CC BY
9MSC 2010: 34K20, 34K40
Sufficient conditions for the asymptotic stability of solutions to one class of linear systems of neutral type with periodic coefficients
T. K. Yskak
Novosibirsk State University, Novosibirsk 630090, Russia. E-mail: istima92@mail.ru
Abstracts. In the paper we consider one class of systems of linear delay differential equations with a parameter and periodic coefficients. They belong to systems of neutral type. We prove an analogue of M. G. Krein's theorem for this class. Using a Lyapunov-Krasovskii functional, we obtain sufficient conditions for the asymptotic stability of the zero solution and establish estimates characterizing the exponential decay of solutions to the systems at infinity.
Keywords: delay differential equations, neutral type, periodic coefficients, asymptotic stability, Lyapunov-Krasovskii functional
1. Introduction
In the present paper we consider a system of linear delay differential equations of neutral type
d
- (;y(t) + Dy(t - t)) = ¡A(t)y(t) + B(t)y(t - t), t> 0, (1.1)
where D is an (nx n)-matrix with constant complex entries, A(t) and B(t) are (nx n)-matrices with continuous T-periodic complex-valued entries, i > 0 is a parameter. We assume that the spectrum of the matrix A(t) belongs to the left half-plane C_ = {X € C : Re A < 0} for all t € [0, T].
Our aim is the study of the asymptotic stability of the zero solution to this system for I » 1.
Note that in the literature there exists a number of results for systems of the form (1.1) in the cases D = 0, B(t) = 0 (ordinary differential equations [2, p. 209]) and D = 0, B(t) ^ 0 (delay differential equations [6]). We formulate some of them.
At first consider the following system of ordinary differential equations with a parameter I> 0
d
-y = iA(t)y, t> 0, (1.2)
where A(t) is a matrix with continuous T-periodic entries. A very important theorem of M.G. Krein holds [2, p. 209].
Theorem 1. If the spectrum of the matrix A(t) belongs to the left-half plane C_ for all t € [0,T], then the zero solution to (1.2) is asymptotically stable for all sufficiently large positive i.
© T. K. YSKAK
Emphasize that in this theorem the asymptotic stability of the solution to (1.2) is guaranteed only for sufficiently large f > 0. This fact is essential, because there exist examples of systems of the form (1.2), the zero solution of which is unstable for arbitrary f> 0. Indeed, consider the following system of ordinary differential equations proposed by R.E. Vinograd (see, for example, [1, p. 124])
d
-y = A(t)y, t > 0,
where
A(t) =
dV
-Ц + 15 cos 12t -6 + 15 sin 12t
6 + 45 sin 12t - Ц - 45 cos 12t
2 ^ 2 2
It is easy to see that A(t) = A(t + . Eigenvalues of the matrix A(t) are constant and equal to —1 and —10, solutions of the system are given by the formula
ла 2t f cos6t \ , -i3i / sin6t \ y(t) = cie2t + C2e 13t
\ sin6t / \ - cos6t J
From the formula it is easy to see that the zero solution to this system is unstable.
Note that a "threshold" value f0 was obtained in [6], beginning from which the asymptotic stability of the zero solution to (1.2) is guaranteed. To formulate the corresponding statement we need to give some notations. According to Lyapunov's criterion, the Lyapunov matrix equation
VA + A*V = -I
has a unique solution V = V* > 0 if and only if the spectrum of A belongs to the left halfplane C_. Here A* is the conjugate transpose of A. Since the spectrum of the matrix A(t) belongs to the left half-plane C_ for all t € [0,T], then by the Lyapunov criterion there exists a unique solution H(t) = H*(t) > 0 to the matrix equation
HA(t) + A*(t)H = -I
for every fixed t € [0,T]. Introduce the following notations
Hmax = max \\H(t)||, vmax = max v(H(t)), te[o,T ] te[o,T]
where v(H(t)) is the condition number of the matrix H(t), \\H(t)\\ is its spectral norm. The following theorem holds [6].
Theorem 2. Let the spectrum of the matrix A(t) belong to the left half-plane C_ for all t € [0, T], let N be a number such that the following inequality holds
maxr \\A(t) - A(s)\\ < 1 (1.3)
NN 4Hma^Y vmax
and
2NHmax , n ,
f0 = -T-ln Vmax- (1.4)
Then the zero solution to (1.2) is asymptotically stable for all f > f0.
