Connective stability of competitive equilibrium

Automanca, Vol 11, pp 389--400 Pergamon Press, 1975 Printed m Great Britain

Connective Stability of Competitive Equilibrium** Stabilit6 Connecuve d'Equilibre Concurrentiel

Konnektive Stabilitht vergleichsf'~ihiger Gleichgewichte D. D. gILJAK~

Necessary and sufficient condmons for asymptonc connectwe stabzhty of competitive eqmhbrmm can be apphed to dynamw models m economtcs, ecology, pharmacokmetics, transistor cwcu~ts and arms races Summary--The purpose of th~s paper ts to derive necessary and sufficient condmons for connecuve stability of nonhnear mamx systems described by the equatmn

matrix A = (d.) are replaced by nonhnear trinedependent functions a . - - a . ( t , x) In the followmg development, we will place certain constraints on the elements a.~(t,x) of the n x n matrix A = (a,j) so that various stablhty properlaes of the nonhnear system (1) can be implied by stability of the linear system (2) and expressed m terms of the elements d 0 of the n x n constant matrix A -- (~j). Stability analysis of nonlinear m a t n x systems was m m a t e d m Automanca by Rosenbrock [4] and was subsequently developed by a number of authors [5-16] The obtained results can be used to study stabihty aspects of mathematical models arising m as diverse fields as economics [14, 17] and biology [18-201, arms races [21], transistor circuits [10] and pharmacokinetlcs [II1 In these various dlsophnes, nonlinear matrix systems are appropriate models for a competmve-cooperatlve interaction a m o n g n agents, these being commodities or services on a market, interrelated species m a given ecosystem, countries involved in an arms race, etc In the present work, we will introduce the connecttve concept of stability [1-31 to study structural properties of competluve eqmlibnum m nonhnear matrix systems We will show that rehablhty of stabzhty m such systems is high and that they remain stable despite ume-varymg, including ' o n - o f f ' , mteractmn among mdwldual agents present in the system. In mulUple market models [14], for example, such structural variations can be used to study the effects on stabihty of variations m taste of consumers, changes m technology, government spending, etc. In model ecosystems [18], the structural v a n a u o n s prowde a proper framework for conslderauon of how complexity of the mteraclaons among species affect community stabdlty [19, 20] As for the arms race problems [211, 'expense-and-fatigue' and 'interrelation' coefficients can be considered as umevarying functions of tame, thus allowing changes m

x = A(t, x) x, where the matrix A(t,x) has ume-varymg nonlmear elements The obtained results can be used to study stability of compctmv¢ eqmhbrmm m as d~vvrse fields as economics and engineering, model ecosystems and arms races I INTRODUCTION IN THIS work, we will consider stability under structural perturbations [1-3] of free dynamic systems described by the differential equation§ ffi A ( t , x ) x ,

(1)

where x ( t ) e ~ '~ is the state of the system and the n x n matrix funclaon A" ~ x ~ ' t - ~ . ~ 'is is defined, bounded and continuous on ~r-x ~'~ so that the solutions x(t; to,Xo) of (1) exist for all initial conditions (to, x0) ~ " x ~ , t and t ~ 0 0 The symbol ~ " represents the Ume interval (% + Qo), where ~is a number or the symbol - 0 % and ~00 is the semi-mfimte time interval [to, + oo) The 'nonlinear matrix system' (1) is an obvious extensmn of a linear time-mvanant system = dx,

(2)

m winch the constant elements d . of the n x n * Recexved 25 February 1974, rewsed 29 July 1974, rewsed 30 January 1975 The original versmn of tlus paper was not presented at any IFAC meeting It was recommended for pubhcatmn m revised form by assoemte editor. I Landau 1"The research reported hereto was supported by the NASA Grant NGR 05-017-010 D. D ~iljak Is with the Department of Electrical Engineering and Computer Science. Umvers~ty of Santa Clara, Santa Clara, Calfforma 95053 § W~th some obwous exceptwns, lower case ~tahc letters denote vectors, capital letters denote matrices and Greek letters denote scalars

389

390

D D

attitude of various countries due to national and mternaUonal government measures to be taken into consideration Extrapolating the stabihty results obtained m the study [14] of multiple markets to these other fields of apphcatmn, we conclude that connecuvity is, m general, an inherent property of stabdity of compeUuve eqmhbnum m models of dynamic systems--stability of such models is always connective Th~s ~s the reason why we will be able to estabhsh not only sufficient condmons for connecUve stabihty as m [1-3], but rather the necessary and sufficwnt condmons for connectwe stab~hty of nonhnear matrix systems Another important d~stinctlon of th~s work ~s that it considers asymptotw stabthty of nonlinear matrix systems By rule, all of the prewous authors [4--16], e~ther exphc~tly or lmphcRly, obtained the condmons for exponentml stabd~ty of the system (1). By using Rosenbrock's type [4] of Lmpunov function in the framework of general companson functmns as suggested in [2, 3], we will w~den the stabd~ty conditions obtained by Pers~dsku [8] and Sandberg [10] to imply uniform asymptotic stabfltty in the large of a broader class of nonlinear m a m x systems Since most of the nonhnear matrix ecosystems [18] are not globally stable, we will provide estimates of stabdity regwns Furthermore, we wall consider forced nonhnear matrix systems and derive conditmns for ultimate boundedness under structural perturbatmns Such cons~deraUons are of particular interest in studies of price-quantity adjustments in multiple markets [22] w~th shifts in demand schedules, as well as the effect of enwronmental changes on stabihty of muluspecaes communmes [18] FlnaUy, we will outhne a more detaded appllcauon of our results to these two different fields and motivate further exploitation of the proposed approach for purposes of obtaining new ~mportant results It is of interest to note that the condmons for connectwe asymptotic stabd~ty of competmve eqmhbnum are expressed in terms of the classical Hmks condmons The conditions were proposed by Hicks when he initiated the stablhty analys~s of competitive markets m his book [45]

2 NONLINEAR SYSTEMS To consider the connectwe aspect of stabd~ty, we write the elements a . of the matrix A In (1) as

a~t(t, x) = - 8. ~b~(t,x) + e~j(t) ~b.(t, x),

(3)

where 3 . is the Kronecker symbol, and ~b,(t,x), ~b.(t,x)ECt°,°)(Y'x~ '~) In (3), e . = e . ( t ) are elements of the n × n mterconnecnon matJ tx

~ILJAK E = (e.), which are e~j(t) ~ C°(if -) and are restricted as e . ( t ) e [ 0 , 1], V t e 3 - [ 1 ] In case of system (1), the element e.(t) reflects the coupling between :~(t) and x~(t) at each instant m time, that is, the t~me-dependent influence of the state x~(t) on the derivative of the state x,(t) Therefore, the raterconnection matrix E(t) represents the structural perturbatmns of the nonlinear matrix system (1) In th~s section, we study asymptotic stability properties of system (1) under structural perturbatmns More precisely, we mvesugate stabd~ty formulated as

