Lyapunov transformation and stability of differential equations in banach spaces
A sufficient condition of exponential stability of regular linear systems with bifurcation on a Banach space is proved.
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1999
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510907626422272 |
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| author | Tran, Thi Loan Тран, Тхі Лоан |
| author_facet | Tran, Thi Loan Тран, Тхі Лоан |
| author_sort | Tran, Thi Loan |
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| description | A sufficient condition of exponential stability of regular linear systems with bifurcation on a Banach space is proved. |
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UDC 517.929
TRAN Till LOAN (Hanoi Teacher's training College, Vicmam)
LYAPUNOV TRANSFORMATION
AND STABILITY OF DIFFERENTIAL EQUATIONS
IN BANACH SPACES
IIEPETBOPEHH$I JIgIIYHOBA
I CTII~KICTb IIHtDEPEHII, IA.TIbHHX PIBHflHI:,
B I~AHAXOBHX IIPOCTOPAX
A sufficient condition of exponential stability of regular linear systems with bifurcation on Banach space
is proved.
BCTaHOI~mUO ltoc'ra'rlli yMonil eKCnOHentfiaJmuoi CTiltKOeTi pel~yJl~lpliHX dlilli~tliHX CHc'reM a 6iqbypKa-
ttiem n 6atlaxonoMy npocTopi.
vd-TransforInat ion and its properties. In this section we shall give the definition,
examples and some properties of a v d-transformation on Banach spaces. It is an
expansion of a v d-transformation on finite-dimensional spaces given by Yu. S.
Bogdanov [1 -5 ] . From that, we shall give the definition of regular linear equations
which are applied to study the stability of regular linear equations with bifurcation on
Banach spaces.
Let E be a Banach space and let G be an open simple connected domain
Containing the origin O of E.
We define H as follows: H = G x R = {r 1 = (x , t ) : x ~ G, tE IR}.
Consider a function
v0: R+-> ~ +
that is continuous, monotone, and strictly increasing and satisfies the following
conditions:
%(0) = 0; r e ( t ) ---> +** as t---> + * * .
Let d be a given real function of two variables:
d: IR+xR + ~ R,
(~I,~2) ~ d(~tl,~/2)"
We assume that d satisfies the following conditions for all y> 0, Y3 > Y2 > Yl > 0 :
dr) d(Y2, YI) = - d ( 7 1 , 7 2 ) ;
d 2) d(~/' 2, ~/) > d(~' l , ~') ;
d3) d(v3,~/2) + d('Y2,~/i) _> d(~/:3,1fl);
d4) U {d(v' Vl)} = r .
yER +
Assume that J is a diffeomorphism from H to H, namely
J: H--->H,
11 = ( x , t ) ~ rl" = ( x ; t ' ) ,
and let it satisfy the following equalities for all t E R :
�9 TRAN THI LOAN, 1999
ISSN 0041-6053. Yxp. ~ra..a'ypu., 1999, m; 51, N'-' I 0 1417
1418 TRAN q['HI LOAN
J(0, t) = (0, t),
J(x, t) = (x" t).
It is easy to prove that the set L of all such transformations L = {J} is a group with
composition of maps.
Consider a real function
v : H * - - - > R §
= ( x , t ) oCt) = Vo(iixll),
where H* = G* x R = ( G \ { 0 } ) x R.
Since the function v : H* ---> R + is independent of t, i.e. v (x, t) = v (x, t') for
all t , t ' e R, we can denote by v(x) thevalueof v ( x , t) for any x e G* and t a
E R .
Definition. The transformation J e L is called a vd-transformation iff
sup Id{v(rl),V[J(ri)]}[ < +oo. (1)
"qEH*
From the definition of the function d, we also have
sup [d{v(~'), v(J -I (tl'))}l < + ~,.
"q' e H*
Consequently, if we denote by Lvd the set of vd-transformations, then it is a
subgroup of L.
Examples. 1. Let Vo(X, t) = [Ix[I, d0(Tl, 7 2 ) ----" l n T l , and let J (x , t) (with
Y2
fixed t) be a linear transformation having a bounded partial derivative with respect to
t. Then J is a vo do -transformation if and only if it is a Lyapunov transformation [6].
