Ii(x), (1)
36 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2008, №7
was investigated by assuming that the Cauchy matrix of the linear system dx/dt = a(t)x without
pulses satisfies the estimate ‖C(t, τ)‖ 6 ϕ(t)ψ(τ) [2], and that the Cauchy matrix of the impulsive
linear system
dx
dt
= a(t)x, t 6= ti, ∆x = Bx, t 6= ti, (2)
satisfies the estimate ‖C(t, τ)‖ 6 ϕ(t)ψ(τ)
∏
τ<ti<t
γiϕ(t+i )ψ(t+i ), where ϕ : R
+ → R
+; ψ : R
+ →
→ R
+, ‖g(t, x)‖ 6 l)(t)‖x‖m, m > 1, l(t) : R
+ → R
+, l(t) ∈ C(R+), ‖Ii(x)‖ 6 γi‖x‖, γi =
= const > 0.
In the monograph by Samoilenko, Borysenko, Matarazzo, Toscano, Yasinsky [5], the problems
of the stability by Lyapunov and the practical stability by Chetaev of solutions of system (1) were
investigated by assuming that the Cauchy matrix of the shorted system (2) satisfies the estimate
‖C(t, τ)‖ 6 η(t)l(τ), η : R
+ → R
+, l : R
+ → R
+, η ∈ C(R+), l ∈ C(R+) and ‖g(t, x)‖ 6
6 l(t)‖x‖m, m = const > 0, ‖Ii(x)‖ 6 γi‖x‖.
The conditions for the stability, as well as for the practical stability, of the trivial solution
of system (1) in the linear approximation (2) were found in [4, 5] by using analogies of the
Gronwall–Bellman–Bihari lemmas for the discontinuous functions for integral inequalities of such
a type:
u(t) 6 c+
t∫
t0
ν(τ)um(τ) dτ +
∑
t0<ti<t
kiu(ti − 0), m > 1. (3)
In report [6], Danylo S. Borysenko found the integral inequality
u(t) 6 ϕ(t) +
t∫
t0
ν(τ)um(τ) dτ +
∑
t0<ti<t
kiu
p(ti − 0) (4)
and obtained new analogies of the Gronwall–Bellman–Bihari lemmas for the discontinuous functi-
ons (see Remark 2).
In works [7, 8] with the use of results obtained in [6], the new estimates of solutions of
impulsive nonlinear systems with nonlinearities on the right-hand side of a system not only of
the Lipschitz type but also of the Hölder one were obtained.
The investigations in [9, 10] are devoted to the problems of the stability and the practical
stability of solutions of impulsive nonlinear systems in the linear and nonlinear approximations
under assumption that the pulse forces are characterized by functions of the Lipschitz type.
In the monograph by Borysenko, Iovane [11] and in works [12–15], the method of integral
inequalities for discontinuous functions and its applications to the qualitative analysis of properti-
es of the solutions of impulsive differential systems with nonlinearities of different kinds on the
right-hand sides have obtained the further development (by including also the impulsive systems
of partial differential hyperbolic equations).
In this article, we use the results of investigations performed by Bainov, Iovane, Lakshmi-
kantham, Leela, Martynyuk, Samoilenko, Simeonov (see, e. g., [1–15]).