Note that the existence of N, for which (1.3) holds, is assured by the continuity of the entries of the matrices A(t), H(t) and the function v(H(t)). In the proof of this theorem it was essentially used the following criterion of the asymptotic stability [4] for the system
ftV = A(t)y, (1.5)
where A(t) is a continuous T-periodic (n x n)-matrix.
Theorem 3. I. If the zero solution to (1.5) is asymptotically stable then, for every continuous matrix C(t) on [0,T], there exists a unique solution L(t) to the boundary value problem
d
—L + LA(t) + A*(t)L = -C(t), 0 <t<T, dt (1.6)
L(0) = L(T),
C(t) = C*(t) > 0, t £ [0,T], (1.7)
moreover, if then
L(t) = L*(t) > 0, t £ [0,T],
II. Let the right-hand side C(t) be continuous on [0,T] and satisfy the conditions (1.7). If the boundary value problem (1.6) has a Hermitian solution L(t) such that L(0) > 0 then the zero solution to (1.5) is asymptotically stable.
For i > ¡0, due to the criterion of the asymptotic stability, the boundary value problem for the Lyapunov differential equation
d
—L + iLA(t)+1A*(t)L = -M, 0 <t<T, dt (1.8)
L(0) = L(T) > 0,
with M = M * > 0 has a unique solution L(t) = L*(t) > 0. Moreover, as was shown in [6], the estimate holds
IILWII < ^ (1 - exp (-^y. (1.9)
Hereafter, we will use the same notation for T-periodic extension of the matrix L(t) for t > 0.
In the case of D = 0, B(t) ^ 0 the zero solution to (1.1) is also asymptotically stable for all sufficiently large ¡i > 0. The authors of [6] found a "threshold" value such that the asymptotic stability of the zero solution to (1.1) is guaranteed for i > fa We now formulate this result.
Theorem 4. Let D = 0, let N be a number such that (1.3) holds, and let fa > satisfy the equation
l( 1 - (vmax)N Jc=4eT/2Hmax (vmax)N max ||B(0I|. V J ]
Then the zero solution to (1.1) is asymptotically stable for all ¡> fa.
In the proof of this statement the authors used a modified Lyapunov-Krasovskii functional of the form
t
V(t,y) = (L(t)y(t),y(t)) + 2 J e-(t-a)Ms)\\2 ds,
t-T
where L(t) = L*(t) > 0 is T-periodic extension of the solution to the boundary value problem (1.8) with M = I. Here the condition on i is also essential, because, based on an example constructed by R. E. Vinograd, we can construct an example of a system of the form (1.1) with D = 0, B(t) ^ 0, for which the zero solution is unstable for i = 1. Namely, consider the following system
d { п ^
-y(t) = A(t)y(t) + B(t)y (t - 3
where
A(t) =
- t + T cos12t -3 + 15 sin 12t
3 + 15 sin 12t -- 15 cos 12t
B (t) =
4 ^ 4 4
e2n/3 / "T + 15 cos 12t -6 + -2
2
2 1 6 + 15 sin 12i -- 15 cos 12i Using the example described above, we obtain that the vector-function
cos 6t
2t ( cos 6t \ y (t) = ce2t . ^ sin 6t
is a solution to this system. Therefore, its zero solution is unstable, though the eigenvalues of the matrix A(t) are constant and equal to — 1 and -5.
It should be noted that Theorem 4 is an analogue of M. G. Krein's result formulated in Theorem 1 for systems of linear delay differential equations of the form (1.1) with D = 0. Our aim is to establish such analogue in the case of D = 0. In Section 2 we formulate sufficient conditions for the asymptotic stability of the zero solution to (1.1) and obtain estimates of solutions to (1.1) with ¡i = 1. In Section 3 we give conditions on the matrix D and a "threshold" value on the parameter beginning from which the zero solution to (1.1) is asymptotically stable.
The author expresses his deep gratitude to Prof. G.V. Demidenko for statement of the problem and valuable advices.
The author is grateful to the anonymous referee for the helpful comments and suggestions.