2. If v ( x , t ) = Ix[ 2, E = ~ ,
1 d ( ' t , , 72) = I.
then all conditions d I ) - d 4) are satisfied.
Thus,
J ( x , t) = ( x
k
is v d-transformation.
if 7t '72 - - I,
if 71"72 < I,
+ 1 sin t sin 2 x, t)
2
ISSN 0041.6053. Yrp. Juam. ~.'ypn., 1999. m. 51, N e ! 0
3. We consider the following linear differential system in the J.-space:
dXk -- - C O S t ' X k + Xk+ ! , k e N . (2)
dt
The transformation J(x', t) = eSmt(x , t) with v(x, t) - II,:ll: d(7 , 72) = in 7__L
72
is a vd-transformation and it reduces (2) to the system
L Y A P U N O V T R A N S F O R M A T I O N A N D S T A B I L I T Y . . . 1 4 1 9
dY.._~k = Y~+l, k r N.
dt
From example 1, we can see that a v d-transformation is an expansion of the
Lyapunov transformation, but it still keeps an important property, namely, the stability
of the trivial solution of the following differential equation on the Banaeh space E:
dx
- - = f ( x , t ) ,
dt
(3)
f (O, t) ~ 0.
We denote by x(t; ~) the solution of equation (3) that satisfies the initial condition
x(t0; ~) = ~ and assume that
~ = lim sup IIx(t;~)ll, ~.l = lim sup v(x( t ;~) ) .
~o + II,~ll<,~ ~---, o + ,,(~)~ ~
t ~ t o t > t o
Definition [7]. The solution x = 0 of the differential equation (3) is said to be
Lyapunov stable iffor any e > 0 there exists 8(e) > 0 such that, for each solution
x( t) of (3 )whose initial value X(to) = ~ satisfies the condition I1~11 < 5(e) , the
inequality II x( t ; ~) II < e hold for all t > t O.
From the definition, we can see that the solution x = 0 of the differential equation
(3) is stable iff ~. = 0.
Proposition 1. 7~ = 0 if and only if 9~ 1 = O.
Proof. By the continuity of the function v, we immediately have lim v(~) =
~--+o
= 0. Since v (II x II) is a monotone strictly increasing function, we can deduce
lira ~ = 0. Therefore,
u(~,) ~ 0
lim ~k = 0 r lim v(~k) = 0. (4)
k ---) .0 k .---I. -0
We assume that 7~ = 0. Then
lim IIx(tk ;r = 0
k -'~' 0"
for all sequences { e k } c R + : e,-:->0, { ~ k } c E : II~kll < ek, and {tk} c R, tk>
>_t 0. By virtue of (4), we have lim IIx(tk;~k)ll = 0 ~ lim v(x(tk;~k)) = O.
k -'-~** k .---> *~
It follows that ~ .=0 r Xt=0 .
Proposit ion 2. A vd-transformation preserves the stability of the trivial solution
x = 0 of the differential equation (3).
Proof. By vd-transformation
(x , t ) ~ Y(x, t) = (y, t),
equation (3) is transformed to the following one:
ay = g(y, t). (5)
dt
By assumption, the solution x = 0 of equation (3) is stable, which means that
lira sup IIx(t;xo)ll = o <=> lim sup v [x ( t ;xo) ] - o.
t~to t>to
ISSN 0041-6053. Yx.p. ,uam, ~."vpn.. 1999.m. 51, A ~ I0
1420
If the solution y = 0 of (5) is unstable, then
lim sup v[y(t, Y0)] > 0.
z~O + U(yo)~
t ~ t 0 ,
This means that there exists a positive number 8 such that
3 { ' q n } C E : tin--->0, 3 { t n } c R + : tn>_.to, V n ~ N:
v[Y(tn, TIn)] > 8.
Since v[x(tn ; ~n)] --4 0 as n ---> **, where (~n, tn) = J- l (rln, tn), onecansay
~ ;~n)] < 8 ~' n ~ N.
From (6), (7) and d 4) we conclude that
Id{u[x(t~;~.)], v[y(tn;n.)]}l = d{v[y(t.;rl.)], v[x( t . ;~ . )]} >
> d{5, v[X(tn;~n)]}-'>+** as n-.-->~.