Preliminary considerations. Let us introduce the impulsive system of ordinary differential
equations of such a form:
dx
dt
= f(t, x), t 6= ti(x), ∆x|t=ti(x) = Ii(x). (5)
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2008, №7 37
Let the following assumptions be fulfilled:
(H1) f , Ii are defined in the domain Ω = {(t, x) : t ∈ J = [t0, T ], T 6 ∞, t0 > 1, ‖x‖ 6 h}
and f(t, 0) = Ii(0), ∀ t ∈ J , ∀ i ∈ N;
(H2) f(t, x) = A(t)x + r(t, x), Ii(x) = Bix + Ji(x), where A(t), Bi are some matrices;
det(Bi + E) 6= 0 ∀ i;
(H3) {ti(x)} : t1(x) < t2(x) < · · · , lim
i→∞
ti(x) = ∞ uniformly for x ∈ Ω;
(H4) ∃ θ = const > 0: inf
‖x‖6h
ti(x) − sup
‖x‖6h
ti−1(x) = θ > 0, ∀ i ∈ N;
(H5) if x(t) = x(t, t0, x0) the solution of Cauchy problem for system (5) and t∗i -moments of
time: t∗i = x(t)
⋂
(t = ti(x)), then inf
‖x‖6h
t∗i (x) > t∗i > sup
‖x‖6h
ti−1(x), ∀ i ∈ N;
(H6) ∃θi = const > 0: θ1τ < i(t, t + τ) < θ2τ ; t0 < t∗1 < t∗2 < · · · , lim
i→∞
t∗i = ∞; where i(a, b)
is a number of points {t∗i } ∈ [a, b] ⊂ [t0, T ], T 6 ∞ (θi depends only on τ);
(H7) ∃L = const > 0:
∥∥∥∥
∂ti(x)
∂x
∥∥∥∥ 6 L ∀ i ∈ N, x ∈ Ω, sup
06σ61
(
∂ti(x+ σIi(x))
∂x
, Ii(x)
)
6 0;
(H8) ∃ r(t) : R
+ → R
+, r(t) 6 r = const < ∞, r ∈ C(R+) : ‖r(t, x)‖ 6 r(t)‖x‖α∗
, α∗ =
= const > 0; ∃ ki = const > 0: ‖Ji(x)‖ 6 ki‖x‖
β , β = const > 0, i ∈ N;
(H9) the Cauchy matrix C(t, t0) of the shorted system
dx
dt
= A(t)x, t 6= t∗i , ∆x = Bix, t = t∗i (6)
satisfies such an estimate: ‖C(t, t0)‖ 6 c exp[(α̃ + θi lnα)(t− t0)][t/t0]
β̃, where the α̃ parameter
is the characteristic index of a Lyapunov nontrivial solution of the system dx/dt = A(t)x, the β̃
parameter is connected with the characteristic Lyapunov degree of this system, and
α2 = max
j
λj [(Bj + E)T (Bj + E)], θi lnα =
{
θ1 lnα, 0 < α < 1;
θ2 lnα, α > 1.
Remark 1. As for the notion of stability and practical stability of solutions of system (5),
see, e. g., [5].
Remark 2. To estimate the solutions of system (5), we will use such results [6–8]:
A. Let u(t) be a nonnegative piecewise continuous function at t > t0, with first-kind disconti-
nuities at the points {ti} satisfying the “integro-sum” inequality
u(t) 6 ϕ(t) +
t∫
t0
g(τ)u(τ) dτ +
∑
t0<ti<t
aiu
m(ti − 0), (7)
where ϕ(t) > 0, g(t) > 0, ai = const > 0, {ti} : t1 < t2 < · · · , lim
i→∞
ti = ∞, m = const > 0, and
ϕ(t) is nondecreasing on J = [t0, T ], T 6 ∞.
Then
u(t) 6 ϕ(t)
∏
t0<ti<t
(1 + aiϕ
m−1(ti)) exp
[ t∫
t0
g(s) ds
]
, (8)
38 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2008, №7
where 0 < m 6 1, ∀ t > t0;
u(t) 6 ϕ(t)
∏
t0<ti<t
(1 + aiϕ
m−1(ti)) exp
[
m
t∫
t0
g(s) ds
]
, (9)
where m > 1, ∀ t > t0.
B. Let the function u(t) be nonnegative piecewise continuous on J with first-kind disconti-
nuities at the points {ti} t0 < t1 < t2 < · · · , lim
i→∞
ti = ∞ and satisfy the inequality
u(t) 6 ϕ(t) +
t∫
t0
g(τ)um(τ) dτ +
∑
t0<ti<t
aiu
m(ti − 0), (10)
where ϕ(t) is a positive function monotonously nondecreasing on J , g > 0, ai > 0, m > 0, m 6= 1.