Defimtwn 1 The equdlbnum state x = 0 of the system (1) is connectively asymptoUcally stable m the large ~f and only if it is asymptoUcally stable m the large for all mterconnectmn matrices E(t) Before we turn to denvatmn of the condmons for the kind of stabdlty expressed by Defimuon 1, we need the notion of the fundamental mterconnectwn matrix ~ [I] The matrix E is a time-mvanant mterconnectlon matrix m which the elements 6. take on binary values, 1 ff the j th state xj influences the tth time denvaUve -~ of the state x,, and 0 ~f xj has no influence on x~ The matrix E IS a binary matrix [23] which reflects the basle structure of the system Therefore, any mterconnecUon matrix E(t) ~s generated from g by replacing the unit elements of E by corresponding elements e.(t) of E(t) The condmons for connective stabdlty are expressed m terms of E, but are valid for all E as reqmred by Definmon I This is an ~mportant qualltaUve result since we show stabdlty of a class of nonhnear time-varying systems by proving stabdlty of one member of that class whlch is a t~me-lnvarmnt linear system To estabhsh condmons for asymptoUc connective stability, we assume that the elements a.(t, x) of the matrix A(t, x) are specified by (3) where ff~(t, x), ff~j(t, x) are bounded functmns o n e ' - x M n, and that there exist numbers oq > 0, ~./> 0 such that

q,,(t,x)lx~l>>.~,,~,(lx~[), q,.(t,x)x,<. %~,(lx,[), Vt,j=l,2,

,n,

V ( t , x ) ~ . f f ' x ~ n (4)

and c(,> o~,, In (4), ~ ~+-->~._ are comparison functions defined as ib,(p)~ C°(..~+), 16~(0) = 0; and ~(Pl) < ~(P2), V Pl, P2 0 ~ P1 < P2 < + oo [24] By A = (d.) we denote the n x n constant mamx with the coefficients d.

=

- 8~j ~ + ~. % ,

(5)

where the elements ~j take the values 1 or 0 accordlng to the matrix E

Connective stabihty of competmve eqmhbrmm We prove the following

Theorem 1 The equlhbrmm state x = 0 of the system (1) IS connectwely asymptoUcally stable m the large if the n x n constant Metzler matrix `4 = (d.) corresponding to (4). satisfies the Hicks condmons*

~1:>0 '

I)L dxl a12 (-

ti~t

d2~

"

Vk = 1,2,

~lkl

dk~

dkk I

,n (6)

Proof Let us consider the function v ~ ÷ , v(x)

= X

d~lx~l,

(7)

391

From (5), . 4 = ( a . ) is a Metzler matrix [22], that is, it has non-negatwe off-diagonal elements d . 0 # J ) t For a Metzler matrix `4, the Sevastyanov-Kotelyansku conditions (6) [25] are equivalent to saying that the matrix .4 is a Hicks matrix [17], that is, all even-order pnncipal minors of`4 are positive and all odd-order pnnclpal minors of.,[ are negatwe.:~ In the following developments, we will usually refer to condmons (6) as the Hicks conditions, or as the Hickslan property of the matnx `4, since most of the results [26] for Metzler matrices are related to the sign condmons on all pnnclpal minors As shown m [3],§ the HJcksmn property of `4 is eqmvalent to saying that for any constant vector c > 0 there exists a constant vector d > 0 such that CT = -- d T `4

as a candidate for Lmpunov's function [4] for system (1) where d,>0, t = 1,2, ,n, are yet unspecified numbers For v(x), we have the lnequalmes

del([lxH)~ Kx)<<. d~rx(llxl[ ), V ( t , x ) e Y ' x , ~ " ,

Therefore, we can rewrite mequahty (12) as D +

v(x) <~- c w w(x) n

-<-cm Z ,(Ix, l)

(8)

where ~i and ~ri are compartson functmns gwen as

(llxll)

H(llxll) =" d,,llxll (9)

= d, llxll,

and d m = ram, d~, d M = max, d,

ffI(llxll)

+

Note that in (9),

as Ilxll-,+o

. since the denvatwe of I x~(t)l need not exist at a point where x,(t) = O, ~t is necessary to calculate the right-hand derivative D+Ix~(t)[w~th respect to equatmn (1) as proposed in [4] and [I0] For this purpose, the functmnal a~ ~s defined as

o~-

1,

If x , > 0 , or if x, = 0 and ~ , > 0 ,

0,

if x, - 0 and 2, = 0,

-1,

(10)

l f x , < 0 , or lfx~ = 0 and ~ , < 0 ,

where x, = x~(t) e C~(~ ") Then,

D+ix,(t) l = odc (t)

(11)

Using the constraints (4), and the expressmn (11), we calculate the desired derivative as

(15) where c m mm,G and m(llxll) s a comparison function From (8), (15) and [24, Theorem 425], we conclude global asymptotlc stabdlty of x = 0 m (1) To show that stablhty ~s also connectwe, we need only to notice that =

A(t,x)x<<.`4w(x),

= Xd,

~--1

which looks similar to the second condmon (4) save for the reversal of the inequality s~gn If condmons (4) are slmphfied to

e t ~O~j(t,X)Xj(t)

~--1

V(t,x)e3"x,~",

(12)

where d = ( d l , d2, ,dO T is a posture constant vector (d>0), and the posmve vector function w: ~ ~ ~+'~ Is defined as

=

(16)

¢,(lx, I),

ft

<~dT`4w(x),

V ( t , x ) ~ # " × ~ '~,

where inequahty (16) IS taken component-wise Therefore, (15) holds for all E(t) This proves Theorem 1 It is a well-known fact that under the condmons of Theorem 1, stablhty o f x - 0 is also uniform [24]. It should be noted that constraints (4) imply ~kt(t,x)>0, V ( t , x ) ~ 3 " x ~ ' k Posmvlty of ~b~(t,x) is absolutely essential for stabdlty of (1) since it is easy to show that Hicks condltmns (6) Imply d ~ < 0 , V , = 1,2, ,n.l[ With this in mind, we can rewrite the first condmon in (4) as

D + v(x) = Z d, o, :ft(t) ft

(14)

xd), ,4,.(Ix.I)F.

