- Email: [email protected]
Tracer identifiability of multiphase transport systems
Tracer Ident$ability
of Multiphase Transport
Systems b’
DAVID
RAPPAPORT
Faculty of Electrical Engineering Technion-Israel Institute of Technology, Haifa,
and JOSHUA
DAYAN
Faculty of Mechanical Technion-Israel
Israel
Engineering
Institute of Technology, Ha;fa,
Israel
Using the two-compolzent Markov procese of Ref. (3) to de-scribe tracer$ow in a multiphase spacidly inhomogeneoua transport (plu.g@w) system a mathematical framework i8 established for the conduction of tracer experiment8 for 8y8tem identification. of the Transport velocities are ass-unzed known. The problem of unique identij2ation
ABSTRACT:
exdange coefficients between various phases ting boundary rnemuremmte is studied. f3~e.8 of different output-boundary con$gurations are considered (acceasibb and inaocesm’ble outputa). Su$?cient condition for identifiability are given.
I. Introduction We consider a conservative mass flow system in a steady state defined on the subinterval of the real line [u,b]. We shall assume the existence of m different interwoven parallel flows, called phases, each of which at any cross-section, i.e. x E [a, b], is characterized by a velocity f (x, i) m/see for i=l , . . . , m. That is, all particles belonging to a particular phase at crosssection x are moving with the same velocity. In chemical engineering this is known as “infinite radial diffusion” (i.e. a “white noise” type assumption which implies that particles are uniformly distributed within each phase and that present location within the phase cross-section is independent of the past). The phases will be restricted to one of the following three categories: forward, backward and rest. Forward and backward phases shall be called dynamic phases. Rest phases are those for which f(~, i) = 0 for x E [a, b]. If i is a forward (backward) phase f(x, i) > 0 ( > 0) for all XE [a, b) ((a, b]). For the case of forward (backward) phases new material enters the system from the outside at x = a (x = b) and exits at x = b (x = a). For the case of rest phases no new material enters from without. At exits the material is collected in some fashion. We also suppose that at any XE [a, b] there exists a mechanism (e.g. chemical, mechanical) by which particles or material move from phase to phase. This mechanism will be represented by a positive exchange coefficient and which accounts for the fraction of the &(x), whose units are se+
59
David Rappaport and Joshua Dayan material in phase i moving to phase j, i # j, at time t and cross-section xE[a,b]. A common method used to identify the spatial parameters of flow systems is by the introduction of marked tracer material into the system. The dynamics of the tracer material are presumed to be described by a firstorder variation about the steady state. Letting C(t, x, i) denote the concentration of tracer material per unit length of material in i at time t and position x by performing a mass balance we obtain the following set of linear hyperbolic partial differential equations (utilizing the obvious shortened notations) :
aCi
at-
-~(fici)+(j=~i;il,cj)-h,,ci~
hij
=
j_g.+ihij
for i = 1 , . . . , m. Systems described by Eq. (1) are commonly called transport or plug-flow systems. We assume that the following situat,ion prevails: (a) The steady-state velocity profiles f (x, i) for i = 1, . . . , m can be measured along the system. This is the case in many physical systems such as fluidized beds (l), in spouted beds (2) and many other flow systems. (b) Injection of tracer material can be made at a number of neighboring points along the system. At each such point injection is in the form of a “pulse” of high tracer concentration. (c) At boundaries (exits), x = b for forward phases, x = a for backward phases, tracer concentrations are measured (e.g. counts in the case of radioactive material). Phase boundaries are classified as follows: an accessible (inaccessible) phase boundary is one at which the concentration in that phase can (cannot) be separately measured. We treat the cases in which either all the phase boundaries are either accessible or inaccessible, i.e. in the latter only the overall concentration without regard to phase is available. Unless stated otherwise we shall refer to the (in)accessible case as being that for which all phase boundaries are (in)accessible. The case of inaccessible boundaries is by far the most common and most important. We assume that the exchange coefficients &(x) are not measurable and hence unknown. The problem of system identifiability will be defined as the ability to uniquely determine the exchange coefficients on the basis of the input and output measurements described above. Basically the idea is that given the phase velocities and certain concentration histories at the boundaries (i.e. outputs) for each of the various injection points, under what conditions does there exist a unique relation between the system described in Eq. (1) and the outputs. The purpose of this work is to provide a logical mathematical framework for the conduction of tracer experiments and to derive conditions under which results can be meaningfully interpreted given the underlying assumption that the idealized plug-flow transport model is a reasonable description of the particular system in question. By taking a probabilistic view of the tracer experiment rather than the deterministic presented above, in (3) it was shown that the behavior of a
60
Joumal of The Franklin Institute
Tracer Identi$ability
of Multiphase
Transport Xystems
typical tracer particle in the system described by (1) is described by a particular two-component Markov process with discrete second component. This model is specified in the next section along with a summary of pertinent system properties. In Section III a precise definition of the tracer experiment is given, Examples are cited of systems which are not uniquely identifiable from input and output measurements. We use these examples to motivate the class of inputs and conditions for identifiability which we use in the sequel. In the fourth section we imbed the system into a larger class of “extended” systems in order to obtain additional system properties. In Section V we prove a result which states sufficient conditions for identifiability of multiphase inhomogeneous plug-flow systems for both the cases of accessible and inaccessible outputs.