Then
u(t) 6 ϕ(t)
∏
t0<ti<t
(1 + aiϕ
m−1(ti))
[
1 + (1 −m)
t∫
t0
ϕm−1(τ)g(τ) dτ
]1/(1−m)
,
∀ t > t0, 0 < m < 1;
(11)
u(t) 6 ϕ(t)
∏
t0<ti<t
(1 + aimϕ
m−1(ti)) ×
×
[
1 − (m− 1)
[
∏
t0<ti<t
(1 + aimϕ
m−1(ti))
]m−1 t∫
t0
ϕm−1(τ)g(τ) dτ
]−1/(m−1)
;
(12)
∀ t > t0 :
t∫
t0
g(τ)ϕm−1(τ) dτ 6
1
m
, m > 1; (13)
∏
t0<ti<t
(1 + aimϕ
m−1(ti)) <
(
1 +
1
m− 1
)1/(m−1)
.
Main results. In this section, we obtain the new conditions of boundedness of the solutions of
system (5) by using the property of boundedness of the shorted system dx/dt = A(t)x, t 6= ti(x),
∆x|t=ti(x) = Bix; in addition, the conditions for the stability by Lyapunov and the practical
stability (uniform, attractive) by Chetaev of the trivial solution of system (5) will be found.
Theorem 1. Let assumptions (H1)–(H9) be valid for system (5), and let the following condi-
tions be fulfilled:
a1) α + θi lnα = β̃ = 0; α∗ = 1, 0 < β 6 1;
b1) ∃ a = const > 0: ‖A(t)‖ 6 a;
c1) L <
1
(a+ r)h
;
d1) ∃m1(t0) = const > 0:
∏
t0<t∗
i
<t
(1 + cβki‖x0‖
β−1) 6 (1 +m1(t0)‖x0‖
β−1) ∀t ∈ J ;
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2008, №7 39
e1) ∃m2(t0) = const > 0:
t∫
t0
r(τ) dτ 6 m2(t0) < ∞ ∀ t ∈ J ;
f1) c(1 + m1(t0)λ
β−1) exp[cm2(t0)] < Λ/λ;
g1) ∃m3(t0) = const > 0: ‖x0‖(1 +m1(t0)‖x0‖
β−1) 6 m3(t0)‖x0‖
β;
h1) λ < β
√
Λ(cm3(t0) exp[cm2(t0)])−1.
Then
I. a2) all solutions of system (5) are bounded in Ω, if conditions a1–e1 hold;
II. Trivial solution (t. s.) of system (5) is
b2) practically stable (p. s.) relative to λ, Λ, J , if conditions a1–f1 or a1–e1, g1, h1 hold;
c2) uniformly practically stable (u. p. s.) relative to t0, if conditions b2 hold and m1(t0) (i =
= 1, 2, 3) are independent of t0.
Remark 3. If ti(x) = ti = const, β = 1, A(t) = A, the result of Theorem 1 is similar to
Theorems 4.3.13 and 4.3.14 in [5], pp. 287, 288); if β = 1, it coincides with Theorem 4.1 in
[11], p. 86.
Theorem 2. Let the following conditions be fulfilled:
a3) assumptions (H1)–(H9) hold;
b3) α + θi lnα = 0, β̃ < 0, α∗ = 1, 0 < β 6 1;
c3) conditions b1, c1, e1 of Theorem 1 take place;
d3) ∃m4(t0) = const > 0:
∏
t0<t∗i <t
{
1 + cβ
[
t∗i
t0
]β̃(β−1)
ki‖x0‖
β−1
}
6 (1 + m4(t0)‖x0‖
β−1)
∀ t ∈ J = [t0, T ];
e3) c(λ + m4(t0)λ
β) exp[cm2(t0)] < Λ;
f3) ∃m5(t0) = const > 0: ‖x0‖(1 + m4(t0)‖x0‖
β−1) 6 m5(t0)‖x0‖
β ;
g3) λ < (Λ(cm5(t0) exp[cm2(t0)])
−1)1/β .