* See Definmon A3 and Theorem A2 of Appendix

03)

,/,,(t, Vt,j= 1,2,

,n, V(t,x)~5"x~",

(17)

t F o r the defimtlons and propertms of Metzler m a t n ~ s , see Appcndvt ,+ See Theorem A2 of Appendlx § See Theorem A6 of Appendlx [[ See Corollary A l of Appendlx

392

D D ~LJAK

where comparison funcuons are chosen as [x~ [, then we can estabhsh exponential property of stablhty as shown m [10] and [14] Furthermore, if we introduce the notion of absolute stab~hty for the nonhnear matrix systems as proposed by Pers~dsku [8], we can prove that Hmks mequahtles (6) become both necessary and sufficient condmons for stab~hty. On the barn of constraints (17) we define the following classes of continuous functions tr/., = {¢t(t, x) ~b,(t,x) t> ~}, ~,~,,

]

{Co(t,x). I¢O(t, x)[ ~ o%},

(18)

where ~), % are numbers as m (4). Then, we state

From (20) and (21), we get the dlfferenUal mequahty

D+v~-zrv,

VteJoo,

Vve~+,

v[x(t)] <~v(x0) exp [ - zr(t- to)], Vte,~'0,

V ( t 0 , x o ) e o q ' x ~ n,

for all (to,x o ) e ~ ' x : ~ ~, all ¢~eW~, ¢ , e W , , and all mterconnecUon matrices E(t). To estabhsh this kind of stabthty, we can use the following

(23)

Ilxll-
IIx(t, to, xo)II n IIXoIIexp [ V ¢. e~.,

H Ilxollexp [ - zr(t- to)], V te~-'oo (19)

VE

Using the well-known relationship between the Euclidean and absolute-value norms

Definmon 2.

Ilx(t, t0,x0)ll

(22)

By integrating (22), we obtain

rr(t - to)],

VtE~7"0, \/(to, x o ) e J ' x ~ '~, The eqmhbnum state x = 0 of the system (1) is connectively, absolutely and exponentially stable if and only if there exist two posmve numbers rl and zr independent of mmal condmons (to, xo) such that

VE

V¢,etF,,

V E,

(24)

II = nt d~ d~-X,

(25)

wRh where ds}t = max, 4 , dm = nun~ d~. Therefore, the Hicks conditions (6) are sufliclent for absolute exponentaal property of connective stabdlty of the eqmhbrmm x = 0 m (1) This estabhshes the hf' part of Theorem 2. To prove the 'only if' part of Theorem 2, we select the particular system (I) specified by

¢,(t,x)=al,

¢.(5x)=%,

V t , J = 1,2,

,n, (26)

Theorem 2 The eqmhbnum state x = 0 of the system (1) is connecUvely, absolutely and exponentmlly stable if and only if the n xn constant Metzler matrix ,4--(do), corresponding to (18), satisfies the condmons (6).

Proof. Let us consider again the functton v(x) m (7). When (4) is reduced to (17), the vector

w(x) m (13) becomes [Ix, l, lxo.I, ,Ix.IF, and from (12) and (16) we get

-

d,I

,lla.l+

Ix, I

41¢,1,

V (t, x) eft-" x N n

(20)

Since ,4 xs a Metzler matrix, the fact that it satisfies the condmons (6) and is a Hicks matrix is eqmvalent [26-28] to saying that there exists a posmve vector d =(dl, d~, ,dn) T and a posmve number zr such that

la.l-d:'Xd,

Vs=l,2,

,n,

(21)

t=].

that is, ,4 is a quasldommant dmgonal matrix [27] * * See Theorem A4 of Appendix

and the fundamental mterconnectmn matrix E That is, the matrix A(t,x) m (1) is taken as the constant Metzler mamx `4, and system (1) is described by (2) A constant Metzler matrix ,4 is a HurwRz matrix, it has all elgenvalues wxth negatwe real parts, ff and only if it is a Hxcks matrix, that is, ff and only If it sausfies condmons (6) 1 Therefore, xf ,4 does not saUsfy (6), the eqmhbnum x = 0 of (1) is not stable V~bteW~, V ¢o e ~F~ This completes the proof of Theorem 2 It is of interest to note that the eqmhbrmm x = 0 of the system (1) is connectwely, absolutely and exponentially stable even when the mamx functmn A(t, x) is replaced by KA(t, x), where K = &ag {kll, k~z,

, k,~}

with posmve &agonal elements, that is, k,, > 0 for all l = 1,2, ,n This is a consequence of Theorem 2 and the fact [28] that if a Metzler constant matrix ,4 = (do) with a negatxve &agonal is a quasldommant diagonal matrix so is the matnx K~ = (k,d,j), which is obvious from (21) It should be noted, however, that the degree 7r of exponentml stabdlty calculated by (21) for K = I, where I~s the n x n ~dentRy mamx. may be d~fferent for any K # I t See Theorems A1 and A2 of Appendix

Connectxve stability of competmve equdlbnum

393

3 LINEAR SYSTEMS

aLj(t) in A(t), we can broaden the class of systems

Let us consider a hnear ume-varymg system described by the dlfferentml equatmn

(27) to include negative off-dmgonal elements in A(t) at the expense of losing the nccesmy part of the Hicks condmons We again consider A(t) with a.(t) defined m (28) where oq and c% are all numbers such that ~ > % i> 0, that is,

x = A(t) x

(27)

which is a s~mphficatmn of system (1) In equatmn (27), the n×n matrix A ( t ) = [atj(t)] has timevarying elements a**(t)~ C°(~ ") defined by

aij(t ) = - 3. ~ + e.(t ) % ,

where ~,, % are numbers In case of system (27), the following definmon of stabd~ty ~s appropriate

Definmon 3 The eqmhbnum state x - 0 of the system (27) is connectively exponentially stable m the large ff and only ff there erast two posluve numbers H and ~r independent of the mmal condmons (t0,x0) such that the lnequahty (19) ~s satisfied for all (to, x o ) ~ T x ~ n and all lnterconnectmn matrices

E(t). Definmon 3 follows from Defimtmn 2 by omitting the 'absolute' aspect of stabdlty which is approprmte to do m case of hnear systems. We consider first a matrix A(t) m (27) defined by (28) where a~ > 0, a~j/>0, a~ > oq~ so that

a~(t) I <0, I>0,

i •j, ~j,

Vt~3".

a,~(t) < O, V t e3"

(30)

(28)

(29)

Therefore, no s~gn pattern Is prescribed to the matrix A(t) except that it is a negatwe dmgonal matnx. To state condmons for connectave stab~hty of the system (27), we denote by A = (d.) the matnx A(t) wluch corresponds to the fundamental raterconnection matrix J~, and form the n x n matrix B = (b.) as b . = "~a'w

[14, 1,

l = J,

(31)

, +J

The matrix B ~s McKenzle's 'dmgonal form' of the matrix .4 [27] We notice that the matnx B has negative dmgonal elements and non-negaUve offdiagonal elements regardless of what is the s~gn pattern of the elements m the matrix A--the matrix B ~s a Metzler matnx. From Theorem 2, we can derive the following