2. Asymptotic stability of the zero solution to the system of neutral type
Consider the initial value problem for (1.1) with i = 1 d
- (y(t) + Dy(t — t)) = A(t)y(t) + B(t)y(t — t), t> 0, y(t) = p(t), t £ [—t, 0], (2.1)
[ y(+0) = p(0),
where D is an (n x n)-matrix with constant entries, A(t) and B(t) are (n x n)-matrices with continuous T-periodic entries. We assume that the spectrum of A(t) belongs to the left-half plane C_ for all t € [0, T]. When studying the asymptotic stability of the zero solution, we use the following modified Lyapunov-Krasovskii functional introduced in [3, 5]
t
V(t,y) = (H(t)(y(t)+ Dy(t - т)), (y(t)+ Dy(t - т))) + j (K(t - s)y(s),y(s)) ds. (2.2)
t—т
We now formulate a theorem which is an analogue of statements from [3, 5].
Theorem 5. Let there exist a smooth T-periodic matrix H(t) = H*(t) such that H(t) > 0 and a matrix K(s) = K*(s) € C1[0,т] such that
d
K(s) > 0, —K(s) < 0, s € [0,т]. ds
Denote by q(t) the minimal eigenvalue of the matrix
Q(t)= H_2(t)(Qu(t) - Qi2(t)Q—iQu(t)^H_1 (t),
where
d
Qii(t) = -dH(t) - H(t)A(t) - A*(t)H(t) - K(0), Qi2(t) = H(t)A(t)D + K(0)D - H(t)B(t), Q22 = K(т) - D*K(0)D > 0. Choose a number k > 0 such that
d
—K(s) + kK(s) < 0, s € [0,т].
Then for the solution to the problem (2.1) the following estimate holds
t t (H(t)(y(t)+ Dy(t - т)), (y(t)+ Dy(t - т))) + j (K(t - s)y(s),y(s)) ds < exp (- Y(s)ds)
t—т
x ^(H(0)(p(0) + D^-т)), (ф(0) + Dip-))) + J(K(-s)p(s),<p(s)) dsj , (2.3)
where
j(t) = min {q(t),k} .
Proof. Let y(t) be a solution to the initial value problem (2.1). Consider the modified Lyapunov-Krasovskii functional (2.2) on the solution y(t). Its derivative has the form
JtV(t,y) = ((dH(t)+ H(t)A(t) + A*(t)H(t)+ K(0)) y(t),y(t))
Denote
where
+ < ( -H (t)D + A * (t)H (t)D + H (t)B (t) ) y(t - t ),y(t)
)) y(t - t),y(t)^
+
d
(v(t), (
-H(t)D + A *(t)H(t)D + H(t)B(t) ) y(t - t)
+ ( D*-rH(t)D + D*H(t)B(t) + B*(t)H(t)D - K(t) y(t - t),y(t - t)
dt
t
-n.
t—T
+ I (-K(t - s)y(s),y(s)) ds.
С(t) = -
( Cn(t) Ci2(tn
V С 12(f) Ö22(t) ) '
Cii(t) = jtH (t) + H (t)A(t) + A * (t)H (t) + K (0),
)
Ci2(t) = dtH (t)D + A* (t)H (t)D + H (t)B (t), d
С22(t) = D*—H(t)D + D*H(t)B(t) + B *(t)H(t)D - K(t).
dt
Then we have
ltV
<c«( y(y<-10 'U^))>
t
-к
t—т
-K(t - s)y(s),y(s)) ds = 0.
)
It is easy to verify the validity of the following identity
<c <4:;) ■(»)>-<(
Indeed,
Qii(t) Q12 (t) Q * 2 (t) Q22
(c К S) •(*))
Cii(t) Ci2(tn / I -D\(zi + Dz2\ (
c*2(t) C22(tw v 0 i Л z2 м
I -D 0I
)(
zi + Dz2 z2
Hence,
(c К s) -U))
I0 D I
A ( Cii(t) Ci2(tn / I -D\(zi + Dz2\ ( ) V C*2(t) C22(tW V 0 I )\ z2 )\
zi + Dz2 z2
(2.4)
)(z i Г2) ^nf2)
Multiplying the matrices, we obtain the required identity. By the conditions of the theorem, Q22 > 0. Hence, the following representation holds
%w T) (Z2)• (Z2)) = «Q»®-QrtvM)«,«)
+ {Q-i(Q22Z2 + Q*2(t)Zl), (Q22Z2 + Q*2(t)Z1 )}.
