DSpace at VNU: Stability radii for differential algebraic equations

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VNU JOURNAL OF SCIENCE, Mathematics - Physics t XVIII n°l - 2002

S T A B I L I T Y R A D I I F O R D I F F E R E N T I A L

A L G E B R A I C E Q U A T I O N S

N g u y e n H u n D u

College o f N atural 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 ) = 0

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 e y W o r d s a n d P h r a s e s : Differential algebraic equation, ind ex o f m atrices pencil,

stability radii.

Introduction.

Ill th e last decade, a large am ount o f works has been d ev o ted to robustness m easures

am ong them there is a powerful tool, nam ely th e stability radius, w hich w as introduced by

Hiiirichsen and Pritchart (see [2] ) It is defined as the sm allest value () o f the norm o f real

or com plex perturbations d estabilizin g th e system If com plex p ertu rbations are allowed

Ị) is called th e com plex sta b ility radius If only real p erturb ation s are considered, the

real radius is obtained A d etailed analysis o f th e stab ility radius for ordinary differential

equations can he found in [2,3.4].

In th is article, we deal w ith th e com p u tation of sta b ility radii o f sy ste m s described

by a differential algebraic equation

w ith constant, m atrices A and B T h is problem has been well in vestigated for th e case

o f nonsingular m atrix A. when (1.1) turns into an explicit sy ste m o f ordinary differential

equations (O D E s for short) m X ' ( t ) = M X ( t), w here the m atrix M = A 1B A ccording to

the works in [2] [3] the sta b 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 in v estig a tio n of the index of

the pencil {A , 13} is necessary but th e situ a tio n becom es more com p licated

It is known th at ill O D E s case, if the original eq uation (1 1 ) is stable, then by

continuity o f spectru m , the sta b ility radius is p ositive However, th is 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<‘ o f différential algebraic eq u ation s (D A E s for sh ort) T h e mail! reason is that tho structure of solution s o f differentia] algebraic equations depend s strongly on the indrx o f (hr pencil {.4 B } and th e so lu tio n s o f (1.1) have som e fixed com ponents I nder the perturbations, th e index o f th e perturbed sy stem s may he changed, th at im plies the chauvin*; o ft hr dim ension o f th ese fixed one and som e o f eigenvalues may 1)0 ’‘disappeared"

which causr.s the stab ility radius o f (1.1) perhaps to he equal to Ü M oreover, it is different

to ODE.S case, in which we alw ays arc able to find a dist urbance w hose norm equals to stab ility radius Ị) and under which our sy stem is unstable, such a m atrix ill D A Es case may not exist

T herefore, to stu d y sta b ility radii o f algebraic differential equations, one* must pay

a tten tion on th e index o f equation or the distu rb an ces m ust having som e special forms that we call stru ctu red perturbations in order to exclu d e “violent factors’*.

T h r article is organized rus follows: In th e next section , we stu d y som e basic prop- (Ttk's o f (inferential algebraic equations S ection 3 deals w ith a form ula for com puting the stab ility radius o f (1.1) w here structured d istu rbances are considered Section 4 is concernrd w ith a special class o f the pencil o f m atrices {A 13} for w hich th e com plex and real sta b ility radii are equal.

2 Preliminary.

C onsider Ĩ 111* equation

where* A (•_ A and IỈ arc» constant m atrices in / \ " ix m ,(/v = c or I \ — /?.), (let A = 0: till* pencil o f m atrices {.4, B Ị is supposed to he regular, index {/1, D ) = k > 1 It is known that there ex ists a pair o f nonsingular m atrices \ \ \ T such th at

where / s is th e unit m atrix in K sXs. Further B \ € I \ VX1\ u is a A:- nil potent m atrix having I.hr Jordan box form i.e u = d ia g Ụ iy J ‘ 2, •//) w ith

/ 0 ° \

\ 0 0 0 / Midi that !n a X |< /< /p, = A: (see [5]) M ultiplyin g both sides o f (2.1) by w ~ l wc obtain

whore T l X( t ) = i *Y( t ) € l \ ’\ Z ( t ) € K m~r Since u is a k — liilpotent matrix.

it is easy to see th a t th e equation U Z '( t ) — Z ( t ) = 0 has only a unique solution z = 0