Theorem 4

That is, A(t) ~s a nine-varying Metzler matrix [14]. By .4--(dO) we denote again the constant Metzler matrix which ~s obtained from A(t) when the elements e~(t) are chosen according to Now, we prove the following

The equlhbnum state x = 0 of the system (27) is connectively exponentmUy stable m the large ff the n x n constant Metzler matrix B = (bo) correspondlng to (28), satisfies the Hicks condmons bll

bz2

blk

Theorem 3

( _ 1 ) 4 bu

b=

bsk. >0,

The eqmhbnum state x - - 0 of the system (27) ~s connectively exponentmlly stable m the large ff and only ff the n x n constant Metzler matrix .4 = (d,~), corresponding to (28), satisfies the H~cks conditions (6)

Proof. The 'if' part of Theorem 3 follows from the hf' part of Theorem 2. To prove the 'only if' part, we need only notace that m Definmon 3 we reqtured stabihty for all E, and thus for J~ Since the Hicks condmons (6) are necessary and sutiic~ent [17] for stab~hty of the constant Metzler matrix .4 which corresponds to ~, failure of A to be a H~eks matrix ~mphes that ,4 ~s not a Hurw~tz matrix, i e the eigenvalues of ,4 are not all with negative real parts, and the eqmhbnum x -- 0~s unstable for E ( t ) - ~ . Thls, m turn, ~mphes that x = 0 is not connectavely stable and the proof of Theorem 3 ~s complete. Since mequahty (20) ~s msensmve to any particular sign pattern m off-diagonal elements 28

bkl bk2

Vk=l,2,.

bkk

,n. (32)

Proof We use again the fact that the Hicks mequahtles (32) are eqmvalent to the quasidominant diagonal property of the constant Metzler matnx B According to (31), the matnx .4 is quasidominant diagonal if and only if B Is, and Theorem 4 follows from the 'if' part of Theorem 2. We can allow negauve signs of the off
Cx'(t)l

[C~a(t)

c,n(t)J'

(33)

394

D D ~ILJAK

where the square submatnces Cn(t) >IO, C2~ >I0 are non-negative and the submatnces Clz(t)<<.O, C~a(t) <<.0 are nonpositive for all t ~ r - . Let us consider the system (27) when

A(t) = C ( t ) - D,

(34)

where C(t)= [e~j(t)oqj] is an n x n time-varying Morishima matnx and D = diag{~,1, o~, , a,~} For A(t) in (34), we assume (30) and form A and B as for Theorem 4. Then, we prove the following

for some numbers c%~>0, ~ , > ~ , , and on some region ~V"c ~,~ defined by

~=

{ x e ~ n [ x , l < / z . g t = 1,2,

,n}

(36)

and numbers t*~> 0 We consider again the function v(x) m (7), and assume that the matnx A with elements d,j defined m (5) satisfies the Hicks conditions (6). From (7) and (15), we have that v(x) and D + v(x) with respect to equation (I) satisfy the inequalities

d ,Ixl< (x)< d Ixl, Theorem 5 The eqmhbnum state x = 0 of the system (27) is connectively exponentially stable in the large if and only if the n x n constant Metzler matrix B = (bo) corresponding to (34), satisfies the Hicks condmons (32).

Proof. The 'if' part of Theorem 5 follows from Theorem 2. In order to show the 'only if' part of Theorem 5, we notice again that the I-hcksian property (32) of the Metzler matrix B Is eqmvalent to saying that it is quasidommant diagonal. Obwously, B is quasidolmnant diagonal if and only if .4 is From [30], we have that a negative diagonal matnx .4 is Hurwitz if and only if it is quasldommant diagonal matrix. Therefore, violation of Hicks conditions (32) implies that .4 is not a Hurwitz matrix and x = 0 is not stable for E = E The proof of Theorem 5 is complete. 4 STABILITY REGIONS If the constraints (4) on the elements a . of the system matrix A hold only in a finite region of the state space, then we have only local stabihty of the eqmhbnum state x = 0 of the nonlinear matrix system (1) A problem of interest in this case is to estimate the regmn of stability using the proposed analysis. For tlus purpose, we can use the Rosenbrock's type of Laapunov function as outlined by Welssenberger [31 ] We start wath the following

Defimtion 4 A regina , A ' c ~ , ~ is a region of connective asymptotac stabihty assocaated with the equihbnum state x = 0e.A¢ of the system (1) if and only if for all interconnectionmatrices E(t) corresponding to E, x -- 0 is stable in the sense of Lxapunov, and for te~o, llmt~+~x(t; t0,xo) -- 0 whenever Xoe..Ct' Let us assume that the nonlinear functions ~b,,~b. in (3) satisfy the conditions

~,,(t,x)lx, l>>.~,V,,(Ix, I), ~bo(t,x)xj<.%,#?#(lx,[), v l , j = 1,2,

,n,

V(t,x)~arx.A ~

(35)

D+v(x)<~ - cm ]~ ~b,(lx, l), V (t, x) e J" x.A/',

V E(t),

(37)

where ¢~'s are comparison functions in (35) A region ..go of connectwe asymptotm stabd~ty ~s given by the following

Theorem 6 The region .A¢ defined by .A' = { x ~

n v(x) < mm d~/~}

(38)

is a region of connective asymptotic stability corresponding to x = 0 and J~ of the system (l).

Proof From (37) we have that v(x) > 0,

D+v(x)
~/E(t)

(39)

Since , ~ ' c j V ' , from (39) we have directly that x0e..~' lmphes both the Llapunov stablhty of x = 0 and the fact that llmt_,+~ x(t, to, x o)= 0 for all mterconnecuon matrices E(t) This proves Theorem 6 The region ../t' provided by Theorem 6 m (38) is the largest region of connective stability that is available with the Llapunov function v(x) m (7) and the constramts specified m (35) Larger stabihty regions could be obtained by other types of Llapunov functmns as &scussed m [31 ] Furthermore, by using the result of Theorem 2, one can determine the regions of connective exponential stability, which have the property that any solution starting m the region converges to the equilibrium state faster than an exponential

5 FORCED SYSTEMS Let us consider the nonlinear matrix system

x = A(t, x) x + b(t, x),

(40)

where A(t,x) is defined as in equation (1), and b ,Y'x~n-~ n is a forcing function whmh has components of the form

b~(t, x) = l~(t) ~z(t, x)

(41)

Connectwe stablhty of competmve eqmhbrmm In (41), vector

l,(t)

are components of the mterconnectmn

lit) = [ll(t), 12(0, , l,,(t)] T,

395

To include the connectwe property of boundedness, we need the norton of the fundamental mterconnectmn vector I The constant vector

i= (il, i~, ,i.?

