DSpace at VNU: On the asymptotical stability for index-k tractable daes

VNU JOURNAL OF SCIENCE, M athem atics - Physics T XVIII, Nq - 2002 O N T H E A S Y M P T O T IC A L S T A B IL IT 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 Nguyen 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 [ - ] In [9] Tatyana Shtykel proposed a numerical parameter x ( A , ft) characterising the asym ptotical stability of the trivial solution of linear system index-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 T h e index-A; tr a c ta b l e D A E s C onsider th e differential algeb raic eq u a tio n A X ' + B X = 0, (1) w here A , B are co n sta n t m a trices o f order 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 with index-k S in ce th e m a trix p en cil is regular index-A; and rank[(cA + f i ) " A]k = r, there e x ist invertible m a trices W) T such th a t A - W {o °u )T" ' u k = o, Ư1 ï o, B = W {~Ề' / I for all I < k , ) ' - 1' w here I s is th e X id en tity m a trix L et u s set Q o = t [° / « ‘-> = r ( o 2) T - , P , = / - Q , = r ( ằ = °o )T T ypeset by Ạạ^S-TIẼX 10 D a o Thi L ie n Let A = A - B Q k- = w ( ị ^ T -\ N\ = ker.Ẩ, S i = {z £ R m : BPk-2z € Im *4} It is cleax th at 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 at A' l - T ( o I m- r + U + + U k- ) W Multiplying ( ) 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, ( Q o* ) ' + Q i * +