T hus, th e ab ove sy stem is reduced to

: "• N g u yen H u u Du, Dao Thi L ie n

Y ' { t ) - D \ Y ( t ) = 0,

where Y { t ) € K ' \ Z ( t ) e K m~ r

T he trivial solution X = 0 o f (2.1) is said to be a sy m p to tica lly sta b le if there are

a certain projection p € C ( K m ) and p o sitiv e constan ts a , r such that th e solution o f the initial value problem

AX'(t) - BX(t) = 0,

P ( X ( 0) - X o ) = 0

is unique and the estim a te ||X ( /) || < c ||n X o ||i,_cvi, t > 0 holds In fact, if th e index of

{A 13} = 1 wc choose p — I — Q w here Q is the projection OI1h e r A along 5 = {z Ç c :

D z £ /m 4}.

We d enote by a ( C , D ) th e spectrum o f the pencil {C\ D }, i.e th e set o f all solutions

of tlic equation (lot(AC - D ) = 0 In case c = / we write sim p ly rt-(D) for ơ ( Ị D ) It

is known that system (2.1) is a sy m p to tica lly stable iff all finite eigenvalues of the pencil {.4 /? } lie w ithin the half left hand side o f com plex plan (see[5]) If ơ (A , B ) = 0 then (2.1) lias only a unique solution X ( f ) = 0 Indeed, ơ (A , B ) = 0 im p lies th at for any

s. d('t(.s/l - B ) = dot w d e t ( s I(Ị fc — z?i)đet(ố*ơ - I ) d e t T ~ l = nonzero constant Thus

(I A* = 0 i.e., th e equation (2.4) m ust be absent Hence (2.1) is equivalent to (2.5) which lnus only a trivial solution X ( t ) = 0 In th is case we also consider (2 1 ) is asym ptotically stable by choosing p — 0.

3 S t r u c t u r e d d is t u r b a n c e s

As is clone ill O D E 's case, one fixes a pencil of m atrices {A , B ) to be stable; a pair

o f m atrices E € F 6 K q*m and consider the distu rb ed sy stem

when* A £ T h e m atrix E A F is called structured d isturb ance D en o te by

V k = {A € K pxq : (3.1) is either irregular or u n stab le } i.e V/v is th e set o f ubacT distu rbance L et d/t = in f{||A || : A € V/v}- W e call rf/v the structured sta b ility radius o f th e quadruple { A , B, E , F } If K = c , we have com plex stability radius and if K = R we have real sta b ility radius.

First, we investigate th e com plex sta b ility radius o f (2 1 ), i.e., K = c Sim ilar as

in O D Es put G ( s ) = F ( s A — B ) 1E and we shall prove th at

d c = [ s u p | | G ( s ) | | ] ~ l

a€C*»

.We point out d c > [sup5eo l|G (5 ) |ll~ l ‘ Taking A € V o, there are tw o cases: a) T h e pencil o f m atrices {A , B + E A F } is regular T h en , we take a value s G

ơ (A , B -f E A F ) (it notes th a t a (A , B -f E A F ) j=- 0 since A £ V c ) Su ppose th at x / 0

S ta b ility radii fo r d ifferen tia l algebraic equations

is its corresp onding eigenvector, that, is s A x — ( B + E A F ) x = 0 or equivalently, X =

( S'/4 - B ) 1 E A F 1 \ which follow s that

F t = F ị s A B ) l E A F x = G ( * ) A F x

Ị | A | Ị > | K ; ( s ) r ' > ( sup ||ơ (.s-)|ir '

f o r a ll A • v< • w h ic h im p lie s t h a t (!(' > Ịs u p sẽ<:> l|G ( 's )ll] *•

h) Tilt* p m cil of m atrices {A B ■+• E A F } is irregular, then for any s r it exists

a vector .V =£ 0 such that sAr - ( B + E A F ) x = 0 B v using a sim ilar procedure we can

prove d e > i.su|>M:.r j|G(ü)i|]

W e now prove th e inverse relation d c < I I G M - ' F o r a n y £ > 0 we find

.S(] G C " such that Ị|G (so)|| 1 < [supNícC;+ ||Cr(s)Ị|]“ l + £ Suppose th a t u € C 7 such that

|w|| = 1 and ||G(#o)tt|| = ||G(*o)|| A corollary of Haln-Banach theorem follows that there

is a lineal' function y* defined on c p such that Ị|y*|ị = 1 and y*G($o)u = |(c?(s{))//II =

|ỊƠ(*o)|| Put A = ||G (s o )||~ l wy* € C V*'I. It is clear that

AG(S[))a = ||G(*0)|| ]uy*G(sị ))u= ||C?(.So)Il ^J-IIG^o)!! = u.