such that 1~ ~-'--> [0, 1],

ls a binary vector the component l, of which 1s 1 xf the forcing function b(t, x) influences x,, that ls, b~(t,x)~O, and it ls 0 lfb,(t,x)-O. Now, we are m a positron to prove the following

and the funcuon ?,(t, x) ~ C (°'°) (~" x o~") satisfy the condmons

I~,(t,x)]<<.3,,

Vt = 1,2,

Theorem 7

,n,

V (t, x) e~'- x . ~ ,

(42)

where fl~'s are non-negative numbers We recall [24, Defimtmn 36 1 l] that the solutmns x(t, to, xo) of (40) are said to be uniformly bounded ff it ~s possible to find an estimate

Ilx(t, to, xo)ll
(43)

for all (t 0, x o ) ~ " × ~t', where the scalar functmn x(to, p) ~s posture and bounded for p>O, and ~ ¢ = ~ n xs an open regmn If ~t' is ~ n , we say that the solutmns x(t; to, :Co)are uniformly bounded m the large The connecUve property of boundedness ~s introduced by the followmg

The soluUons x(t, to, Xo) of the system (40) are connectwely umformly bounded m the large if the n x n constant Metzler matrix A = (d,s) correspondlng to (4), satisfies the Hicks condmons (6)

Proof We again use the funcuon v(x) defined m (7) and compute D + v(x) using constraints (4), (42), the notauon b, = l, fl,, and the fact that A = (d.) satisfies condmons (6), that is, condmons (21), D+v(x)= ~__ld,a,~la.x~+b, ) =

aj%xj+~

a uo~

+

¢cr,

<~- ~_l(d,, q- d,-l ~id, d,,) d, e~;q-,~ld, '.

Defimtwn 5 The solutions x(t, to, :Co) of the system (40) are connectwely umformly bounded m the large if and only ff they are umformly bounded m the large for all mterconnectmn matrices E(t) and all mterconnectmn vectors i(t) To derive condmons for the property of system (40) specified by Definmon 5, we recall [24, Theorem 46 1] that the solutmns x(t, to, Xo) of (40) are uniformly bounded ff there exists a functmn v(x) e C°(Xpc) such that

,/,~(IIx II) -< v(x) .< ~,~~(IIx II), D + v(x) < 0,

V (t, x) e ~ " x £ae,

(44)

where .L~ae ~s the complement of the regmn .L~ = { x e ~ '~ 114 ~<~:}, and ~: ~s a posture number. Using (44) and the propemes of the comparison functmns

,~-~(,')-< ~,~-~[¢H(llxll)], eH-~(,') ~>¢H-'[¢~(11 #11)], where ex-~(p) and q~zr-X(p) are the reverse functmns of ¢I(9) and ez](P), we have from (44),

CH-~(o
v(t,x)eer×.z¢

(45)

When x o ~.~e, v[x(t; to, x0)] decreases as t increases, at least as long as the solutmn x(t; to, Xo) is m . ~ From (46), we have

Ilx(t;/o, x0)ll<¢~-~U~(x0)], vt~o,

(46)

for all (to, X0)~'x~f~ae whmh ~s an eslamate of type (43)

<-¢m(llxll)+n, V(t,x)eg'x.~', where em(ll x lt) and ,~ are defined as

(47)

n

¢m(llxll)=,,r, dj¢,
Defimtwn 6 The solutaons x(t; to, xo) of the system (40) are connectively exponentmlly and ultimately bounded m the large with respect to the region

.~ -- {x~-:

Ilxll~< ¢}

396

D D

if and only if there exist three posmve numbers 3' < ~:, 17 and rr independent of the Initial condmons (to, xo), such that

Ilx(t; to, Xo)l[< ~,+ 1-IIIxoll exp [ - 7r(t- to)],

vt

o,

(49)

for all (to, :co)~ " x .o~°~, all mterconnection matnces E(t) and all mterconnection vectors fit). We can show again that the Hickslan property of the matrix 3 guarantees the kind of boundedhess described In Defimtion 6, That is, we prove the following

~ILJAK of (53) converge to each other as the time goes on desplte the structural perturbations descnbed by the matnx E(t) We prove the following

Theorem 9 For any two solutions xl(t ) and x2(t ) of the system (53), there exist two positive numbers I-I and ~r independent of initial condmons (to, Xx0), (to, x~o) such that

Ilxx(t)- x (t) ll n IlXl0-X,ollexp [-

r(t- to)],

V t ~.~00, (54)

Theorem 8 The solutions x(t, to, x o) of are connectively exponentially bounded in the large if the n x n matrix 3 = (do), corresponding condmons (6).

the system (40) and ultimately constant Metzler to (17), satisfies

for all (to, Xxo), (t0,Xzo)~o~-×~n and all raterconnection matrices E(t), if the n × n constant Metzler matnx 3 = ( ~ ) corresponding to (28), satisfies the Hicks conditions (6).

Proof We Introduce the new variable

Proof We use again the funcUon v(x) defined m (7), and calculate D+v(x) with respect to the system (40) as

y(t)

= xl(/) - xg.(t),

and use again the function v(y) defined in (7) compute D + v(y) using (53) and (55),

D+ v(x) = ~td, °i(~la,,xj + b, )

~=I

<<.-=~(x)+,7,

a,a',s a, lx, l+

a,b,

v t~, v (to, :co)~ J " × ~ " ,

(50)

where ~r Is defined in (21) and ~7is specafied m (48) By integrating the last mequahty (50) we get

,Ix(t)]

'7 + [,,(Xo)-

exp [-

Applying the mequalmes Ilxll
17 = nt dM dr~-L

(51)

(50, (52)

Tlus proves Theorem 8 The ultimate boundedness regmn ~ of Definmon 6 can be estimated by choosing ~ = d m - t rr-~ ~7+ e, where e > 0 ~s an arbitrarily small number The ume tx necessary to reach the region .g' can be estamated from (49) and (52) as tt = to + ~-11n (17 IIxoll In case of the hnear system *

= A(t) x + b(t),

(53)

where the elements a,~(t) of the matrix A(t) are defined by (28), and the function b ~'-->~'* is contanuous for all t ~ ' , we can strengthen our results and show that any two solutions xt(t ) = xx(t, to, Xxo), xz(t) = xu(t, t o, X~o)

(56)

where Y0 = Xxo-X2o- From (56), we get inequality (54) with rr and 17 defined m (21) and (25), respectively. Since inequality (56) Is vahd for all E(t), we have connective convergence in (54) This proves Theorem 9

6 APPLICATIONS

- to)],

V t ~.~0.

