Asymptotic Behavior of Nonlinear Compartmental Systems - CiteSeerX

372

IEEE TRANSACTIONS

ON CIRCWTS

AND

SYSTEMS,

VOL.

CAS-25, NO. 6,

JIJNR

1978

Asymptotic Behavior of Nonlinear Compartmental Systems: Nonoscillation and Stability HAJIME MAEDA, MEMBER, IEEE, SHINZO KODAMA, MEMBER, IEEE, ANDWZo

Abstrurci-This paper di.9cwm propertics related to the stabiity of a class of nonliuear compartmental systems. SpecificaUy, mathematical conditions which guarantee Qe same qualitative behavior inherent in linear compartmental systenw are considered. We first consider the nonoscillatory property of solutions and show that the system bas no periodic oscillation under a mild condition. ‘Ibe result is then used to derive a necessaryand sufficient condition for every solution to convergeto a set of equilibrium points which may depend both on the input and the initial stati. A sufficient condition is also given for an equifiirium state to depend only on the input. The asymptotic behavior of the free system9 is also considered, and a sufficient condition is given for the origin to be globally asymptoticaNy stable. Furthermore, for a closed compartmental system it is shown that for each given initial state, unique equilibrium state, if it exisQ, dependsonly on the total sum of the components of the initial state. Finally a sufficient condition is given for solutions to converge to the unique point.

I. INTR~O~JCTI~N HIS PAPER discussesproperties related to the stability of a class of nonlinear time-invariant compartmental systems. The compartmental system has been widely used as a mathematical model for expressing dynamical behaviors appearing in the study of tracer kinetics in clinical medicine, chemical reaction, and ecology [l]-[4], [19]. A general form for a linear time-invariant compartmental system is given by i =Ax + u (A =[a,] E Rnx”; x=col [x,;. - ,x,] denotes the state of the system; u the input), with, constraints uij > 0, i #j; a, + Xjzia, = -a,, < 0, 1 < i < 12[2]. It is well known that the matrix A satisfying the constraints has no purely imaginary nonzero eigenvalue (though it may have complex ones with negative real parts), and the system has the following properties: When a constant input u is applied, a) the system has no periodic oscillations, and b) a unique equilibrium point (i.e., x* = -A -‘u) exists and is asymptotically stable if and only if det A #O. One naturally suspects that nonoscillation and asymptotic stability are intrinsic properties not only of the linear class but of more general compartmental systems. For example, a conjecture is given in [lo] that a free nonlinear compartmental system of the form ii = Cy= ,a,(~)?~; uU(x) > 0, i #j, Cl= @x) = 0, has no sustained oscillation; that

T

Manuscript received Se tember 29, 1977; revised December 28, 1977. H. Maeda and S. Ko Bama are with the Department of Electronic Engineering, Osaka University, Osaka, Japan. Y. Ohta is with the Department of Electrical Engineering, Fukui University, Fukui, Japan.

OHTA, MEMBER IEEE

is, every solution with nonnegative initial state approaches a steady-stateconstant solution. This conjecture, however, is false as we will show in Section III. In this paper, we confine ourselves to a class of nonlinear time-invariant compartmental systems zii = XT=,a,(~~) + Ui, 1 < i < n, where the input ui is constant. The main purpose of the paper is first to establish a condition for nonoscillation of solutions of the system, and then give sufficient conditions for stability which are sharper than those in [5]--[9]. We remark here that once sustained oscillations are ruled out, the asymptotic stability problem amounts to boundedness of solutions and uniqueness of equilibrium points. We would thereby be able to obtain sharper conditions for stability. In Section II, we describe the nonlinear compartmental system in more detail and give necessary and sufficient conditions for solutions to be nonnegative for all t > 0, and in Section III we propose a sufficient condition for the nonoscillatory property mentioned above. We then give an example which shows that if the condition for nonoscillation is violated then a periodic oscillation may occur, which shows that the conjecture in [lo] is generally false. Note that if Q(+)= a,(~~), we can show that the nonlinear compartmental system under consideration belongs to a class of reciprocal networks; hence the nonoscillation property is easily derived by making use of well-established methods in the-theory of ;-nonlinear -reciprocal network [11]413]. But in general u,(xj)#uji(xi), and the system under consideration may be thought to correspond to nonreciprocal networks. Hence a different approach will be needed to derive such a property. In Sections IV and V we assume that the condition of Section III for nonoscillation holds. In Section IV we give a necessaryand sufficient condition for the boundedness of systems; if a solution is nonoscillatory and bounded, then it approaches an equilibrium state which in general depends both on the input and the initial state. A sufficient condition is also given for an equilibrium state to depend only on the input. In Section V, we consider free systems and show that if the system is closed (i.e., Cl= ,uU(xj)= 0, 1< j < n), then the total concentration Zy= ,xi(t) is invariant for t > 0. We give a sufficient condition for solutions to approach a unique steady state determined by X;=ixi(0). For a system which is not closed,