Hence ||A || > ||G (S ())||~ 1 On the other hand, from A = ||G(.S())||_1 uy* we have? ||A || <

||6'(«o)|Ị 1 • Therefore IIAll = ||G(.Ç())|| V Further, since AG(s{))it = a. we obtain

£A G («o)m = E v t 0 Let X :== (.SoA - then (.So/4 - z?).r = E u which fol­

lows E A F r = (.So/4 - J3)t or (*{)A — B — E £ F ) x = Ü i.e .So £ ơ { A A Ỉ -f* E A F ) T his

m eans that th e sy stem

A X ’{ t ) - ( B + E ă F ) X ( t ) = Q

is unstable Therefore A 6 vv* Further,

dc- < ||A || = |i ơ (« o )~ l < [ sup \\G (s )\\)-1

s£C+

Because £ is arbitrary then d c < [supt, €^+ Ị | G » r ' T hus,

dc = [ sup |ỊG (s)|ị]_ l

«6 c+

We n o te th a t th e function G ( s ) is analytic on the half plan then by maxim um

principle, it only a tta in s m axim um a t 6* = oc or on i R Thus

dr = (sup||G(5)||]-‘

«€ IỈỈ

Follow ing th e above argum ent, we bee that if there ex ists So 6 c y such that

l | G f o ) l l = [ s u p ’ t r - ! I G ( * ) I I ] t h e n

d r = l|G (.S(,)ir1 = [ m a x ||G ( s ) ||] - 1.

N guyen H u u Du, Dao T hi L ien

Moreover, if tho m atrix A is given I)V

ă = \\F ( s 0A - B ) - ' E \ \ V

then A is '‘had'' matrix with 11 All =

dc-However, t he above argum ent d oes not allow us to com p u te a “bad” m atrix A w hose norm oquals to <i(- as ill O D E s case oven we take th e lim it as s -» oc Wo now show that

if CV(.s) i does not attain its m axim um over half plan e C" then there is no m atrix A such that (!(' — A and the system A X ' - ( B 4- E A F ) X = 0 is unstable Su ppose, ill contrary, there is such a m atrix A Let So € ơ ( A , B + E A F ) n C 4 and X is its eigenvector, i.e s()Ar - ( B 4- E A F ) x - 0 which im plies th a t ||AỊ| > ||G (so)ir~1 > + IK*(«o)||] 1 rr

dc- T his is contradiction.

M oreover, for any sequence (s ,i) in c * w hich m axim izes ||G(«sk)|Ị at oc and associated to s„ is constructed as above (w e can supp ose th at there ex ists iim,,-»-* A ,, = All if not wo take a subsequence), then th e sy stem ( A X ' — (13 -f E A ( ) F ) X — 0 is stable Since the set o f m atrices A such th a t th e pencil o f m atrices { A B + E i \ F ) has the index

1 is open then the index o f { A , B + E A ( ) F } must, bigger then 1.

We consider a special case where E = F = / (unstructured disturb ances) A s is

s m i till* stab ility radius w ith unstructu red distu rb an ces is

where G ( s ) — (sA B ) l We prove th a t if in d (i4 ,/? ) = k > 1 then th e m atrix function G(.s) is unbounded on i.R. Indeed.

as s —> “X Therefore, ill th is case, d c — 0 T h is m eans t hat under a very sm all disturbance, the DAEs w ith the index greater then 2 is no longer stable.

If ind(>4, D ) — 1 it is easy to prove th a t |ịơ (.s)|| is bounded oil i.e., d(7 > 0 but perhaps it (loos not exist any "bad” m atrix A such th at !|A|| = d c.