We

3ffil

<~- roy(y), V (t, y) ~$7" × ~ , ,

a ,+a-x

(55)

In the present section, we will show how the obtained stability results can be Interpreted in the context of multiple market models in economics [17], and multispecles community models in biology [18] Economics

In economic studies, the linear constant system

, = ix

(2)

is widely used to model n interrelated markets of n commodmes (or services) which are supphed from the same or related sources and which are demanded by the same or related industries In (2), x ( t ) ~ n Is the price vector and 3 = (d,) is an n x n constant matrix. When all commodmes are gross substitutes, A is a Metzler matrix, that is, the coetficlents d** are specified as m (29), and the market (2) is stable if and only if A is a Hicks matnx defined by con&Uons (6) This is a classical result established in 1945 by Metzler [33]. Going a step further, Metzler showed that Hicks conditions in this case

ConnecUve stabdlty of competmve equlhbnum imply stablhty on all subsets of markets and, thus, the markets are totally stable m the sense of Lange [34] In the spirit of Lange's total stabihty, we showed by Llapunov's direct method that under certain constraints, the nonlinear t~me-varylng models

(1)

x = A(t,x)x

can be stable despite the time-dependent structural perturbatmns whereby the market is decomposed into subsets of markets dunng the adjustment of prices--this is connective stabdlty established by Theorems 1 and 2 Such structural perturbations include the cases when the kth commodity disappears from the market e,k = elo~ = O, l,j = I, 2 , .

, s,

or when the slopes of excess demand represented by coefficients a , of A change their values due to changes m tastes of consumers, as well as changes m technology, etc. Furthermore, since m Theorems 1 and 2 no sign pattern is assumed in the offdmgonal elements a,3 (5 # j ) of A, the model (1) may represent a 'nmxed' market of substatutes and complements The conditions of Theorems 1 and 2 allow the changing of a commodity from a substatute to a complement for another commodity over the tame interval. It is important to note that this result obtained for a nonhnear market model does not rest on Walras' law [17] (or any other assumptaon concerning market excess demand) for validity In the case of the linear tame-varying model Y: = A ( t ) x ,

(27)

when the commodmes follow the pattern 'substatutes of substatutes are subsUtutes, complements of complements are substitutes, and complements of substitutes and substitutes of complements are complements', we defined the tame-varying Monshima case following [29]. Theorem 5 provides necessary and sufficient conditmns for stabihty of a tame-dependent Monstuma market under structural perturbatmns Theorems 1-5 have their mstablhty counterparts [15] When a market has an inferior commodlty, Glffen's paradox [17], then the instability version of the obtained results estabhshes condmons under which the market is unstable for all possible structural vanataons Of particular interest is the forced matrix model considered in Section 5. The forcing functaon represents the shifts an demand schedules on some or all markets, wtuch were studied by Arrow in [22] He showed that under the assumptaons that the demand is shifting upward in time and that the supply curve may do the same but never more rapidly than the demand, the prices rise with a rate of increase that approaches a limiting value The

397

positive linear tame function was chosen to represent trends in otherwise stable, linear and constant market models In Sectaon 5, we considered stufts in demand and supply functions wtuch had no specified sign or form save that they are bounded We showed that m a stable market under bounded stuffs, the role of the eqtuhbnum is played by a compact regmn, and prices on all markets are globally ultimately bounded with respect to that region That is, all prices reach the regmn m finite time and once m the regmn, they stay there for all future times Ttus property of the price-adjustment process was estabhshed for a nonlinear tame-varying model studied in the preceding seehon We also provided an upper estimate of the mentioned regmn by the same Llapunov function used to determine the global stabihty properties of the model. The estimate is chrectly proportmnal to the size of the shiftsin the excess demand functions Furthermore, the esUmate of the region ~s mvanant to structural perturbations, and the adjustment process is again exponentaal--pnces on all the markets reach the regaon faster than an exponential despite structural changes m the model. In summary, the results obtained in ttus work show a wide tolerance of stable competitive market systems to a large class of gross nonhneantaes and ume-varymg structural perturbataons in the interacuons among markets. This important quahtatavc property is estabhshed by the classical Hicks condmons so far exclusively apphed to stability analysls of competmve equilibrium m hnear constant models of multiple markets Ecosystems

In p o p u l a t w n ecology, one of the central problems is the relataonship between complexity and stability of multispecms communities [18]. The questaon is. Does a h~gher degree of trophm web complexity lead to a higher degree of commumty stablhty, or is It the other way around and the complex c o m m u n m e s are less stable than the simple ones It was shown [19] that the problem "complexity vs stability" can be appropriately formulated m the context of nonlinear matrix systems. We start with the linear constant model (2) where x ( t ) ~ '~ represents the population vector and A = (~,~) Is the n x n constant community matrix We assume that all species are density dependent which is represented by negatave dmgonal elements of A (d~<0, i = l , 2 , ,n). Such an assumption reflects the resource hmitataon m the commumty [35]. No assumptions are made on the signs of the off-diagonal elements o f / [ , thus allowing for 'mixed competltive-predator-symbmtm-saprophmc interactions among specms In the context of connective stability, we can study how much of lnteractmns among species can

398

D D S~LJAZ

be tolerated by stable communmes From Theorem 2, we conclude that the ecosystem model (2) is stable ff the commumty matrix .4 is quasldomlnant diagonal, that is, there exist an n-vector d = (al, as,

, d.)r,

wtth components all posmve such that 7~

d, la.l>

dda.l, j-- 1,2,

,n

(57)

Condltton (57), which Is eqmvalent to Hicks condiuons (6) when .4 is a Metzler matrix, includes the usual dmgonal dominance n

la.l>

J = 1,2,

,n

as a specml case when d - - { l , I, , l} Although condition (41) is easler to interpret m the context of system (2), it Is more restncUve than that of (57) From condiUons (57), It Is posslble to draw conclusmns about the tolerance of stabd~ty to increasing complexaty of the system So long as the magmtude of interactions does not violate mequahues (57), stahihty of the community is preserved. This conclusmn is compaUble wlth the empmcal evldence noUced by Margalef [36] 'ht seems that sp¢cles that interact feebly w~th other species do so wxth a great number of other species. Conversely, species w~th strong mteracUons are often part of a system w~th a small number of species", and it does not contradict the statement of Levms [37] that "there is often a hm~t to the complexity of systems" If condmons (57) ts satisfied, then the hnear t~mevarying model (27) is stable for all lnterconnectmn matnces E ( t ) as shown by Theorem 4. This means that, m parttcular, a removal (or destruction) of a number of species from the commumty would not affect stabdity estabhshed by dmgonal dominance of the commumty matrix .4 Such structural changes that alter the number of species are analyucaUy considered by MacArthur [38] assuming a specml symmetric form of the commumty matrix A Not only that we d~d not require this speoal symmetric form of A, but we also d~d wave the assumptmn of constancy of the system (1), thus allowing for t~me-varymg structural perturbations which take place at disequthbrlum populatmns and g~ve rise to a Ume-dependent matrix A f t ) Dmgonal dominance (57) as only sufficient for stabdity of 'm~xed' communmes Condmon (57), however, ~s both necessary and sufficient for stabdity of (27) ff the mteracUons obey the rules "friends of friends are friends, enemies of enemies are friends, and friends of enemies and enemies of friends are enemies" (Theorem 5) Th~s result can be used to verify the Gardner-Ashby [39] and May [18] computer experiments