0098-4094/78/0600-0372$00.75 0 1978 IEEE

'

MAJ3DAetd.:

NONLINEAR

we give a sufficient condition for the origin x =0 to be globally asymptotically stable. These conditions are given in terms of the graphical structure of the interconnection of the individual compartments and the characteristics of the rate functions, and thereby can be easily applied. Quite recently, Sandberg has independently dealt with the related material in [21], and given the sufficient condition which guaranteesthat the unique equilibrium point is globally stable for a more general form of differential equations. The major differences of our results from [21] are that the nonoscillatory property is established-once this is established, further implications mentioned above follow-and that the conditions given here are stated in terms of the interconnection of the individual comparf ments. II.

SYSTEM

373

COMPARTMENTALSYSTEMS

DESCRIPTIONAND NONNEGATIVENESS OF SOLUTIONS

x(t;O,u)>O, for any t>O, and u(t)>O; iv) x(t;x’,O)>O, for any t > 0, and x”>O; and v) x(t; x0,0) Z 0, for any t > 0, and x0 > 0. Proof: The equivalence i)eiv)tiv), and the implication i)+ii) are shown in [6], [14]-[16], and ii)+iii) is clear. We only show that iii) implies iv). For an arbitrarily given x’>O, find t^>O small enough so that x0 - if(axO) > 0,

O/t, for 0 < t < i and the input just constructed, transfers the origin to the x0 at time i. By iii), x(t; 0,~) > 0, for t > 0 holds; and from the time-invariance property of the equation it follows that x(t+ t;O,u)=x(t;x’,O)> 0, for all t > 0. This completes the proof of Theorem 1. In the subsequent sections, we will assume that each uii( *) is continuous, smooth enough to ensure the uniqueness of solutions, and nonnegative, i.e.,

Consider a compartmental system consisting of n compartments in which x,(t) denotes the amount or the concentration of a certain material of interest in the ith compartment at time t. Let the ith compartment receive the material through the input at the rate ui and from some other, (say the jth) compartment at the rate uii. At fora>O (lO (O 0, x0 > 0, and u(t) > 0; iii) constantsteady state.

374

IEEE TRANSACITONS

ON CIRCUITS

AND

SYSTEMS,

VOL.

1x-25,

NO.