Consequently,
TRAN THI LOAN
(6)
(7)
I$SN 0041-6053. YtqJ. ~utm. ~'vpn., 1999, m . 5 1 , 1 ~ 10
sup IdMxft , ;~n)l . o[J (xCt , .~ , ) ) ] } l = +r
n E N
which contradicts the definition of J.
Regular system.
Def'mition. A transformat(on J ~ L satisfying the condition
d{v(TI),v[J(T1)]} = o(t) as t--->+**
for aU rl E H* is called vd-transformation.
Definition. A transformation y = L( t )x is a generalized Lyapunov one if
;~[L(t)] = ~ [ L - l ( t ) ] = 0, (8)
where z [L ( t ) ] := li-'m 11n II ~ (t)ll ~ called the characteristic exponent o f L(t).
t ~ * * t
By definition, we immediately have following remarks:
Remark 1. Generalized Lyapunov transformations preserve Lyapunov expo-
nents [6].
R e m a r k 2. A generalized Lyapunov transformation is a generalized vd-
transfoimat~on if
v(x) = IIxll, d(`/~,`/2) = In `/_L1,
"/2
and J is homogeneously linear for x (here, J(x, t) = (L( t )x , t)).
We consider the following linear differential system:
dx
- - = A ( t ) x , (9)
dt
where x E R n, A (t) ~ 5s (R n, I~ n) and is real and continuous for all t E R , and
sup [Ia(t)ll < **.
t
Let X(t) be a normal fundamental matrix of (9) and let a x = ~ '~= 1 nk txt be the
sum of all its exponent numbers [6 ] .
LYAPUNOV TRANSFORMATION AND STABILITY ... 1421
Def in i t ion [6]. The linear system (9) is said to be regular iff
1 t
= lira : f SpA(x )dx . fix
t ....~ on
[ - -
tn
We know the following proposition [6] :
Lemma. A necessary and sufficient condition for system (9) to be regular one is
that there exists a generalized Lyapunov transformation that reduces system (9) to
the system with constant matrix B ~ ~(]R n, ]R n) :
dy
= B y . (10)
dt
Definit ion. A linear differential system
dr
- - = A ( t ) x , (11)
dt
where A ( t ) e ~ ( E , E ) a n d is continuousforall t ~ R and supllA(t)ll < **, is
t
said to be a regular one iff there is a generalized Lyapunov transformation y = L ( t )x
that reduces it to a linear differential equation with constant operator:
dy = By . (12)
dt
We now present the main theorem for regular differential equations on Banach
spaces.
Consider the differential equation
dr
- - = A( t ) x + f ( x , t), (13)
dt
where A ( t ) e ~s E), sup IIA(t)ll < **, f e CCI '~215 f (0 , t) - 0, and
teR /
IIf(x,t)ll <- w( t ) l l x l l m, m > l , z [V( t ) ] = o.
Under these conditions, the following theorem is true:
Theorem. If equation (11) is regular and all its characteristic exponents are not
larger than - ~ < O, the trivial solution x = 0 of equation (13) is exponentially
stable [7], i.e., there exist N > 0 and A > 0 such that
IIx(t)ll <- A e -N( t - t~ IIx(t0)ll
for all solutions x( t) of(13).
Proof. We denote by X(t ) (X ( t o) = Jde) the Cauchy operator of equation (11)
[7, p. 147].
1. First, we estimate the resolvent operator K(t, x) = X( t )X -l (x), "Co< x < t.
By virtue of the regularity of equation (11), there is a generalized Lyapunov
transformation y = L(x )x that reduces equation (11)to equation (12).
The operator Y(t) = L( t )X( t ) is the resolvent operator of equation (12).
If we put H(t, "c) = Y(t) y - t (x), then K(t, "c) = L( t )H( t , x)L - l (x).
Assume that all characteristic exponents of equation (11) are not larger than r
Hence, all characteristic exponents of equation (12) are not larger than ct, i.e., for
every solution y(t) = Y( t )y o and e > 0, there exists c > 0 such that
ISSN 004 ] '6053, Y~. ~am. ucypu,. 1999. m, 5 I. N e I0
1422 TRAN THI LOAN
z[L(t)] = z[L-~(t)] = 0
It follows that
I ly ( t ) l l -< ce (a+e/2)t V t>-to.