ZZ. The Markov
Tracer Model for an m-Phase
Plug-flow
System
(A) We denote the space of sample functions of the process by R. The letter z denotes the vector [x, y], where x takes values in the closed interval of the real line [a, b], a< b and y takes values in the set of m points Y = (1, . . . . m>. Each point in Y is called a phase; H symbolizes the state space of the process which is the product space [a, b] x Y. Cl is the set of all the functions z(t, w) taking values in H, where t belongs to the semi-infinite interval [0, co) and where y(t, w) is a function in the space of all right continuous step functions on [O, 00) taking values in Y with a finite number of isolated jumps (i.e. changes in value) on every bounded subinterval [tl, tz], 0
w) = f [W, w), $4, w)l
(2)
with the initial condition ~(0, w) = x,, in [a, b]. We shall arbitrarily assume that the phase j for 1 0 for some finite number fmin, for all 2 G [a, b) withf (b, j) = 0; for I, + 1 o.
Vol. 300, No. 1, July 1975
2
61
David Rappaport and Joshua Dayan For each w we shall interpret each z(t, w) for t 2 0 as one of the possible trajectories of a tracer particle with the jump times and the phases visited given by the function y(t, w). (B) We take as the u-field of events on Q, denoted 8, the one generated by all finite base cylinder sets of the type {w: z(t,, o) dI, . . ., z(t,, w) EA,) for every finite subset of times ti belonging to [0, co) and for arbitrary events A, belonging to the u-field in H, denoted H, generated by product sets of the form [c, d] x (‘}z w h ere aO. (C) We assume the following properties for the exchange coefficients &: for the case of a rest phase i we require &(x) to be continuous for all x E [a, b]. For i a forward (backward phase) we take &(x) to be piecewise uniformly continuous on [a, b) ((a, b]), with piecewise uniformly continuous derivative with respect to x on [a, b) ((a, b]). We assume that at most there are a finite number of discontinuities. At a point of discontinuity we require &(x) and its derivative to be continuous from the right (left). At x = b (x = a) h,(b) = 0 (&(a) = 0). Let hii =
$ h,(x) j=l, jsi
and
h,,,
= sup h,(x).
(D) Consider a time stationary strong Markov process defined on (a, /3, P) and state space H where P is characterized by the properties stated below : P(Y~+~ = yl each 7 E [0, CL]1x1 = x, yt = y) = exp
IS
1
t+CX &&,)ds
where 9’s = f(xs, y), s E [t, t + a}, X~= x. In view of the assumption If@, i) I>, fmin, equivalently, (backward) phase and for a
1, Pa)
for y a forward
x2 without jumping 1x1, y) = exp [ - /;Hdv].
(3b)
Further, P(Yt+, = j z i,
one jump on [t,t+a]lxl
= x, yt = i)
where 8, =f(x,G),
rE[t,Sl,
2* =f(x*,j),
pE[8,t+a-j,
x, = x, cc,*= 5,.
Consider the Banach space B of bounded measurable functions (in x) g(x, y) on H with the norm supH 1g(x, y) I. Let D be the subset of B in which g is continuous and has finite piecewise continuous derivatives with respect to
62
Jonmalof
The Franklin Institute
Tracer Identifiability of Multiphase Transport Systems x, with at most a finite number of discontinuities. These derivatives are right (left) continuous at the points of discontinuity for y a forward (backward) phase. For gE D define the operator A by MGY)
=f(l,Y)~(2.Y)-/\uu(Z)9(5.Y)+~_lfiiyn,(x)o(x.j).