Slim m ing up we obtain

T h e o r e m 3 1 n) The com plex sta bility ra dius o f System (2.1) is given by

where G ( s ) = F(.s/1

-bj T h ere exists a “b a d ” m a trix A such that ||A || = d ç i f and o nly if G ( s ) attains its m axim um over ỈR.

= [su p ||G (s)||] 1,

sÇiK

d o = [su p ||G («)||] \

A' £ i f?

c) hi th e a ISC E = F — I, d c > 0 i f a n d o n ly i f i n ả ( A s D) = 1

A question risrs Ihtc: w henever t he function ||G («)|| a tta in s its m axim um at a finite value *(, \Y< firstly remark t hat th e answer depends strongly on th e chosen norm o f C ' n

sinct* G ( s ) \ has m axim um 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 E uclid norm in th e 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 th e way to obtain t he decom p osition (2.2) First we decom pose (A — B ) ~ l A into Iordan form by a nonsingular m atrix 5 , that is (A - B ) l /l = S d i(u j( M , V ) S " X. where

r is a nilpotent m atrix o f the form (2 3 ) and M is nonsingular T h e m atrix w and T in (2.2) ih given by

i r (.4 D ) S d i a g ( M J ) : T = S d i.a g { I,( V - u = V ( V - I ) 1. (3.2)

If G ( s ) is unbounded oil c \ then A c = 0 and there is no th in g to say The assum ption G’(.s) to he bounded im p lies that F T d ia q ( Q U J ) \ V ~ [ E = 0 Vj > 1 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 / ( * ) = :: ơ ( i / s ) ị | ,J i f 6* Ỷ u a n d / ( 0 ) = 1ÌI1Ì.V-+OC | | G ( l / s ) | | 2 ( w e r e m a r k t h a t t h is

limit always exists) It is easy to see that f ( s ) = ||FTdm//(s(/ — si?i) 1, —I ) W 1 E\\~.

Since all entries o f the' m atrix G ( s ) are only rational functions which are analytic then by th e m axim um principle, th e m axim um o f G ( s ) takes place on ly at s = oc or

s G iR T herefore, G (s ) a tta in s its m axim um at s = 00 iff /(.<?) has the m axim um value

at s = 0 (o f course we consider on ly s in c t+) T hus, taking a ray Í —> t • e, t > 0 where

= (c o s n sin a ) < a < “ , th e attainm ent of m axim um value a t 0 o f /(.s ) implies that //( () ) < 0 for every c It is ea sy to see that

/ / ( 0 ) = 2 cos a [FTrftasf(0, - / ) W r * £ ] * [ í T d t a p í A O ) ^ " 1^ ] = 2 c o s a C * D , where c = [F7Vi/a<7(Q, — I ) W ~ l E ] : D = [F T ,d ia ịf(/, 0 ) i y “ ! £ ] and C' * D denotes the Frobenius inner product o f tw o m atrices C , D

In using the expressions o f w and T ill (3.2) we obtain

c : = F T d ia g [0 , - I ) \ v l E

= F T d ia y iO , ư ( ư - / ) ■ ' - / ) w ' £ = F T d ia g { 0 , V - I ) W XE

= F S d ia g ( J ( V - I ) ~ l ) ■ diag(Q, V - I ) d i a g { M ~ l , I ) S ~ l { A - B ) ~ l E

= F S d ia g (0, 7 ) 5 “ *(i4 - B y 1 E

and

D : = F T d i a g ( 1 ,0 ) V I 1 £ = F T W ~ l E + C

= F S d ia ợ( I , ( V - / ) - 1 )d ia $ (M ~ l J ) S ~ l [A - B ) ~l E + c

= F[Sdì ag( M, V - I ) S ~ l ] - l {A - B ) ~ l E + C

= F[(.4 - B ) l A - Sdi ag(0, /) 5 - l )-1(i4 - JB)-1JE + C

= F U - (/1 - ổ ) S í / m ổ ( 0 , / ) S - 1] " 1£ ’ + C'

2 2 Nqxtyen H u u D u, D ao Thi L ien

Su m m in g up, we have: if c * D > 0 th en G ( s ) has m axim um a t a finite value s. In the case c * D = 0 we can com p u te higher derivatives o f / to ob tain th e answ er but the formula is com p licated and we do not realize here.