Then, the operator family { e -(a+r Y(t), t >- tO} is point-bounded. By virtue
of the Banach - Stenhauss theorem, there exists c I >0 such that
Ile-("+~/2)'y(t)ll <- ci ** II Y(t)ll < c l e c~+~/2)'
Therefore, II H(t, x)l l = II Y ( t - x)l[ -< c I e(ct+e/2)(t-x) for the equation with
constant operator (12).
On the other hand,
IIg(t)ll <-- c2 eetl2,
IlL - l (t)ll -< c3 eex/2.
IIK(t,'c)ll . <- IIL(t)ll IIH(t,x)ll IIL-~(x)ll -<
< C I C2C3 e(ct+e)ft-'c) e e~ = c(~, to) e (ct+~)(t-~),
where c = c l c 2c 3.
Since K(t, to)= X(t), we have I Ix ( t ) l l -< ce ~+~)'.
In the case where tx < 0, there exists a positive number e such that t~ + e < 0,
whence
IIK(t,x)ll <- ce",
II g(t) ll < c.
2. We now prove the theorem. Denoting y = x e u176 where -[ is a positive
number such that 0 < "t< k, we transform equation (13) to the form
dy = B( t )y + g ( t , y ) (14)
dt
with B(t) = A(t) + TJdE
g(t , y) exp ( y ( t - to)) f( t , y e-~l(t-t")). (15)
Let us show that the equation
drl = B(t)'q (16)
dt
is regular. Indeed, by virtue of the regularity of (11), there is a generalized Lyapunov
transformation z = L(t)~ that reduces (11) to an equation with constant operator
dz
= Cz,
dt
where C = L ' ( t ) L - l ( t ) + L ( t ) A ( t ) L - t ( t ) .
The transformation ~ --" L( t )~ implies the following:
= [L ' ( t )L - I ( t ) + L ( t ) B ( t ) L - l ( t ) ] ~ = (C + yJdE) ~.
dt
The regularity of (16) is proved.
ISSN 0041-6053. Yxp. ~am. ~. pu.. 1999, m. 51, N e 10
L Y A P U N O V T R A N S F O R M A T I O N AND STABILITY ... 1423
- - We denote by TI (t) a solution of (16) and then e -~'(t-t~ rl (t) is a solution of (11).
This yields
Z [ Tl(t)e -Y(t-t~ < - ~
Z [ ~ q ( t ) ] < z [ e Y(t-t~ + Z[ ' q ( t ) e -Y(t-t~ <_ -~ , + y < O.
By virtue of the estimation of the resolvent operator, the following inequality is
true:
II K(t, x)ll -< Ne ~*,
where K(t, "r is the resolvent operator of (11).
Now considering the solution of (14)
we have
to < "C < **,
y ( t ) =
t
K(t, t o) y( t o) + ~ K(t, "0 g(x, y(x)) dx,
to
t
II y(t)ll --- II K(t, to)ll Ily(to)ll + ~ IlK(t, x)ll IIg(x, Y(x))ll dx <.
to
t
<-- Ne et~ Ily(to)ll + J" Ne~X e't(x-t~ m e-m't(x-to) dx <-
q~
t
< Nee' ' Ily(to)ll + J" NeeX e(l-m)rfx-t~ eer Ily(x))ll m d'~ =
to
l
= ct Ily(to)ll + ~ c2 e [2e-(m-l)'t](x-t~ IlY(x))llmdx,
lO
where c I = N e et~ c 2 = c N e -2et~
Hence,
t
Ily(t)ll <-- q Ily(to)ll + ~ c2 e-8(x-t ') IlY(x))llmdx,
to
where 8 = ( m - 1 ) y - 2~.
Let us find a positive number g such that 8 > 0.
Since
i e -8(x-t~ = 1 1 -8(t-to) 1
8 5 e < 5
t0
there is A > 0 such that
provided: that
N =
t
(m- 1)c~ -t Ily(t0)ll m-I ~ c2e'SC'-'~ < 1,
to
!1 y(to) II < a .