(4)
In Eq. (4) for y a forward (backward) phase the derivative is the right (left) derivative. In Eq. (4) we invoke the notational convention that f(x, i) = 0 means that the first term on the right side (i.e. f(ag &r) is zero. A is the weak infinitesimal operator for the process. By Dynkin (4) paragraph 1.15, p. 40, for gE D the expectation E,(gIz,) is continuous from the right for every x,,EH, and in addition the time derivative from the right exists, is continuous for every .z,,EH, and satisfies
=
wherelb,&(g
%Ag Izo),
(5b)
Izo)= s(z,).
It will prove convenient in the sequel to represent the infinitesimal operator in vector form defined below. We shall use the scalar (4) and vector forms (6) of A interchangeably without much comment. Let
and
Therefore, Ag(x) = P(x)
‘2 +r(x)
g(x),
(6)
where we take Ag(x) to mean
A as defined in (4). We remark at this point that for i a forward (backward) phase the state (b, i) ((a, i)) is an absorbing state, i.e. Ag(b, i) = 0 (Ag(a, i) = 0).
Vol.300,No. 1,July 1975
63
David Rappaport and Joshua Dayan 111. Precise
Definition
of the Tracer
Experiment
We now precisely define the tracer experiment : (4 The tracer process is Markovian in nature and its description is given by the Markov process represented by the generator A with domain D. (b) The set of possible initial states (at time zero) for the process will consist of points (2, i) where i = 1, . . . , m and x belongs to a small closed set 1~ (a, b). The interpretation of this is that the tracer pulse may be injected at any of these initial states. In a control context this may be interpreted as the ability to exert a distributed control in an arbitrary small neighborhood of (01,i) where 01E I. The following arrival time distributions (outputs) are available (cl (measured) : (i) For the case of accessible boundaries all the distributions P&x = b, iIxO,j) (P(t,x = a,i lx&) for the case of i a forward (backward) phase i = I, + 1, . . ., I, (i = I, + 1, . . ., m), for all t > 0, all X~EI andj = 1, . . ..m. (ii) For the case of inaccessible boundaries the available distributions (outputs) are 2
P(t,x = b,ilxO,j)
i=t1+1
The problem of identifiability profiles under what conditions (i.e. the exchange coefficients) ? Example
and
$J P(t,x = a,i]z,,j)
for all t>O,
i=za+1
all ~,fl and j = 1, . . ..m. is paraphrased as follows : given the velocity do the outputs uniquely define the process
1
Consider two systems (A and B) defined on [0,5-5771, each composed of two phases. Both systems have the same velocity profiles while the exchange coefficients of I3 are a translation of those of A. Let f(~, 1) = 2 + sinx, and f(~, 2) = 2 - sin x for x E [0,5+57-r].Let X&(x) and vi(z) be as shown in Fig. 1.
FIG. 1.
Ley A;(X) = X#X - 2n) for 47-r6 5 d 5~r and zero otherwise on [0,5-5771. We can easily show that for each X~E [0,2r] that for both the cases of accessible outputs and inaccessible outputs the two systems are indistinguishable on the basis of the input-output measurements described above. However, if we were to allow for initial states in an arbitrarily small neighborhood of 01= 2x, for X~EI, x0 > 27r, we see that the tracer particle
64
Journal
of The Franklin
Institute
Tracer IdentiJiability of Multiphase Transport Systems entering system A no longer sees on his right exchange coefficients which are simply a translation of those of system B. This is, in fact, the ciux of the matter ; for a system to be identifiable the exchange coefficients in the region I on which the inputs are distributed must have a functional relationship t’o the exchange coefficients in other parts of the system; in particular, we shall further on prescribe analyticity. In the next example it is demonstrated that some restriction must be placed on the relation between the velocity profiles of a given system in order to ensure identifiability for the case of inaccessible outputs. Example 2 Consider a two-phase
system with forward phases. Let f(x) = f(G 1) = f(G 2),
all x E [a, b]. Let &(x) and &(z) be arbitrary. structure of the sample space implies that
Then for any x0 E [a, b] the
P(t,x~b?lxo,j)= p(P[x,,+/;.j-(x&ds]), where
% =f(x8),
x,I,,~ =
x0
and
&?) =
1, pa0 0,
B
Obviously in the event of inaccessible outputs no information about the exchange coefficient is contained in the outputs. IV.