E x a m p l e 1 Let us calculate sta b ility radius o f th e structured perturbed equation

A X ' ( t ) ( B 4- E A F ) X ( t ) = 0 w here A is distu rb ance and

E =

- ầ - 1

tot.icallv Stable By a direct com p u tation we obtain

It is St»011 t hat hid ( A B ) == 2 and <r(,4 fl)

G ( s ) = F ( s A - B ) ~ l E =

3s-f 1 3.s-f 1 3s4 1

3*4-1 .s 3.s-f 1$ 3s +1 Al

3*-f 1 3.9+1 3 ü ' f 1

T hus \\G{s)\\ — 3 m a x {| |, ị ị Ị} w hich a tta in s its m axim um at So = 0 and ||C (0 )|| =■

3 Hence d(' = 1 /3 C hoose u = ^ 1 ^ th en ||G (0 )u || = (7(0) = 3 L et y* = (0 1 0) we

/ 0 1 /3 0 \ have A = ijC7(0)II 1 uy* = I 0 1 /3 0 I M oreover, đ et(.s/l - B - JSA F) = 2.S = 0 for

\() 1/3 0 /

s = Ü.

E x a m p le 2 Let us consider th e equation /LY'(£) - J 3 X (i) = 0 w here A = ^ 2*

and B = ^ *2 ()**)’ ^ seen ~ Ơ{ A , B ) — - 1 and G (s) — (.S/1 - Ữ) 1 = f ,s/ ^Sỵ2 ^ 1/4) ' H ence- \\G(S)\\ = m a x { 3 / 4 , l / 2 + | s / ( s + 1)1} which

d o esn ’t attain its m axim um on c 4 Further, lim.'-too ||G (s)|| = 3 /2 i.e A c = 2 /3 If we choose Ü = ^ y , it is clear th a t ||u|| = 1 and ||G (s)tz|| = |ị<ST(.s)ỊI when

is large T hus, w ith y* = ( 1 0 ) we have

A = ||G (s)|Ị“ Siy* = converges to ^2 ^ 3 o ) 30 6 00 ■

It is easy to verify d e t ịs A - D - A ) = - 8 / 3 for all s, i.e., ơ ( A, - ( / ? + A )) = 0 and

th e equation A X f(t) — ( B + A ) X ( t ) = 0 i.e th e sy stem

—XI 2j* > — ~X ] -f* 2 x 2 = 0

3

4

2 x\ — 4 4 + - X j = Ü

«J

sta b ility radii fo r d ifferen tia l algebraic equations

has a unique solution /'I - 0: /*•> = 0 which is asym p totically stable.

4 T h r e q u a li t y o f real a n d c o m p l e x s t a b i l i t y r a d iis o f D A E s

In I his section , wo are concerned with a special case where* the com plex sta b ility radius is equal to real stability radius For DAEs this is a difficult qu estion because under

th e action o f the pencil of m atrices ị A D ) the positive cone ÌĨ+ is no longer invariant even both A and B arc positive We arc able to solve problem under a very strict hyp othesis Suppose th at A 11 c IV "* '".

A m atrix I I ~ (n, j ) c f í " íXn‘ is said to 1)0 p ositive if (\,ị > 0 lor any i j D en ote

[ho nbsolutr o f tin- m a t r i x M = by \M = (IM ill) a n d o f th e vector /• by I./•; — { /*!» ./ J Ị.r„, ị) Wo define a partial order relation ill ftmxnt by

A/ < N & M - N < 0

L e t / / ( / 1 , I Ỉ ) b e t h e a b s c i s s a s p e c t r u m o f t h e p e n c i l { - 4 / ? } i e f ỉ ( A B ) : = 111H X { ‘}ỈÀ : A €

a ( A J i ) }

We consider the* equation

where w4 /Ỉ are constant m atrices in /?,,ixm the pencil {.4 £?} is regular ĩf’ iii(H A B ) > 1 then thero is nothing to say b ecause d c = (Ifi = 0 So we su p p ose that in d (A , B ) = 1 and the following conditions are satisfied:

ii) T here ex ists a sequence (/.„): t„ > 0; tu oc such th a t ( t „ A - D ) 1 > 0 for all

iii) T h e equation (4.1) is a sy m p to tica lly stable.