ISSN 0041-6053. YsO. zlam. ~y. pu., 1999. m~ 51, N e 10
1424 T R A N T H I L O A N
We apply here the Bihari lemma [8] and find
Ily(t)ll < - q [ly(to)ll ct
[ l _ N ] l l ( m _ l ) = A Ily(t0)ll, A = [ l _ N l l / ( m _ l )
IIx(t)ll < Ae-V('-'")llx(to)ll) h, X(to) = y(to),
which means the exponential stability of the solution x = 0 of (13), and the proof of
the theorem is completed.
1. Bogdanov Yu. S. Application of generalized exponent numbers to the investigation of the stability
of equilibrium point II Dokl. Aead. Nauk SSSR. - 1964. - 158, N -~ 1. - P. 9 - 1 2 (in Russian).
2. Bogdanov Yu. S. Generalized exponent numbers of nonautonomous systems II Different. Equat. -
1965. - 1, N TM 9. - P. 1140-1148 (in Russian).
3. Bogdanov Yu. S. Onthe reveal of asymptotically stability by means of little v d - n u m b e r s / / I b i d . -
1966. - 2, N ~ 3. - P. 309-313 (in Russian).
4. Bogdanov Yu. S. Approximate generalized exponent numbers of differential systems II Ibid. - N ~ 7.
- P. 927-933 (in Russian).
5. Bogdanov Yu. S., Bogdanova M. P. A nonlinear analog of Lyapunov t ransformation/ / Ibid. - 1967.
- 3, N 9 5. - P. 742-748 (in Russian).
6. Demidovitch B. P. Lectures on the mathematical theory of stability. - Moscow: Nauka, 1967. -
472 p. (in Russian).
7. Daletski Jm L., Krein M. G. Stability of solutions of differential equations in a Banach space. -
Moscow: Nanka, 1967. - 534 p. (in Russian).
8. Bihari J. Ageneral izat ion of a lemma of Bellman and its application to uniqueness problems of
differential equat ions/ /Acta math. Acad. sci. hung. - 1956. - 7, N ~ 1. - P. 81-94 .
Received 14.04.97
ISSH 0041-6053. Yxp. ~.uJm. ~. pn.. 1999. m. $J. N e l 0
|
| id | umjimathkievua-article-4741 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:04:27Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/14/eda23c1a46428176c189bfe79fb0d814.pdf |
| spelling | umjimathkievua-article-47412020-03-18T21:12:54Z Lyapunov transformation and stability of differential equations in banach spaces Перетворення Ляпунова і стійкість диференціальних рівнянь в банахових просторах Tran, Thi Loan Тран, Тхі Лоан A sufficient condition of exponential stability of regular linear systems with bifurcation on a Banach space is proved. Встановлено достатні умови експоненціальної стійкості регулярних лінійних систем з біфуркацією в банаховому просторі. Institute of Mathematics, NAS of Ukraine 1999-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4741 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 10 (1999); 1417–1424 Український математичний журнал; Том 51 № 10 (1999); 1417–1424 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/4741/6183 https://umj.imath.kiev.ua/index.php/umj/article/view/4741/6184 Copyright (c) 1999 Tran Thi Loan |
| spellingShingle | Tran, Thi Loan Тран, Тхі Лоан Lyapunov transformation and stability of differential equations in banach spaces |
| title | Lyapunov transformation and stability of differential equations in banach spaces |
| title_alt | Перетворення Ляпунова і стійкість диференціальних рівнянь в банахових просторах |
| title_full | Lyapunov transformation and stability of differential equations in banach spaces |
| title_fullStr | Lyapunov transformation and stability of differential equations in banach spaces |
| title_full_unstemmed | Lyapunov transformation and stability of differential equations in banach spaces |
| title_short | Lyapunov transformation and stability of differential equations in banach spaces |
| title_sort | lyapunov transformation and stability of differential equations in banach spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4741 |
| work_keys_str_mv | AT tranthiloan lyapunovtransformationandstabilityofdifferentialequationsinbanachspaces AT tranthíloan lyapunovtransformationandstabilityofdifferentialequationsinbanachspaces AT tranthiloan peretvorennâlâpunovaístíjkístʹdiferencíalʹnihrívnânʹvbanahovihprostorah AT tranthíloan peretvorennâlâpunovaístíjkístʹdiferencíalʹnihrívnânʹvbanahovihprostorah |