Consequently, from (2.4) we obtain d
JtV (t, y) < -{[Qll(t) - Q12 (t)Q^21Q*2(t)](y(t) + Dy(t - T)), (y(t) + Dy(t - T))}
t
+ j (ddtK(t - S)y(S),y(S)) ds
t-T
or
— V(t, y) < -{Q(t)H2 (t)(y(t) + Dy(t - t), H1 (t)(y(t) + Dy(t - t))}
t
+ J {jK(t - S)y(s),y(s)} ds.
t- T
Then, by the definitions of q(t), k, and Y(t), we have
d
-V(t,y)+ Y(t)V(t,y) < 0.
The estimate (2.3) immediately follows from here.
Theorem is proved. □
Remark 1. It is easy to see that Y(t) is a T-periodic function.
Remark 2. Generally speaking, the asymptotic stability does not follow from Theorem 5 because the conditions of the theorem do not guarantee that exp ^- f j(s) ds^ ^ 0 as t —y TO.
Hereafter, we assume that the conditions of Theorem 5 hold. Note that from the conditions on the matrix K(s) and Lyapunov's criterion it follows that the spectrum of the matrix D belongs to the unit disk {X £ C : |A| < 1}. Indeed, since Q22 > 0 then
K(0) - D*K(0)D = P> 0;
i.e., the Hermitian positive definite matrix K(0) satisfies the Lyapunov discrete equation with a Hermitian positive definite right-hand side. According to Lyapunov's criterion, this equation has a unique Hermitian positive definite solution if and only if the spectrum of D belongs to the unit disk.
Lemma 1. Let x(t) be a T-periodic function. Then
t
J x(s)ds > T;S + q, o
where
T / i T
S = J x(s)ds, q = mm ^ ( J x(s)ds — T J x(s)ds 1 .
Proof. Let t £ [nT, (n + 1)T]. Then,
t nT t
x(s)ds=1x(s)ds 4x(s)ds■
o o nT
Because of T-periodicity of the function x(t) we have
t t - nT
У x(s)ds = nS + J x(s)ds.
oo This is equivalent to the equality
t-nT
J x(s)ds = T;S + J x(s)ds — t—^nTS.
From here the statement of Lemma 1 follows.
Lemma is proved. □
Denote
Ф = max te[-T,o]
K = max IIH-1 (C)|1/2 ^H(0)II(1 + ID2) + J IK(s)| dsj , (2.5)
° = exP (iS|oa,T] ( T j ^ d - j ^ d-]
In the next theorem we establish estimates of solutions to the initial value problem (2.1), which are similar to estimates from [3, 5].
Theorem 6. Suppose that the conditions of Theorem 5 hold. Let l be the minimal positive
cT jH Jo 2
integer such that IIDl || < 1 and let A = T fT ds > 0.
t
1. If \\Dl|| exp(A^T) < 1, then for the solution to the problem (2.1) the estimate holds l|y(t)|| < $ — ID\\eAlT)-1 £ ID\\eAjT + max{\D\eAT,..., \\Dl\\eAlT}| e-At.
2. If \\Dl\\ exp(A^T) = 1, then for the solution to the problem (2.1) the estimate holds < $ I «□( 1 + ^ly \\Dj\\eAjT + max{\\D\\eAT,..., \\Dl-1 \\e(l-1)AT, 1} | e-At.
3. If \\Dl\\ exp(A^T) > 1, then for the solution to the problem (2.1) the estimate holds
l1
< ф(ka\\Dl\\-1(1 - \\Dl\\-le-AlT)-1 g \\Dj\\eAj7 \ j=0
7 1 7 1 \ M\\Dl II).
+ \\Dl\\ max{1, \\D\\,..., \\Dl-1\\}) t.