The Extended
whatsoever
System
Defined in this section are systems on the real line subinterval [c, d] where c ,< a < b < a?, the “projection” of which on [a, b] is the original system which will enable us to explore further properties of the original system. Consider the m phase Markov system on [c, d] processing all of the properties given in Section II. The parameters of this system will be identified by the symbol “. The system on [a, b] will be distinguished by the notation used until now. We shall assume that for: (a) Rest phases: &#) = h,,(x) for i,j pairs and for all XE [a, b]. i&(x) arbitrary, continuous on [~,a] and [b,d]. (b) Forward (backward) phases : (1) .h,i) =f(x,i) for x~Ca,b) (bbl); (2) f(b, 4 = lim,&(x, i) (fb, i) = lim,J(X, 9); (3) &i(X) = &j( x ) f or x E [a, b], for both forward and backward phases ; (4) Aii(x) = 0 on (b,d] and [~,a). Summing up, on the interval [a, b] the velocity profiles and the transfer coefficients are identical for both systems. On the intervals (c, a] and [b, d) the velocity profiles of the extended system are continuous extensions of those of the original. On theee latter two intervals the transfer coefficients for
Vol. 300, No. 1, July 1975
65
David Rappaport and Joshua Dayan dynamic phases of the extended system are zero whereas those for rest phases are continuous extensions of those of the original. Thus, exchange is prevented outside of the interval [a, b]. Note that the transfer coefficients of the extended system as chosen are right and left continuous, respectively, for the cases of forward and backward phases. Thus the extended system possesses all of the required system properties, and one identifies with it the generator whose form is given in Eqs. (4) and (6). We remark that the imbedding of the original system in the above class of extended systems is possible by virtue of the fact that the probabilistic plug-flow model allows for the existence of internal discontinuities in the transfer coefficients. This was verified in Ref. (3). the Consider the following two results which relate probabilistically extended system to the original. Lemma 1 Let (x,, y,,) be a state of the original system, then for i a forward (backward) or rest phase, for each a, p such that a < 01< /3 < b (a < 01< /3c b) and for all t>o:
P(t,a
= ilxo,yJ =P(t,a
= ilzg,yJ, = P(t,ol
= ilx,,y,)).
Theorem 1 For all t > 0, for each i, and (x,,, y,,) a state of the original system P(t,
x 2
b, y = i j x,,,
ZJ,,) =
P(t,
x =
b, y = i 1x,,, yo)
(74
and B(t,z
= il2,,y,)
= P(t,x
= a,y = ilx,,y,).
W)
Rigorous proofs for these intuitively obvious results are available in Ref. (5). The basic idea is that for every initial starting point (x0, y,), a state of the original system, there is a one-to-one correspondence between the trajectories of the original and the extended systems. The trajectories are identical until an absorbing boundary of the original system is reached. In the original system the trajectory remains fixed at the barrier thereafter. In the extended system the trajectory leaves the confines of the original system never again to return. Note that by Theorem I and the path properties for t 2 0, b
= b,y = ilx,,y,),
where t,, 1 = 1,2, are the unique times satisfying xl
=
b+
s
:If^(x,,i)ds.
If t - t, < 0 the corresponding probability is taken to be zero. A similar result exists on the subinterval [c, a]. Hence, there is a one-to-one relation between
66
Joomal
of The Franklin
Institute
Tracer IdenliJiability of Multiphase Transport Systems the boundary arriva,l time distributions associated with the original system and with “spatial” probabilities, for a fixed time (t), on the subintervals [c, a] and [b,d]. In particular “spatial” discontinuities on the subintervals [b,d] and [c, a] corresponds to “time” discontinuities in the boundary distributions of the original system. Therefore, by using the arrival time distributions associated with the original system we can evaluate expectations associated with the extended system of the form, &(gIz,, yO) where g(x, y) = 0 for x E [a, b]. In the sequel it will be convenient to use a particular form of the extended system. Below we shall classify the various cases and state their properties. (1) Case of accessible outputs If i is a forward (backward) or a rest phase” let f”(z, i) = limzraf (2, i) (lim ,,,f(z,i)) for all x~[b,d) ((~,a]). Also let f&i) = f&i) (f(b,i)) for x E [c, a] ([c, d]). The extended system exchange coefficients are given at the beginning of this section. (2) Case of inaccessible outputs We extend the velocit,ies of the system in two stages. Choose an arbitrary velocity v> 0. Using elementary arguments one can show that on the subinterval [b, b + p] ([a-p, a]), where p > 0 is arbitrary, that it is possible t,o “accelerate” all forward (backward) phases to velocity v (-v) on the subinterval [b,b+p] ([a-p, a 1) such that the time At (also arbitrary) it takes to traverse this subinterval is the same for each phase. In addition, for i a forward (backward) phase we choose f(x, i) constant and continuous on [a - p, a] ([b, b + p]). Thus after the first stage we have a multiphase system for which the output velocities are the same for all forward (backward) phases. We then extend the system to [c, d] from [a-p, b + p] exactly as in the case of accessible outputs. Notice that here this means that for all forward (backward) phases for 2 E [b + p, d] ([ c, a - p]) the phase velocity is v ( - v). We extend the exchange coefficients as described at the beginning of the section. Clearly from the above description, if i is a forward (backward) phase, p and At as above, for x0 E [a, b] by the strong Markov property :
= ilx,,y,)
P(t-At,x>b,y
= &,z2b+P,y
(fi(t-At,x
= ilz,,y,)
= il x,,,y,,), = p(t,x
= ijx,,y,)).