Then (t. s.) of system (5) is:
a4) (λ,Λ, J)-stable; moreover, attractive practically stable (a. p. s.) relative to (λ,Λ,Λ∗, J),
here, λ < Λ∗ < Λ; if only a3–e3 or a3–d3, f3, g3 take place;
a5) (u. p. s.) relative to (λ,Λ, J), only if m2, m4, m5 are independent of t0; moreover, attracti-
ve (u. p. s.) (a. u. p. s.) relative to (λ,Λ,Λ∗, J).
Remark 4. Theorem 2 gives new conditions of (p. s.) (t. s.) (uniform, attractive) of the
perturbed system (5) on some hypersurfaces by using only the second scale of increasing functions
(characteristic degree by Lyapunov), which was first considered by Demidovich [1] (where new
conditions for the stability of solutions in the linear approximation were found). Theorem 2
generalizes the idea to consider the two-parametric scale of increasing functions including both
the Lyapunov indices and the characteristic degree by Lyapunov.
Theorem 3. Let condition a3 of Theorem 2 be valid, and let the following conditions be
fulfilled:
b4) α + θi lnα < 0, β̃ 6 0, α∗ = 1, 0 < β 6 1;
c4) condition c3 of Theorem 2 is fulfilled;
d4) ∃m6(t0) = const > 0: D(t0, t)
def
=
∏
t0<t∗
I
<t
(
1 + cβ
[
t∗i
to
]β̃(β−1)
exp[(α̃ + θi lnα)(β − 1)(t∗i −
− t0)]‖x0‖
β−1ki
)
6 (1 + m6(t0)‖x0‖
β−1), ∀ t ∈ J = [t0, T ];
40 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2008, №7
e4) c(1 + m6(t0)λ
β−1) exp[cm2(t0)] < Λ/λ;
f4) ∃m7(t0) = const > 0: ‖x0‖(1 + m6(t0)‖x0‖
β−1) 6 m7(t0)‖x0‖
β ;
g4) λ < (Λ(cm6(t0) exp[cm2(t0)])
−1)1/β .
Then (t. s.) of system (1) is:
a6) (λ,Λ, J)-stable; moreover, (a. p. s.) relative to (λ,Λ,Λ∗, J) if only (H1)–(H9), b4–e4 or
b4–d4, f4, g4 take place;
a7) (u. p. s) relative to (λ,Λ, J), if m2, m6, m7 are independent of t0; moreover, (a. u. p. s.)
relative to (λ,Λ,Λ∗, J).
Remark 5. Estimate (11), in which we use the two-parametric scale of increasing functions,
generalizes the estimate by Demidovich [1] to the case of the impulsive differential system (5).
Remark 6. For 0 < β 6 1, the constants ki must be sufficiently small, and lim
i→∞
ki = 0.
Analogously, for Theorems 1–3 (case 0 < β 6 1), the next statements take place:
Theorem 4. Let us assume that
a1∗) α̃ + θi lnα = β̃ = 0, α∗ = 1, β > 1;
b1∗) assumptions (H1)–(H9) hold;
c1∗) b1, c1, e1 of Theorem 1 take place;
d1∗) ∃m∗
1(t0) = const > 0:
∏
t0<t∗i <t
(1 + cβki‖x0‖
β−1) 6 m∗
1(t0) < ∞, ∀ t ∈ J ;
e1∗) cm∗
1(t0) exp[βcm2(t0)] < Λ/λ.
Then
I. All solutions of system (5) are bounded in Ω, if conditions a1∗ − d1∗ hold;
II. (t. s.) of system (5) is:
b2∗) stable by Lyapunov, if conditions a1∗–d1∗ hold (stable uniformly, if m∗
1, m2 are inde-
pendent of t0);
c2∗) (u. p. s.), if d1∗), b2∗) hold and m∗
1(t0), m2(t0) are independent of t0.
Theorem 5. Suppose that conditions a3, c3 of Theorem 2 take place and
a3∗) α̃ + θi lnα = 0, β̃ < 0, α∗ = 1, β > 1;
b3∗) ∃m∗
2(t0) = const > 0:
∏
t0<t∗i <t
(
1 + cβ
[
t∗i
t0
]β̃(β−1)
‖x0‖
β−1ki
)
6 m∗
2(t0) < ∞, ∀ t ∈ J ;
c3∗) cm∗
2(t0)λ exp[βcm2(t0)] < Λ/λ.