Proof. Since
{H(0)(^(0) + D^(—T)), (<p(0) + Drt—T))) < $22\\H(0)\\(1 + \\D\\2)
and
) 0 (K(-s)tp(s), ф)) ds < Ф2 y \\K(-s)\\ ds,
-T -T
then we have
2 ( 2
V(0,<p) < $ | 2\\H(0)\\(1 + \\Dr) + J \\K(s)\\ds From (2.3) and (2.5) for the solution to the problem (2.1) the following estimate holds
{H(t)(y(t) + Dy(t — t)), (y(t) + Dy(t — t))) < exp j7(s)dsj $2 Since H(t) is a Hermitian positive definite matrix, then
\\y(t)+ Dy(t — t )\\ < exp I—y
к2
max \\H-1 (£)\\ ?e[o,T]
t
y(S) 2
0
Taking into account the inequality
< \\y(t) + Dy(t — t)\\ + \\Dy(t — t)
< ФЯ exp (-j ^ dsj + \\Dy(t - t )\\.
we obtain
t
Y(s) 2
o
Let t £ (kT, (k + 1)t], k £ N. Then, as above, we have
IIDy(t - t)II < IIDy(t - t) + D2y(t - 2t)I + IID2y(t - 2t)I
Hence,
t-T
\\Dy(t - t)\\<\\D\^«exp i- j ^ds) + \D2y(t - 2t)\\,
and therefore,
< Фкехр i-J ds ) + |\D|\$Kexp ( - J ^ ds ) + \\D2y(t - 2t
dsj + m^exp / ^
Repeating similar reasonings, we obtain
t-jT
< Фк^Г \\Dj\\exp (- J ^ ds) + \\Dk+ly(t - (k + 1)t )
j=0 \ 0
Using the definition of matrix norm, it is easy to show the validity of the following inequality
k I t-jT
' lis).
. „ 2
j=0 \ 0
< $ (K J2 IDj II exp (- J ^ dsj + IIDk+1H In virtue of Lemma 1, the definition of A, and (2.5), we obtain
к
< Ф ( «□ £ \\Dj \ \ exp (-A(t - jT)) + \\Dk+l \ \ j=o
which is equivalent to
к
< Ф ( \\Dj\\eAjT + \\Dk+1\\eAt ) e-At. (2.6)
j=o
Case 1: HDzHeZAT < 1. Taking into account that t £ (kT, (k + 1)t], from (2.6) we have
oo
Then,
< Ф ( \\Dj\\eAjT + \\Dk+l\\eA(k+1)T ) e-At.
j=0
< Ф ( \\Dj\\eAjT + max{\\D\\eAT\\Dl\\eAlT} | e-At
j=0
Rewrite the series
— 1 2—1 3—1
VII eAjT
E d \\eAjT = E d \\eAjT + E d \\eAjT + E d \\eAjT + • • • j=0 j=0 j=l j=2l
and estimate it
i-i —1 —1 jT l AlT j AjT l AlT 2
E D\\eAjT < E \\Dj\\eAjT + \\Dl\\eAlT E D\\eAjT + (\\Dl\\eAlT)2 E D\\eAjT + .... j=0 j=0 j=0 j=o
Hence, we obtain the required estimate.
Case 2: \\Dl\\elAT = 1. Taking into account that t £ (kT, (k + 1)t], from (2.6) we have
j=0
< Ф ^ кПу ] \\D\\e*3T + max{\\D\\eAr, • • •, 1} | e"
It is easy to see that
к / t\ 1-1 E ID\\eAjT < (l + ¿JE D\\eAjT•
j=0 j=0
From here we immediately obtain the required estimate. Case 3: \\Dl\\elAT > 1. Let k £ (ml, (m + 1)1]. Then
к —1 —1
jT l
E D l|eAjr < E \\Dj\\ej + \\Dl \\eAlT E \\Dj \\eAjT + ... j=0 j=0 j=0
l-l
+(\\D\\eAlT\\Dj\\eAjT j=0
or
к l-l
E \\Dj\\eAjT < (\\Dl\\eAlT)™E D\\eAjT(1 + (\\Dl\\eAlT)-1 + • • • + (\\Dl\\eAlT)-m). j=0 j=0
Hence,
к l-l E \\Dj\\eAjT < (\\Dl\\eAlT)m(1 - \\Dl\\-le-AlT)-1 E \\Dj\\eAjT. j=0 j=0
Since t £ (mlr, (m + 1)1т] and \\Dl\\ > e-lAT, from (2.6) we have
\\y(t)\\ < Ф ^ kD(\D1 \\eAlT)m(1 - \\Dl\\-le-AlT)-1 J D\eAjT + \\Dk+l\\eAtj e-At.
Taking into account that k £ (ml, (m + 1)l], we obtain
\\Dk+l\\ < \\Dl\\m\\Dk+l-ml\\ < \\Dl\\m max{1, \\D\\,..., \\Dl-l\\}.
