DAEs arise in various problems in the natural sciences and technology.. In this paper we study the same param eter for linear system of index-A: DAEs... It is cleax th a t Q/c -1 is can
Trang 1VNU JOURNAL OF SCIENCE, M athem atics - Physics T XVIII, Nq4 - 2002
O N T H E A S Y M P T O T IC A L S T A B I L I T Y F O R INDEX-A: T R A C T A B L E D A E s
D a o T h i L ien
Teacher's Training College, Thai N guyen U niversity
A b s t r a c t DAEs arise in various problems in the natural sciences and technology The stability of DAEs was studied by many authors [ 3 - 9 ] In [9] Tatyana Shtykel proposed
a numerical parameter x ( A , ft) characterising the asym ptotical stability of the trivial
solution of linear system in d ex-1 DAEs
A X ' + B X = 0,
with constant matrix Ay B , where A is singular In this paper we study the same param
eter for linear system of index-A: DAEs.
1 T h e index-A; t r a c t a b l e D A E s
C onsider th e d ifferential a lg eb ra ic eq u a tio n
w here A, B are c o n sta n t m a tr ic e s o f ord er m sa tisfy in g
( le tA = 0, r a n k [ ( c A + B ) ~ xA] k = r.
D e f i n i t i o n l ( s e e [3 ]) T h e equation (1) is called index-k tractable i f the m a trix pcncil { A, B } is regular w ith in d ex-k.
S in ce th e m a trix p e n cil is regu lar index-A; an d r ank [ ( cA + f i ) " 1 A]k = r, th ere e x ist
in vertib le m a tric es W) T su ch th a t
A - W { o ° u ) T " '
u k = o , Ư 1 ï o , for all I < k ,
B = W { ~ Ề ' / I ) 7 ' - 1'
w here I s is th e 5 X 5 id e n tity m a tr ix L et u s set
Q o = t [° 0
/ 2) T - , P , = / - Q , = r ( ằ
T ypeset by Ạạ^S-TIẼX
9
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Let
A = A - B Q k- 2 = w ( ị 0 ^ T - \
N\ = ker.Ẩ, S i = {z £ R m : BPk-2z € Im *4}.
It is cleax th a t Q/c -1 is canonical projector onto JVi along S\ and jP/c- 1 is canonical projector onto Si along N\ Denote
A , = A + B P k- 2Q k -i = w ( ị I m ° _ ư ) r -1
It is easy to see th a t
A ' l - T ( o I m- r + U + + U k- 1 ) W
M ultiplying ( 1 ) by P k -\A l \ Q0A Ỉ s Q iA x 1, ,Qk~iAl \ respectively, we obtain:
r ( P ^ X Y + P k - i A ^ B P k - i X = 0,
Q o X = 0,
k ( Q k - ĩ X Ỵ + ( Q k - z X ) ' + + (Q o X )' + Q k - i X 4- Q k -2% + 4- Q q X — 0.
Because of
P k -I + Qo + + Q k - 1 = t (Jú Im _ r + u ° " ' + V k - 1 ^ T ~ l = K
is in v ertib le ,h en ce th e s y ste m (2) is eq u iv a le n t to (1); an d from th e s y s t e m (2) we have
r ( P k - x X Ỵ + P k - i A ỵ l B P k - i X = 0,
\ Q fc_ i X = 0.
X is a so lu tio n o f (1) i f a n d o n ly if P k - i X is th e o n e o f (3 ).
D e f i n i t i o n 2 ( s e e [9]) A m a t r i x v a lu e d fu n c tio n Q( t ) = Ç ( t , A , B ) € c 1 is called
th e G reen m a trix o f e q u a tio n (1) i f i t satisfies th e in itia l va lu e p r o b le m ( I V P )
r ịG { t ) = M Ç (t ) (í > 0),
where M = —P k - i A ị 1B.
It is easy to verify th a t M = P k - \ M = M Ffc-ii and consequently Q(t) — Pk-\etM
is the unique solution of the r v p (4).