\ \ v remark th at the above con d ition s ensure a positive system oil O D E s case Let us choose the m onotonous norm in R ,n. T h at is if \x\ < ịt/l th en I'./’ll |Ị/y||.

L e m m a 4 1 Let the system (4 1) satisfies above conditions, then for till X such that

'RA > fi(A , D ), we have I (A.4 - B ) XI < (5iAi4 - 2?) Ịarị for any X € /?"*.

Proof. Let us take an t, tn € /? such th a t / > /í(A , £?).and - 1 > iu {A y D). Su ppose that A = /.-H u, we have to prove th a t !((/: -f i u ) A — B ) 1 rị < (tA - B ) 1 \x\ for all X € R ni

By sim ple calcu lation we have

((t + iaj)A B ) 1 = ( t „ A - B ) - l [ I - { t n - t - k j ) A ( t „ A - B ) - i \ i

P uttin g G ( t „ ) — (t „ A B ) 1 we ob tain

Ị(í + iw )A - B ) } - 1 = G ( t n) [ I - (tn - t - iü j) A G ( tu) } J

X

< ? ( * „ ) £ ( « „ - i - « j ) n(A G (i„ ))" (4.3)

7 1 = 0

T he above series ab solu tely converges if we can prove th a t ||(£„ - t - y’u;)r(i46t(f/,))|| < 1 where r ( M ) d enotes tlu* sp ectru m radius o f M

21 N g u y en H uu D u , Dao Till L ie n

First, it is easy to see th a t lim ttÈ-4oo(tn - \tn - t =■ t. Therefore, for e =

i fi{A B ) > 0 we have in - I ỉn - t - iuĩ\ > t - e = ụ (A B ) for tn sufficiently large, i.e

t„ ụ (A B ) > \tu - i — On th e other hand, by h ypotheses i) and ii), A G ( t „ ) i‘s p ositive matrix, tIll'll I)V Perron-Frobenius theorem : r ( A G ( t n )) = n ( A G ( t fl)) € ơ ( A G i1 fl)). T his means that c le tjr (/4 G (/„ ))/ — A G ( t f1)) = 0 H ence.

(let[ r ( A G ( t n ) ) I - A G ( t n )} = 0 « d e l[(tnA - B ) - A / r ( A G { t n ))\ = 0

« det[(in - i / r ( A G ( t n ) ) ) A — J5] = 0.

Thus t„ - (-(/t(.4(-?-yy 6 a ( A B ) Therefore, r i À G ù - t" ~ M-4 ’ # ) - w hich im plies |/„ / - 7u;|r(i4G (in )) < 1 Hence, by (4.3)

|( (i + L i ) A - B ) - ' xI < G ( t n) J 2 \tn - t - úư)|” (> 4 ơ (í„ ))M|3:|

nssO

= G ( t „ ) \ I - |*„ - t - i< j\(A G (t „ )} ] \x\ = \(tn - |t„ - t - M M - B ] - ' |x |.

Let t„ ~> oc we obtain

|[(/ + iu>)A - Ơ]' ‘x| < ( t A - B ) 1 |;r|.

Lemma 4.1 is proved Ộ

L e m m a 4 2 G (t ) = (M - B y 1 > 0 for any t > fi(A D ) M oreo ver, G (/) is decreasing

on (fi(A B ), oc).

Proof. Let t() > fi(A , D). By using Lem m a 4.2 we see th a t |G (*o)| 5: G(ỈR/u) = G (tị))

t hen G ịtị)) > 0.

T he decreasing uf G (t ) on J3), oc) follows from th e first part o f th e Irmma and the fact that for s t > ụ ( A % D ) one has G ( s ) — G (t ) = (t — s)G (.s)i4G (£) Ộ

B ecause the function G ( s ) is an alytic oil h alf plane c + then it on ly a tta in s maxim um

at s — oc or s £ iR Furtherm ore, m on otonou s norm is chosen then by Lem m a 4.1 G(.s)

a ttain s the m axim um on [O.oo) On th e other hand, by L em m a 4.2, it follows th at G (s )

hap its m axim um at t = 0 i.e ||G (0)|| = rnax{||G (A )|| : ?RA > 0} = IIB 1II.