Therefore,
< kn\\Dl\\meAmlT(1 — \\Dl\\-1e-AlT)-1 ^ \\Dj\\eAjT ^ j=0
+ \\Dl\\m max{1, \\D\\,..., \\Dl-1\\}eAt^J e-At.
Since t € (mlr, (m + 1)1т], we have
l-1
< Ф( kD(1 - \\Dl\\-1e-AlT)-1^ \\Dj\\eAjT ^ j=0
+ max{1, \\D\\,..., \\Dl-1\\})\\Dl\\£-1.
From here the required inequality follows. Theorem is proved.
□
Remark 3. From Theorem 6 the asymptotic stability of the zero solution to (1.1) follows for
I = 1.
3. An analogue of M. G. Krein's theorem for the system of neutral type
Consider (1.1). We assume that the spectrum of the matrix A(t) belongs to the left-half plane C_ for all t € [0, T], the spectrum of the matrix D lies in the unit disk [X € C : | A| < 1}. Take K(s) = Me_l3s and H(t) as a solution to the Lyapunov differential equation
d
—H + ¡HA(t) + ¡A*(t)H = -2M, H(0) = H(T) > 0, where M is a solution to the Lyapunov discrete equation
(3.1)
M -D*MD = I,
ß = — max — ln i 1--,,1,...
и !ф,Т\ 2т V 2\\M\\
1 — \\D\\2(64\\A(0\\2\\M-1\\\\M fHlax (vmaxr + \\M\\
+16\\A(0\\\\M\\Hmax (vmax)
N\
From Theorems 2 and 3 it follows that the boundary value problem (3.1) has a Hermitian positive definite solution for i > ¡0, where ¡0 is defined by (1.4). Extend it T-periodically keeping the same notation.
Thereafter, we will use a well-known lemma (see, for example, [7, p. 495]).
Lemma 2. Let P\\, P\2, and P22 be (n x n)-matrices, then the following statements are equivalent:
» P = ( % %) > 2)
P11 > 0,
P22 — Pi2P-11P12 > 0,
3)
P22 > 0,
P11 — P12 p— 1P12 > 0
Theorem 7. Let N be a number such that (1.3) holds and
\\D\\ < min ((64\\A(£)\\2\\M-1\\\\M f^ Kax)2N + \\M \\ ?e[o,T] V
+ 16\\A(0\\\\M\\Hmax (Vmax)^ ' .
Denote by ¡2 the maximal root of the following equation
128\\B(t)\\\\M-1\\\\A(t)\\\\D\\ (WM||Hmax(Vmax)N)2
¡
N /miiiu /.. \NX 2
+16\\B(t)\\\\D\\WM\Hmax (Vmax) + 64\\B(t)\\2\\M-1 \\l\\M\Hmax (Vmax)
)
¡¡ = 1 — (\A(t)\2\M-1\\(8\\M\\Hmx (Vmax)N )2 + \\M \\
+ 16\\A(t)\\\\M\\Hmax (V max ) ;,
and
2Hmax(N ln(v max ) + ln(2)) ¡3 = -T-•
Then the zero solution to (1.1) is asymptotically stable for all ¡i > max{i2,i3}.
Proof. In virtue of Theorems 5 and 6, it is sufficient to show that Q(t) > 0 for t £ [0,T]. By Lemma 2, this is equivalent to the inequality
Q22 — Q12(t)Q-11(t)Q12(t) > 0, t £ [0,T].
Taking into account the definitions of the matrices Q11(t), Q12(t), Q22 and the positive definiteness of the matrix H(t), it is sufficient to establish that
K (t ) — D *K (0)D — (d *iA * (t)H (t) + D *K (0)
—B*(t)H(t))M-1 (H(t)iA(t)D + K(0)D — H(t)B(t)) > 0.
According to the choice of the matrices H(t) and K(s), we can present the expression in the left-hand side as a sum
Me-l3T — D *MD — (d *iA * (t)H (t) + D *M —B * (t)H (t)) M-1 (H (t)iA(t)D + MD — H (t)B(t)) = S1 + S2 + S3,
where
Si = M - D*MD - (dVA *(t)H(t) + D*MjM-1(H(t)ßA(t)D + mDj,
S2 = -(1 - e-ßT)M,
S3 = B*(t)H(t)MH(t)ßA(t)D + MD - H(t)B(t}) + (D*ßA*(t)H(t) - D*MjM-1H(t)B(t).