As shown by Theorems 1 and 2, stable communmes can have wide tolerance to nonhnearmes m the mteractmns among species. The nonhneanty constraints (18) include saturatmn of predator attack capacity, nonhneanty m the food supply, nonhnearmes m the precompetmon, etc. The restnctmn that each species ~s densitydependent can be removed by considering species m the stable 'blocks', the subsystems, and using the decomposmon-aggregatmn methods [1-3] The methods can take advantage of specml structural 'block' properties of ecosystems, and conclude stabdlty of the commumty by dmgonal dominance of the aggregate commumty matrix [20] Forced matnx systems considered m Section 5 can be used as models where the effect of environmental changes on commumty stabd~ty are of interest. Multispecies communmes m randomly varying environments were considered m [40] using this general approach and [41]

7 CONCLUSIONS Relying on stabthty results obtamed m studies of linear tlme-mvanant models of competauve markets [17], the classical Hicks conditions and dmgonal dominance were shown to be suflioent, and somet~mes necessary, condmons for connecUve asymptouc stabdlty of nonhnear matrix systems The results obtained m this work can be used to study nonhnear and time-varying effects m pharmacokmeUcs [l l] and arms race problems [21], multispecies commumty models as constituted in [18-20], as well as muluple markets w~th substatutecomplement commodltaes or services [14, 17] Furthermore, multdevel feedback control stabdlzatlon [42, 43] and opumlzatmn [44] schemes were developed for synthesmng competmve structures of large-scale dynamic systems Acknowledgement--The author is grateful to Professor

G S Ladde, Department of Mathematics, New York State Umverslty at Potsdam, for his valuable comment on th~s work REFERENCES [1] D D ~UJA~ Stabthty of large-scale systems under structural perturbauons 1EEE Trans SMC-2, 657663 (1972) [2] D D ~ItaAK On stabdlty of large-scale systems under structural perturbations IEEE Trans SMC-3, 415417 (1973) [3] LJ T GRuJI¢ and D D ~[UAK Asymptotic stabthty and mstabdlty of large-scale systems IEEE Trans AC-18, 636-645 (1973) [4] H H ROSENnROCK A Lyapunov functmn for some naturally occurring hnear homogeneous Umedependent equations Automatwa 1, 97-109 (1963) [5] H H ROSENSROCK A method of investigating stabdlty Proc 2nd lnt Federatwn of Automatw Control Congress, pp 590-594 Basle, Swltzerland (1963)

Connecuve stabthty of compeutwe eqmhbnum [6] H H ROSENBROCK Some condmons for the stabflzty of nonlinear time-dependent dffferenual equations J S I A M Control 2, 171-180 (1965) [7] S K I~RSrDSKn InvestlgaUon of stabdity of soluuons of some nonlinear systems of dlfferentml equauons Pr:kl Mat Mekh 32, 1122-1125 (1968) (In Russmn ) [8] S K PERSIDSmI Problems of absolute stabdlty Avtom Telemekh 29, 5-11 (1969) (In Russian ) [9] S K PERStDSrJI Investigating the stabdlty of solutions of systems of d~fferentlal equations Prtkl Mat Mekh 34, 219-226 (1970) (In Russian ) [10] I W SANDnERG Some theorems on the dynamic response of nonhnear trans:stor networks Bell Syst Tech J 48, 35-54 (1969) [l I] R BELLMAN Topics m pharmacokmeucs, I Concentration-dependent rates Math Bwscl 6, 13-17 (1970) [12] J C G~rlNA and P BORNE Sur une condmon d'apphcatlon du cr~tere de stabiht6 hndaire a certames classes de systemes contlnus non hneaires Compt Rend Acad Sc~ Parts, 275, 401-403 (1972) [13] LJ T GRuJ~t~ On pracUcal stabd~ty Int J Control 17, 881-887 (1973) [14]D D , ~ A K Competmve economic systems stability, decomposition, and aggregauon Proc 1973 Conf on Decision and Control, pp 265-275 San D~egu (1973) [15] D D ~LJAK On connective stability and instablhty of competmve eqmhbnum. Proc 1974 JACC Conference, pp 223-228 Umvers~ty of Texas, Austin, Texas (1974) [16] E J DAVmON The decentrahzed stabdizaUon and control of a class of unknown non-hnear t~me-varymg systems Automatwa 10, 309-316 (1974) [17] J QUIRK and R SAI'OSNIK Introduction to General Eqmhbrmm Theory and Welfare Economics McGraw. Hill, New York (1968) [18] R M MAX', Stablhty and Complexity in Model Ecosystems Monographs m Populauon Biology, No 6 Princeton Umverslty Press, New Jersey (1973) [19] D D ~tJAlC Connective stability of complex ecosystems Nature 249, 280 (1974) [20] D D g~taAK When is a complex ecosystem stable ~ Math Bwsct to be pubhshed [21] L F R~CHARDSON Arms and Insecurity Boxwood Press, Pittsburgh (1960) [22] K J ARROW Pncc-quahty adjustments in multiple markets w~th nsmg demands Proc Symp on Mathematical Methods in the Social Sciences, pp 3-15 Stanford Umverstty Press, Stanford Calfforma (1966) [23] F HARRARY, R NORMAN and D CARTWmOm" Structural Models An Introduction to the Theory of Directed Graphs Wiley, New York (1965) [24] W HAHN Stabthty o f Motion Sprmger-Verlag, New York (1967) [25] F R GANTMACHER The Theory o f Matrices, Vol II Chelsea, New York (1960) [26] M F~EDL~R and V PT~,K On matrices with nonposmve off-dmgonal elements and posmve principal minors Czech Math J 12, 382--400 (1962) [27] L McKENZ~ Matrices w~th dominant dmgonals and economic theory Proc Symp on Mathematical Methods in the Social Sciences, pp 47-62 Stanford Umvers~ty Press, Stanford, Caltforma (1966) [28] P K NEWMAN Some notes on stability condmons Rev Econ Studtes, I-9 (1959) [29] M MO~aSHIMA On the laws of change of the pricesystem in an economy which contains complementary commodmes Osaka Econ Papers 1, 101-113 (1952) [30] L BASSETT,H HABIBAGAHIand J QUIRK Quahtauve econormcs and Morishlma matrxces Econometrica 35, 221-233 (1967) [31] S WE~SSENBERGER Stabd~ty regions of large-scale systems Automatwa 9, 653-663 (1973) [32] J LASALLEand S LEFSCHETZ Stab:hty by Liapunov's Dtrect Method with Apphcatwns Academic Press, New York (1964)