6, JUNE 1978

where g(x2)=x2(xz- 12x,+41)= ~~{(x~-6)~+5} > 0, x2 > 0. It is shown that g( .) has negative slope in (12 - fl)/3 < x2 < (12 + fl)/3, and condition (5) is violated. As shown in Section V, the system is closed and xl(t) + x2(t) + x3(t) = k (constant), for all t > 0. Choose xi(O) > 0, I< i < 3 to satisfy the above relation with k = 34. Then the trajectory is restricted to lie on the hyperplane h!={XlXl+~2+~3=34, xi>O, lO,orvi(t)=O,zji(t)>O} (x:,x;) = (12,4). Linearization about this point shows that J=J(t)= {ilvi(t) 0. solutions and the uniqueness of equilibrium points. From The right-hand side derivative of V(t) is, thus given by Theorem 2 we know that the boundedness of solutions atuomatically implies that the system is completely stable e’til(t)-e’tiJ(t) $(t)= with respect to the positive orthant, that is, every nonnegative solution tends to the equilibrium set. = e’4dtMt) + e’&(th(t) Theorem 3-Boundedness of Solutions: Let each u~(*) - e’& (h(t) - e’AJJ(th (t) satisfy (4) and (5) and let u be a constant nonnegative = -(2e’A,(t)+e’A,(t)+a,(t)‘)v,(t) vector. Then for every nonnegative initial state x0 > 0, the solution of (3) is bounded if and only if there exists at +(2e’A,(t)+e’A,(t)+a,(t)‘)v,(t) least one equilibrium point in K= {xix b 0). t>O < 0, Proof: The “only if” part is obvious from Remark 1 (8) of Theorem 2. We will prove the “if” part. Let x* E K be where the vector e denotes the unit vector, e= co1 an equilibrium point, i.e., f (x*) + u =O, then we have, I], with appropriate size, and c& (&) denotes the [I I. *-. row vector consisting of a,,(t), i E Z (i EJ). To derive (8) from (3) we have used the fact that each column sum is nonnegai=g(t)=f(t+x*)-f(x*) (9) tive, i.e., e’ArI + e’AJ, + e’AKI = - c& < 0, and e’AJJ+ g(z)=col [g,(z);. . ,g,,(z)], gi(Z)= e’A, + e’AU = - & < 0, which follows from (2) and (5). where z(t)=x(t)-xx*, (I 0. In Appendix I it EJ!=,gy(zj) (1 0 satisfies lim,,, f (x(t)) + u = 0, or equivalently, lim r+mf(t) =O, the w-limit set of x(t) is contained in the equilibrium set {x] f (x) + u =0} [ 181, and Theorem 2 follows. We prove that lim,,,f(t)=O for nonnegative bounded solutions. For this, let x(t) be a nonnegative bounded solution, and define V(t) = E.;=tlii(t)l = XyGllvi(t)(, where vi(t)=&(t), for t>O (1 O, or zi(t)=O, Qt)>O}, .Z=.Z(t) ={iJz,(t) 0 if and only if there is a path from every compartment to the outside world which consists of rate functions satisfying a,(o)++ 00, as u-00. Proof We first prove the “only if” part. Let P, = {+]+(a)++ co, as a++ oo}. Suppose that the condition does not hold, then there is at least one compartment from which there is no path to the outside world consisting of a,(*) E P,. Denote such compartments by integer set I, then a&.) @P, for all i E I; and a,(.) @P, for all iEZ and for alljEJ={1,2;.. ?n} - I. Note from (5) that each ag( *) is bounded or a,(*) E P,, hence a,(.) @P, implies that it is bounded. Let a,, and aor be the vectors consisting of Z iEIaji (j E J), and a,, (i E I), respectively. Then, since each element of a,, and a, is bounded; there is a constant K > 0 such that

all xE{x>O]e’x=k}. The result-of [15, p. 132, Theorem 4.91shows that the continuous operator A(.) satisfying i) and ii) has a fixed point x* E&, A(x*)=x*, which shows that f(x*) + u = 0. This completes the proof of Theorem 4. Remark I: The hypothesis (5) in Theorem 4 may be replaced by the condition: each a,(u) is either bounded or a,(u)+ + 00, asa++co

(O 0 is weakened by considering the sequence such that u(“)>O (n Z l), and u(“)+u > 0 as n-co. Let xc”) be a solution corresponding to u@)> 0, f (x(“)) + I.@)= 0. Then {x@)} is uniformly bounded and contains a convergent subsequence,since the sets X,,., and X, are bounded by the Lemma. Denote also the convergent subsequence by {x(“)}, and denote the limit point by x*, then by letting n+co, and from the continuity off (-), we have f (x*) + u = 0. Thus Theorem 4 remains correct under (4) and (12). Remark 2: The theorem could be shown in the alternative way [13], [21], which is based mainly upon the fact that (3) has a periodic solution with period T for every for x1 > 0. e’dx,) + e’a,,(x,) < K < 00, T > 0 (this is shown by Brouwer’s fixed-point theorem), Consider an input u satisfying e’u, > 2~, then for any and each solution of (3) lies in a compact set; with these . shown that (3) has a constant solution. Note x* E K satisfying f (x*) + u =0, we arrive at a contradic- f act s, 1.t 1s tion that in this alternative proof, we must impose the assumption, in addition to (4) and (12), that each a@(*) is smooth 0= e’f,(x*) + e’u, enough to guarantee that the solution of (3) is continu=- e’a,,(x,*) - e’aJl(xl*) + e’aIJ(xJ*) + e’u, ously dependent on the initial state, which is inessential to > -K+2K>0 the existence of solution for such an algebraic equation. since each element of a,,(~,*) is nonnegative. This proves Remark 3: Applying Theorem 4 to linear compartmenthe “only if’ part. We next prove the “if” part. To show tal systems: i =AX + u, where A = [a,]; aii > 0 (i+j), this we use the following result. Zl= laii = - aO,< 0 (1~ j < n), we know that when (and Lemma: Let each a,(*) satisfy (4) and (5), and let each only when) there is a path from every compartment to the compartment possess a path satisfying the condition of outside world consisting of positive-rate coefficients, the Theorem 4. Then equation Ax + u =0 has a solution in K for any u > 0. (1 1) While it can also be shown that the equation has a x,={x>0~f(x)+24>0} solution in K for any u > 0 if and only if det A #O. Thus is a bounded set for each u > 0. we obtain a property of matrix A, which will be used in The proof is given in Appendix II. Now we return to the the sequel. proof of Theorem 4. Let Corollary: Consider A = [a& E R”x”; aU > 0 (i #j), Cy=,a0= - ao,< 0 (1 < j < n). Then det A #O if and only A(x).=x+a[f(x)+u], a>0 if there is a path from every compartment to the outside fi,={x>Ole’x 0. then for a sufficiently large k >O, f(x)+ u 2 0 for all Proof: If det ;Q# 0, then Ax + u = 0, u > 0 has a soluxE{x>Ole’x=k} (such a k exists by the Lemma). Once tion x=[-A]-‘u>O for any u>O [17], and the path the k is determined we can choose an (Y> 0 sufficiently condition follows. Conversely, if det A = 0, then there small to satisfy l/a >maximaxa~,~k{(-aii(u)-ui)/u}, exists a vector c > 0 such that c’A = 0 [ 171,and hence the since each aii( *) is continuous, and satisfies aii(0)=O, equation Ax + u =0 has no solution when c’u > 0, which -a,,(c) > 0, for u Z 0, and, from (5) - da,(u)/d~],=~< cc implies that the path condition does not hold. (of course, if u >O, such an a > 0 exists by assuming (4) If the equation f(x) + u =0 does not have a unique but not (5)). Then the operator A( .) is i) positive in 52,,by solution for a given U, the steady-state solution of (3) (4), i.e., A(x) > 0, for x EQ,, and satisfies ii) A(x) Y x, for depends both on u and the initial state.