From Perron - Frobenius theorem , there e x ists u > 0, \\u\\ = 1 such th a t ||G’(0)//|| =

;jCV(0) By using once more H aln-B anach theorem for positive sy stem th ere exists positive linear functional y* satisfying y * (G (0 )n ) = ||G (0 )u || ^ 11^(0)11 all(ỉ \\y*\\ = 1- Let A ỊỊGT(O)II { utj* > 0 Following th e way as above we can prove th a t A is “bad” m atrix and

|ỊA|| = d r Therefore,

T h e o r e m 4 3 Suppose that the system (4.1) satisfies hypotheses i); ii) and Hi) and a

monotonous norm in R m is chosen, then the com plex sta bility radius d c and the real stability radius (in are equal and (In = (IIB - 1 )!!- 1

As is m entioned above, assu m ing the p o sitiv ity o f G ( t n) for a sequence (tn ) is strong and it is difficult to verify We know give a sufficient cond ition to ensure the

S tability radii fo r d ifferen tia l algebraic equations

*»Ih>\v l iv p o t 1|(‘S<\S It is s a id t h a t th e s y s te m ( 2 1 ) is p o s it iv e i f f o r a n y :/•() € IV " = { y = ( / / I / / J //,!,) : //; '• n w — 1 2 ?/ / } t h e s o lu t io n X f w i t h X () = i f i t e x is ts , s a tis fie s

the condition V/ > 0 for all / > 0 Lot Q he a projection on K e r A th en it is known that (2.1) is equivalent to

w lin r ộ Q (,4 ÜQ) 1 /?; p = 1 Q and D = P (i4 - We note th at Q (loos not depend on thí» choice o f th e projection Q and P ( A - B Q ) ~ l = P (A — Ỉ.ÌQ) Ỵ which implies tlirif 13 is independent o f the choice o f Q. So it is seen th at (2.1) is positive if and only if p > 0 and D is a P - M etlez m atrix, i.e all entries o f B are positive except for rut ries h,j w ith p,j > 0 w here p = (p ij). Indeed, from (4.4) th e general solution o f (2.1)

is Xf — exp(/ữ )P À '<| Thus, the p o sitiv itv condition im plies th at p > 0 (w ith t = 0) On tilt* other hand, for / is sm all wo h a w

/ft-I

which follows that if P ' J = 0 then bij > 0 Conversely, if B is a P — M etlez m atrix then ill noting that p is a projection which com m ute w ith B then for an a such th at o P + D > 0

wo lia VO

e x p( t B ) P = e x p( - c t t P + t ( B + a P ) ) P = e x p( - a t P ) e x p( t ( B + ( \ P ) ) P

= e x p (“ Or£)exp(f(B + a P ) ) P > 0.

w v now supp ose that the system (2.1) is positive In add ing con d itions that P ( A -

B Q ) 1 > 0 and Q { A B Q ) 1 > 0 we can prove that G (t) = ( tA - B ) ~ { > 0 for any t > 0 and / large* To verify this a ttesta tio n we have only to remark that

(/1 B Q ) ‘ (//1 - B ) = t P - (yl - B Q ) ~l B = t P + Q - B

= ( P + £ / / ) [ ' / - ( P - M Q )i? ] = ( P + Q / t ) ( t I B )

Thus.

ơ ( / ) = [(,1 £ ( - } ) ( M - / ; ) ] ■ ' ( / ! - B Q ) - ‘ = [ ( P + Q / f ) ( / / - Ỗ ) j ‘ ( / 1 B Ộ ) 1

= ( / / Ồ) 1P ( i 4 - B Q ) - I + í ( í / - Ỗ ) “ 1ộ ( i 4 - / ĩ ộ ) 1

= (/ - B Y 1 P ( A - B Q ) - ' + Q ( A - B Q ) - 1. (4.5)

Since B is a p - M etlez then there is a f() such th at (t — B ) 1 p > 0 for any t > t()

T hus, under the hypotheses P (i4 — B Q ) ~ l > 0 and Q ( A - B Q ) ~ l > 0 the relation (4.5) tells that G (f ) > 0 for t > t{).

However, it is easy to give an exam ple w here th e system is p ositive but the resolvent

(tA - B ) 1 is not positive So far we do not know if th e p o sitiv e con d ition ensures the equality o f (!(' and (//,» An answer o f this problem is welcom e.