399

[33] L A METZLER Stabihty of muluple markets the Hicks con&tions Econometr:ca 13, 277-292 (1945) [34] O LANGE The stability of economic equilibrium Prwe Flexibd~ty and Employment, pp 91-109 Cowles Commission Monograph, No 8, Bloomington, Ind (1945) (Also in Readings in Mathematical Economics (P NEWMAN,ed ), Part I, pp 178-196 Johns Hoplons Press, Baltimore (1968) [35] J T TANNER EffeCtS of population denslty on growth rates of animal populauons Ecology 47, 733-745 (1966) [36] R MARGALEF Perspectwes in Ecology Theory Chicago Umverslty Press, Chlcagu (1968) [37] R LEVINS Ecological engineering theory and practice Quart Rev Bwl 43, 301-305 (1968) [38] R H MACARTHUR SpecJes packing and competmve eqmhbnum for many species Theor. Popl Bw 1 1-11 (1970) [39] M R GAmaNERand W R AsHEY Connectedness of large dynarmcal (Cybernetic) systems crlucal values for stability Nature 228, 784 (1970) [40] G S LAX)BEaod D D ~ILJAK Stabdlty of muluspecies communities m randomly varying environment J Math Bzology, to be pubhshed [41] G S LABDE and D D ~tLJAK Connective stabihty of large-scale stochasuc systems Int J System Set, to be pubhshed [42] D D ~ A K and M B VUK~Evld On hierarchic stablhzatlon of large-scale hnear systems. Proc Eighth Asdomar Conf Circuits, Systems, Computers, Paofic Grove, Calff, December (1974) 503-507 [43] D D ~UJAK Stabilization of Large-scale Systems A Spmmng Flexible Spacecraft To be presented at the 6th IFAC Congr, Boston, Massachusetts, August (1975) [44] D D ~ILJAKand M K SUNDARESHAN On hierarchic optimal control of large-scale systems Proc Eighth Asdomar Cot~f Circuits, Systems, Computers, Pacific Grove, Cahf, December (1974) 495-502 [45] J R. HicKs Value and Capital Oxford Umversity Press, Oxford (1939) APPENDIX Certain properties o f M e t z l e r matrices T h e Metzler m a t n x was first i n t r o d u c e d m e c o n o m i c s b y J L M o s a k m 1944, as r e p o r t e d b y N e w m a n [28], b u t was n a m e d after Metzler [33] since he gave ~ts essentml d e v e l o p m e n t m 1945 T h e e c o n o m i s t s d o n o t agree on the defimtton o f the M e t z l e r m a m x a n d we have t w o & s t r e e t d e f i n m o n s D e f i m t w n A 1 (Newman [28] ) A c o n s t a n t n × n m a t r i x A - - ( a ~ : ) is called a Metzler m a t r i x if a n d o n l y if its d i a g o n a l elements a . are all negaUve a n d its off-diagonal elements a . (: # j ) are all nonnegaUve. Definition A2 (Arrow [22] ) A c o n s t a n t n x n m a t r i x A - - ( a , j ) is called a Metzler m a t r i x i f a n d only if its off-diagonal elements a~: (t ~ j ) are all nonnegaUve. O b v i o u s l y DefimUon A 2 i m p h e s D e f i n m o n A I , b u t n o t vice versa Since a necessary conchtaon for s t a b d l t y o f a M e t z l e r m a t r i x Is negativity o f the & a g o n a l elements a . , the two defimUons are equivalent wlth respect to stablhty This fact Is e s t a b h s h e d next

400

D D. gm~AK

In 1939, Hicks introduced m lus book [45] the following: Defimtton A3

A constant n ×n matrix A = (a~j) is called a Hicks matrix if and only if all even-order pnnclpal nunors of A are posRtve and all odd-order pnncapal minors of A are negaUve We have the well-known [28] Theorem A 1

A Metzler matrix A is a Hurwitz matrix (all elgenvalues of A have negatave real parts) ff and only ff A is a Hicks matrix From Theorem A1 and Definmon A3, we get jmmediately

Theorem A3

If A is a Metzler matrLx (Defimtmn AI), then A is a Hurw~tz matnx if and only if it is a quas~dominant diagonal matrix. From Theorem A 1 and A3, we have Theorem A4

A Metzler matnx A (Definition A1) is a quasidominant dmgonal matrtx if and only if ~t is a Hicks matrix or, equivalently, it satisfies the Sevastyanov-Kotelyanskal condiUons (6). There ~s still another property of Metzler matnces used m Theorem 1 of the paper, which we get from the well-known [22, 26] Theorem A5

If A is a Metzler matrix (DcfimUon A2), then - A - 1 Is nonnegattve element by element ( - A - 1 > / 0 )

Corollary A1

A Metzler matrix A ~s a HurwRz matrix only if its diagonal elements a . are all negative It is surprising to find out that the following Kotelyanskn extension of a Sevastyanov result reported by Gantmacher [25] Is not acknowledged m econonuc literature: Theorem A2

A Metzler matrix A is a Hurwltz matrix ff and only if it satisfies the Sevastyanov-Kotelyanskal con&uons (6). Therefore, to test stabthty of a Metzler matnx A, one needs to examine signs only of the leading pnncapal rmnors of A In this paper, we also used the following [28] Defimtion A4

A constant n ×n matnx A = (ao) is called a quas~dommant (maan) dmgonal matrix ff and only ff there exist posmve numbers d, (~ = 1,2, ,n) such that either n

41a,,l>:Sd, la.,I, vs= 1,2, ,n,

(A 1)

or

d, la.l>:S41a,,I, lml

V,= 1,2,

,.

The following result is the well-known [28]

(A.2)

If and only if A Is a Hicks matnx or, eqmvalently, ~t Is a Hurwaz matnx The property of Metzler mamces used in Theorem 1 Is given now by the following [3] Theorem A6

If A is a Metzler matrix (DefimUon A2), then for any constant vector c > 0 there erasts a constant vector d > 0 (posmvlty taken element by element) such that cT = - d T A (A.3) if and only If A is a Hicks matrix From (A 3), we get d r = - cx A-X.

(A 4)

Since - A -I ~s nonnegatlve and A ~s a H~cks matrix, every column of - A -1 has at least one posmve element guaranteeing posltivlty of d Conversely, ff A is not a Hicks m a t r t x , - A - 1 is not nonnegatlve, and there are vectors c > 0 for wbach a vector d > 0 cannot be found to satisfy (.4, 3) A = I where / is the n x n ~denttty matrix, confirms the statement Simdar statements gaven above for Metzler matrices are obtained for Monsluma matrices [17, 30] The present paper extends definitions of Metzler and Monshlma matrices to matnces with Ume-varymg elements and estabhshes stabthty results that parallel those gwen for the constant matrices [17]