376

IEEE TRANSACTIONS

Theorem 5-Uniqueness of Steady States: Let each au( *) satisfy (5), and suppose that the equation (3) with a nonnegative constant input u has a steady-state solution. Then, the steady state is independent of initial state and uniquely determined by u if each compartment possesses a path to the outside world consisting of strictly monotone increasing rate functions. Here a,(*) is strictly monotone increasing if

ON CIRCUITS

AND

SYSTEMS,

VOL.

~~-25, NO. 6, -

hyperplane restricted to the positive orthant Hk={x>Oje’x=e’xo=k}.

(18) From the discussion above and Theorem 2 we have the following. Theorem 6-Closed Systems: Let each ati( .) satisfy (4) and (5), and let aoi(.)=O, 1 < i < n. Then for any initial state x0 > 0, the solution of (15) tends to the equilibrium set in Hk. Moreover, if the matrix

for any xi > yj > 0. au(3) - av(uj) > 0, (13) B (x,Y) = [ b,] Proofi Suppose that there are two steady states x*# where y* in K for some u.2 0. Let Z= { ilx,* =yT}, and J= {iJxF [ [ ag(Q - ao(Yj>]/(Xj-Yjh #y,*}. Then ZuJ={1,2;~~,n}. Since f(x*)-f(y*)=O, it follows that O=~jn=l[au(xj*)-a~(yi*)]=~j”,J[a,(xj*)b,= au(yj*)] = 0, for all i E { 1,2,. * * , n}, or equivalently ~.&,*-Y3=0 4Jw-Y,*)=o (14) where B,,=[bii] (i~z,jEJ), B,,=[bti] (iEJ,jEJ); and bu= (ag(x,*) - a,(yj*))/(xj* - yj*), for i E J. It is obvious from (14) that det [ BJJ]= 0. Note, from (5), that b0> 0, for all iE{1,2,. . . ,n} andj E J (ifj). And note also the fact that if the real square matrix has nonnegative off-diagonal elements, the eigenvalue with maximal real part is real, and to this maximal eigenvalue A,, corresponds a nonnegative eigenvector; furthermore, if each column sum is nonpositive then A,, < 0 [2], [ 171.Then it is easily shown that the matrix BJJ has the eigenvalue X,,=O, and from the fact that xj* -y,?#O, for i E J, it is assumed that x;* - yJ*> 0, which implies, by (14), that BIJ = 0, since each element of BIJ is nonnegative. Namely, no .a?(i E I, j E J) is strictly increasing. From the path condition and from the Corollary of Theorem 4, it then follows that det [ BJJ]# 0, which is a contradiction. Remark: Referring to the proof, the condition (5) is replaced with the weakened condition: each av( *) (i#j) is nondecreasing, i.e., forany+>yj>O, (i#j). ag(xj) - ati > 0, An analogous result is given in (211.