Therefore the general solution of equation (1) is of the form
tM
X (t) = g(t)Xo = P k -ie tMXo,
Trang 3O n t h e a s y m p t o t i c a l s t a b i l i t y f o r 11
w h e r e Xq is an arbitrary c o n s ta n t v e cto r T h u s, w e have proved th e fo llo w in g
T h e o r e m 1 L e t { A } B } b e a r e g u la r p e n c il w ith in d e x - k , Q k ~ \ th e c a n o n ic a l
projector onto N\ 'cûong S\, aiid Pk- 1 = I - Q k -1 - Then the initial value problem
r A X ' + B X = 0,
1 P*-i(*(0)-Jfo) = 0t
[or all X o € R m has a u n iq u e solution X( t ) given by X ( t ) = P fc - \ e t MXq w ith the m a trix
M = - P k- l A i i B.
T h is th e o r e m seem s n o t n e w b u t th e m e th o d o f p ro o f is a p p ro p ria te for stu d y in g
th e a sy m p to tic a l s ta b ility o f in d ex -k tra cta b le D A E s.
2 T h e c r ite r io n o f a s y m p to tic a l s ta b ility o f t h e tr iv ia l s o lu tio n o f D A E s w ith index-A:
2 1 T h e a s y m p to tic a l s ta b ility o f th e tr iv ia l s o lu tio n o f D A E s w ith in d e x -k
D e f i n i t i o n 3 ( s e e [3 - 7 ] ,[ 9 ] ) T h e trivial solution X = 0 o f (1) is called stable in
th e sense o f L iapunov i f for certain projector II along the m a xim a l invariant subspace o f the m a trix pencil {A , 8 } associated w ith the infinite eigenvalue th e I V P
f A X ' + B X = 0,
1 U ( X ( 0 ) - X o ) = 0,
for all X o € R™ has a so lu tio n X ( t j X o ) defined on ( 0, + o o ) M oreover, for each e > 0 there exists a ỗ = ổ(e) > 0 such th a t ||X ( i , X o ) || < e for all t > 0 a n d for all Xq e R m
w ith u n r o l l < Í Here we choose II = P i t - 1.
D e f i n i t i o n 4 ( s e e [6] ,[ 9 ] ) T h e trivial solution X = 0 o f (1) is said to be a sym p totica lly stable in th e sense o f L iapunov i f it is sta b le and there is a ỏo > 0 such th a t for dll X q £ H171 sa tisfyin g th e in eq u a lity ||ILXo|| < ÔQ one g ets X ( t, X q ) —► 0 as t -4 4-00.
L e m m a I f Ư is a k -n ip o te n t m a trix then d e t ( x u + I m - r ) ^ 0 for all X £ c
T h e o r e m 2 T he trivia l solution X = 0 o f (1) is a sym p to tica lly sta b le if and only
i f all fin ite eigenvalues o f th e m a tr ix pencil { A, B } have negative real parts.
2 2 T h e c r i te r i o n o f a s y m p to tic a l s ta b ility
L et all fin ite eig e n v a lu es o f th e p en cil { A ì B } w ith index-fc h a v e n e g a tiv e real p arts
A ssu m e th a t th e m a tr ic es M a n d jPfc- 1 have th e str u ctu re s d e scrib ed a b o v e W e consider
th e L ia p u n o v eq u a tio n
w ith a n u n k n o w n m a trix X T h e m a trix F is su p p o sed t o b e h e n n itia n and p o sitiv e
d efin ite S in ce
| | F f c - i e t M | < 7 ( r ) ( M ) r - l e - t a / 2 i
Ơ
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H k = / é M ' P'k _ xF P k ^ e t M di + Q l _ xF Q k - x
J o
converges On the other h an d Hk is herm itian and positive definite and Hk satisfies the
eq u a tio n (5 ).
T h e o r e m 3 I f the m a tr ix p c n c il { A } B } has ind ex-k and all its fin ite eigenvalues belong to th e n eg a tive co m p lex h a lf p la n e, th en x ( A } tì) = 2\\A — B Q /c - 2||||/3||||iĩfc|| < oo
Inversly we assu m e th a t x ( A yB) < o o and H k is a solution o f (5 ).T h en th e follow ing
in e q u a lity is v a lid
-til
< V ữ Ã 7 t ì ) e \\Pk-iXo\\.
T h is m ea n s th a t th e trivial so lu tio n X = 0 is a sym p to tica lly stable.
A c k n o w l e g m e n t W e w ish t o th a n k P rof P h a m K y A n h an d Dr N g u y e n H u u
D u for th eir v a lu a b le s u g g e s tio n s d u r in g th e p rep aration o f th is paper.
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