Then (t. s.) of system (5) is:
i) asymptotically stable by Lyapunov, only if a3∗, b3∗ take place (stable uniformly, if m∗
2(t0),
m2(t0) are independent of t0);
ii) (p. s.) if only a3∗−c3∗take place (stable uniformly, if m∗
2(t0), m2(t0) are independent of t0).
Theorem 6. Assume that
a4∗) assumptions (H1)–(H9), conditions b1, c1, e1 of Theorem 1 hold;
b4∗) ∃m∗
3(t0) = const > 0: D(t0, t) 6 m∗
3(t0), ∀ t > t0;
c4∗) α̃ + θi lnα < 0, β̃ 6 0, α∗ = 1, β > 1;
d4∗) cm∗
3(t0) exp[βcm2(t0)] < Λ/λ.
Then (t. s.) of system (5) is
iii) asymptotically stable by Lyapunov, if only a4∗–c4∗ take place (stable uniformly, if m∗
3(t0),
m2(t0) are independent of t0);
iv) (a. u. p. s.) (if m∗
3(t0) = m3, m2(t0) = m2), iii and d4∗ hold.
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2008, №7 41
Remark 7. For the case α∗ = β > 0 (α∗ 6= 1), it is possible, by using result B of Remark 2,
to make a similar qualitative analysis of the properties of the solutions of system (5).
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Мат. сб. – 1965. – 66, № 3. – С. 344–353.
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Received 15.01.08International Mathematical Center
of the National Academy of Sciences of Ukraine, Kyiv
National Technical University of Ukraine “KPI”, Kyiv
University of Naples “Federico II”, Naples
42 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2008, №7
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| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1025-6415 |
| language | English |
| last_indexed | 2025-11-27T15:17:56Z |
| publishDate | 2008 |
| publisher | Видавничий дім "Академперіодика" НАН України |
| record_format | dspace |
| spelling | Mitropolsky, Yu.A. Borysenko, S.D. Toscano, S. 2009-12-29T15:41:26Z 2009-12-29T15:41:26Z 2008 Investigations of the properties of solutions of impulsive differential systems in the linear approximation / Yu.A. Mitropolsky, S.D. Borysenko, S. Toscano // Доп. НАН України. — 2008. — № 7. — С. 36-42. — Бібліогр.: 15 назв. — англ. 1025-6415 https://nasplib.isofts.kiev.ua/handle/123456789/4969 517.911 Проведено якiсний аналiз властивостей розв’язкiв нелiнiйних iмпульсних систем диференцiальних рiвнянь iз збуреннями на деяких гiперповерхнях {ti(x)}. Отримано новi умови обмеженостi, практичної стiйкостi за Четаєвим (рiвномiрної, стискуючої), стiйкостi за Ляпуновим (рiвномiрної, асимптотичної), притягування розв’язкiв збурених систем за лiнiйним наближенням. en Видавничий дім "Академперіодика" НАН України Математика Investigations of the properties of solutions of impulsive differential systems in the linear approximation Article published earlier |
| spellingShingle | Investigations of the properties of solutions of impulsive differential systems in the linear approximation Mitropolsky, Yu.A. Borysenko, S.D. Toscano, S. Математика |
| title | Investigations of the properties of solutions of impulsive differential systems in the linear approximation |
| title_full | Investigations of the properties of solutions of impulsive differential systems in the linear approximation |
| title_fullStr | Investigations of the properties of solutions of impulsive differential systems in the linear approximation |
| title_full_unstemmed | Investigations of the properties of solutions of impulsive differential systems in the linear approximation |
| title_short | Investigations of the properties of solutions of impulsive differential systems in the linear approximation |
| title_sort | investigations of the properties of solutions of impulsive differential systems in the linear approximation |
| topic | Математика |
| topic_facet | Математика |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4969 |
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