1978

(19) xi+& xj=yj

is irreducible for any x,y > 0, i.e., there is no permutation matrix P such that PBP-‘= then the equilibrium point is unique in Hk for each k > 0. Proof We only show’the last statement. Supposethat there exist two equilibrium points x*#y* in Hk, i.e., B(x*-y*)=O, e’x* = e/y*. Since the matrix B= B(x*,y*) is irreducible, and has nonnegative off-diagonal elements and each column sum is zero, it follows that the eigenvector corresponding to the maximal eigenvalue A,, =0 is unique in the sense that there is no independent eigenvector [2], [17]. Hence we may assumethat x* - y * > 0 (#O), which contradicts e’x* = e’y*. Remark: The irreducibility of the matrix B means that the system is “fully connected.” This condition may easily be replaced by the .weakened one. The system is fully connected in the sense that for each x,y > 0 there is a permutation matrix P such that

V. ASVMPTOTICBEHAVIOROFFREESYSTEMS In this section we consider the asymptotic behavior of free system ci=f(x).

(15) When a&.)-O (1~ i(n), the system (15) is called a closed system; otherwise, it is an open system. To show the boundedness of nonnegative solution of (15), choose V(x)=Ey=,xi as a Lyapunov function. Then V(X)=e’f(x)=

-

i

a,(Xi)GOp

for xEK

(16)

i=l

if all leakagesaoi( *) are nonnegative. In particular, if (15) is closed, we have, from (16) i xi(t)=i~lxi(o), +?>O (17) i=l which is considered the conservation of mass. In other words, the trajectory starting from x(0)=x0 lies on the

where the diagonal blocks are square irreducible matrices and in each column, Bi+,,i,Bi+l,i,.‘*,Bsi (IGiGs), at least one matrix different from zero. Since each off-diagonal element of B is nonnegative, and each column sum is zero for closed system, it is shown that for each diagonal block Bii (1 =Gi < s - 1) there is at least one nonzero (negative) column sum, which implies, from the irreducibility of B,, that det [B,]#O, for 1 < i < s - 1. Noting this fact and the irreducibility of B,, we can show that Bx =0 has a unique solution x > 0 in the sensethat there is no independent solutions, which is used in the proof of Theorem 6. This theorem shows that the location of the equilibrium point is fixed for each k, and it does not depend on the initial distribution in each compartment. This is similar to the property of a class of Markov processes.

MAEDA

et al.: NONLINEAR COMPARTMENTAL

377

SYSTEMS

We finally consider open system. From (4) the origin is obviously an equilibrium point of (15). We consider the asymptotic stability of the origin. The stability in the sense of Lyapnov with respect to the positive orthant may be shown by (16). To show that every nonnegative solution approachesthe origin, we only prove, by Theorem 2, that f(x) = 0 implies x = 0. This is easily shown as in Theorem 5, and we obtain the following. Theorem T--Open System: Let each ag(*) satisfy (4) x = 0 is globally asymptotically and (5). Then the ori& stable with respect to the positive orthant if each compartment possessesa path to the outside world which consists of positive rate functions such that a,(u) > 0, for u > 0. Remark: The hypothesis (5) concerning to the monotonicity of ag( .) may be replaced by the nonnegativeness condition on the leakage functions foru)O (l 0. Let t-f we arrive at a contradiction: 0= o,(t”) ) +(t”, t’)ni(t’) >O. Similarly, by using ~jE1+~ij~~O (i E J-), we can show that J-(t) is monotone non&creasing. Since I+(t) and J_-(t) are n.ndecreasing, there is a T_> 0 such that Z+(t)= I, J-(t) s J, for t > T, where Z and J are independent of t. From (A.2) and (A.3), we have e’xZ(t) = e’xZ(T) + (Ge’cZ(r) dr = e’xZ(T) + (t - T)( v+ 17~)/2, and e’x,-(t) = e’x,-( T) + (t - T)( v- ~,)/2, for t 2 T. Since x(t) is

bounded, it follows that v= V, =O. APPENDIX 11

Here we complete the proof of the Lemma in Section IV. Let P, = {+]+(a)-,+ co, as a*+ co}. Note, from (5), that each aij is either bounded or aijE P,. Define the disjoint sequence of sets of integers, Z,,Z*,. . . , in the following manner: I, = { ilaoi E P,}, Zk = {i B I, u Z2 u - - - u Z,- ,lai E P,, for some j E Zk- i}. Then the sequencehas the following properties: a) by the hypothesis, there is an integer .s such that I, u Z,u . . . u Z,= { 1,2; * * ,n}; b) each aoi(*) is bounded if iEZ,, 2O}. From O