DSpace at VNU: Stability radii for differential algebraic equations tài liệu, giáo án, bài giảng , luận văn, luận án, đồ...
VNU JOURNAL OF SCIENCE, Mathematics - Physics t XVIII n°l - 2002 S T A B IL IT Y R A D II A L G E B R A IC FOR D IF F E R E N T IA L E Q U A T IO N S N guyen H un Du Facility o f Mathematics, Inform atics and M echanics College o f Natural Sciences - VNƯH Dao Thi Lien Faculty o f M athem atics Teacher'S Training College, Tlm inguyen University A b s t r a c t In this article, we deal with the problem o f computing stability radii f o r systems described by differential algebraic equations o f the fo rm A X ' ( t ) + B X ( t ) = where A B are constant matrices A computable fo rm u la f o r complex radii is given and the key difference between O D E s and D A E s cases is pointed out A special casr where the real stability radii and complex one are equal is considered K ey W ords an d P h ses: stability radii Differential algebraic equation, index o f m atrices pencil, Introduction Ill the last decade, a large am ount o f works has been devoted to robustness measures am ong them there is a powerful tool, nam ely the stability radius, which was introduced by Hiiirichsen and Pritchart (see [2] ) It is defined as the sm allest value () o f the norm of real or com plex perturbations destabilizing th e system If com plex perturbations are allowed Ị) is called the com plex stab ility radius If only real perturbations are considered, the real radius is obtained A detailed analysis of th e stability radius for ordinary differential equations can he found in [2,3.4] In this article, we deal w ith the com putation of stab ility radii o f sy stem s described by a differential algebraic equation A X ' ( t ) - B X { t ) = 0, (1.1) with constant, m atrices A and B T his problem has been well investigated for the case of nonsingular m atrix A when (1.1) turns into an explicit sy stem o f ordinary differential equations (O D E s for short) m X ' ( t ) = M X ( t ), where the m atrix M = A 1B According to the works in [2] [3] the stab ility radii can be characterized by th e m atrix A/ and it is com puted ill principle If the m atrix A is singular, then th e investigation of the index of the pencil {A , 13} is necessary but the situation becom es more com plicated It is known that ill O D E s case, if the original equation (1.1) is stable, then by continuity of spectrum , the stab ility radius is positive However, this property is no longer T y p e s e t by 16 sta b ility radii fo r d ifferen tia l algebraic equations IT valid in tin* cas It is known that there ex ists a pair o f nonsingular m atrices \ \ \ T such that A H -Ự ' “ ) r - , B = w Ợ 0' /m°_ r ) r - , (2.2) where / s is the unit m atrix in K sXs Further B \ € I \ VX1\ u is a A:- nil potent matrix having I.hr Jordan box form i.e u = d ia g Ụ iy J ‘2 , /,= /0 •//) w ith °\ () e F l then the matrix function G(.s) is unbounded on i.R Indeed as s —> “X Therefore, ill this case, d c — T his m eans t hat under a very small disturbance, the DAEs with the index greater then is no longer stable If ind(>4, D ) — it is easy to prove th a t |ịơ(.s)|| is bounded oil perhaps it (loos not exist any "bad” m atrix A such th at !|A|| = d c i.e., d (7 > but Slim m ing up we obtain T h e o r e m n) The com plex sta bility dius o f System (2.1) is given by d o = [sup ||G («)||] \ A' £ i f? where G ( s ) = F(.s/1 bj T h ere exists a “bad” m a trix A such that ||A || = d ỗ i f and o nly if G ( s ) attains its m axim um over ỈR Stability radii fo r differential algebraic equations 21 c) hi the a ISC E = F — I , d c > if a n d o n ly i f i n ả ( A s D) = A question risrs Ihtc: whenever t he function ||G («)|| attains its maxim um at a finite value *(, \Y< firstly remark t hat th e answer depends strongly on th e chosen norm of C ' n sinct* G ( s ) \ has maximum values in one norm but has not in another one To sim plify the situat ion W(‘ solve the problem w ith A, B G R w and w ith a Euclid norm in the set of m X HI m atrices, that is if M - ( u i i j ) i-s 3- w x m ~ m atrix then ||A /||2 = (leal wit 11 the way to obtain t he decom position (2.2) First we decom pose (A — B ) ~ l A into Iordan form by a nonsingular m atrix , that is (A - B ) l /l = S d i(u j( M , V ) S " X where r is a nilpotent m atrix of the form (2.3) and M is nonsingular T he m atrix w and T in (2.2) ih given by ir (.4 D )S d ia g (M J ): T = S d i.a g { I,( V - u = V (V - I) (3.2) If G ( s ) is unbounded oil c \ then A c = and there is no th ing to say The assum ption G’(.s) to he bounded im plies that F T d ia q ( Q U J ) \ V ~ [ E = Vj > Thus G {s ) := F T d i a y { ( s I - B ị ) [ { U - / r 1) ^ - 1# = F T d i a g { ( s I - B i ) - \ L e t / ( * ) = :: ( i / s ) ị | ,J i f 6* Ỷ u a n d / ( ) = 1ÌI1Ì.V-+OC | | G ( l / s ) | | (w e r e m a r k t h a t th is limit always exists) It is easy to see that f ( s ) = ||FTdm//(s(/ —si?i) 1, —I ) W 1E\\~ Since all entries of the' m atrix G ( s ) are only rational functions which are analytic then by th e maxim um principle, th e maxim um of G ( s ) takes place only at s = oc or s G iR Therefore, G (s ) attain s its m axim um at s = 00 iff /(.