We now show that S! > 0, \\S2W < ^T1> \\ss\\ < ^IT Since n > from (1.9) the estimate follows
\\H(t)\\< mWH™X {Vmax)N. (3.2)
Show that Si > 0. Taking into account the definition of M, we have
(Siv,v)> u(t)\\v\\2,
where
u(t) = 1 - \\D\\2^2\\A(t)\\2\H(t)\\2\\M"l\\ + \\M\\ +2№\\A(t)\\\\H(t)\\). Using the condition on D and (3.2), it is easy to verify that
u(t) > 0, t £ [0,T],
Therefore Si > 0.
" < V
Prove that ||S2|| < • It is easy to see that
(1 - e-*)\\M\\ <U2). Indeed, this inequality can be rewritten in the following form
1 / 1 - \\D\\2(№2\\A(t)\\2\\H(t)\\2\\M-i\\ + \\M\\ +2№\\A(t)\\\\H(t)\\Y
R <--\Q I 1--V-J
P< T 2\\M\\
Taking into account the choice of R and (3.2), we obtain that this inequality is satisfied.
St 2
Finally, we prove that \\ S3 \ < -^p-. This inequality is satisfied if the following inequality
is true
2\\B(t)\\\\M-i\\№\\A(t)\\\\D\\\\H(t)\\2 + 2\\B(t)\\\\D\\\\H(t)\\ + \\B(t)\\2\\M-i\\\\H(t)\\2 < ^ Taking into account (3.2), it is sufficient to prove the estimate
-inn 4^1111 n u(8\\M \ \ Hmax (Vmax)N )2
2llB(t)llllM-i||A(t)||D|
ß
+2UB(t)llllDll8llM llHmx {Vmax)N + llB(t)ll2llM-if 8lM lHmx (Vmax)N) 2
ß \ ß J
< 1 - ^DtC llA(t)ll2 llM^PM llHmax (Vmax)N )2 + № Ц
+ 16llA(t)llllM 11 Hmax (V max J
But this inequality is satisfied because ¡i> ¡i2.
Theorem is proved. □
Remark 4. Note that the lower bound on i is essential. Indeed, consider the following example, which is an analogue to R. E. Vinograd's example
d (y(t) + ay (t — |)) = A(t)y (t) + aA(t)y (t — , t> 0, where a is a number,
A(t) =
- J2T + 15 cos 12t -6 + -f sin 12t
6 + -f sin 12t - Ц - f cos 12t
Eigenvalues of A(t) are constant and equal to —1 and -10. It is easy to see that the vector-function
2t / cos6t
y(t) = ce2 , . -, yK ' 1 sin 6t
is a solution to this system. Hence, the zero solution is unstable.
References
1. Bylov B. F, Vinograd R. E, Grobman D. M, Nemytskii V. V. Theory of Lyapunov exponents and its applications to problems of stability. — Moscow: Nauka, 1966. [in Russian]
2. Daleckii Ju. L., Krein M. G. Stability of solutions of differential equations in Banach space. — Moscow: Nauka, 1970. [in Russian]; English transl.: Vol. 43: Translations of mathematical monographs. — Providence: Amer. Math. Soc., 1974.
3. Demidenko G. V. Stability of solutions to linear differential equations of neutral type //J. Anal. Appl. — 2009. — Vol. 7, no. 3. — P. 119-130.
4. Demidenko G. V., Matveeva 1.1. On stability of solutions to linear systems with periodic coefficients // Sib. Math. J. — 2001. — Vol. 42, no. 2. — P. 282-296.
5. Demidenko G. V., Matveeva 1.1. On estimates of solutions to systems of differential equations of neutral type with periodic coefficients // Sib. Math. J. — 2014. — Vol. 55, no. 5. — P. 866-881.
6. Demidenko G. V., Matveeva 1.1. Asymptotic stability of solutions to a class of linear time-delay systems with periodic coefficients and a large parameter // Journal of Inequalities and Applications. — 2015. — Vol. 2015, no. 331. — P. 1-10.
7. Horn R. A., Johnson C. R. Matrix analysis. — Cambridge: Cambridge University Press, 2013.
Получена 15.11.2015 Переработана 03.12.2015