In the light of Theorem I and the above chosen extended system we shall interpret the description of the experiment as stated in Section II as follows : (1) For the case of accessible outputs availability of the distributions: P(t,z
= 6 y = ilz,, yO) (P(t,x = a,y = il~O,yO))
for each x0 E.Z and each t > 0, for i a forward (backward) phase, will be understood to mean that for each integrable g(x, y) for which g(x, y) = 0 for
Vol. 300, KGO.1, July 1075
67
David Rappaport and Joshua Dayan x E [a, 61, the expectation $,(g[ x0, y,,) can be calculated for t 2 0, X~EI. If further g(x, y) E Da then also the partial derivatives of the expectation with respect to t and x,,, x,, E I, are calculable. (2) For the case of inaccessible outputs the availability of the dist,ributions P(t, x = b 1x,,, y,,) and P(t, x = a 1x,,, y,,) for each x,, EI and each t > 0 shall be understood to mean that for each integrable g(x, y) for which g(x, y) = 0 for XE [a- p, b + p] and for which g(x, i) = g(x,j), for all i # j, the expectation 8,(g 1x0, go) can be computed for t >, 0, X,,E I. If in addition g(x, y) E Dg then also the partial derivatives of the expectation with respect to t and x,,, z,,EI are computable, The essence of the interpretation stated above is that the availability of the output distributions implies the availability of the solution to the partial differential equations generated by the weak infinitesimal operator.
where t > 0, x~EI, g EDa_, g(x, y) = 0, for x E [a, b]. In the event of an inaccessible output system we further require that g(x,j) = g(x, i) all i,j. As &(g] x,,, y,,) is a functional of the arrival time distribution the identifiability problem can be paraphrased as follows: “Given a set of solutions on the boundary of a plug-flow system under what conditions does this set uniquely specify the unknown system parameters”. V. Identifiability
In this section we present the main result, Theorem II, the identifiability theorem, which gives conditions under which the transfer coefficients are uniquely related to the output distributions. Establishment of this result utilizes two lemmas which are stated prior to Theorem II. Lemma 2 is used to prove Lemma 3, while Lemma 3 is needed in the proof of Theorem II. Essentially, the lemmas maintain that under certain conditions the only discontinuity in the process probabilities, i.e. P(t, z < CL, y = i ]x0, y,,), is that which propagates from initial point (x,, yO) through phase y,, in accordance with the respective phase velocity. That is to say, these conditions preclude the transference of discontinuities from one phase to another, or to another trajectory in the same phase which is a time shift of the one above commencing from (.x0,y,,). Physically, these conditions mean that if two particles enter the system at the same time, one in phase yl, the other in phase yz Z yl, and each propagates through its respective phase without jumping to another, then the t’otal time that these two particles spend abreast of one another is zero. For each t,>O, cuE[a,b] and i~[l, . . . . m], we shall denote the trajectory starting from position a, at time t,, generated by the dynamics of phase i by x$,(4
ilf(x:,r.(4, i) d7, t>t,. s The time t, should not be confused with the initial state (x,, y,,) at time zero.
68
= a+
Journal of The Franklin Institute
Tracer IdentiJiability of Multiphuse Transport Systems For the original system we define the field of characteristic curves (6) of phase i on the rectangle [a, b] x [O,co) in euclidean two space, to be the collection of the following curves, each of which is specified by an initial point on the boundary of the rectangle: (a) For t, = 0, olE [a, b], and i = 1, . . . . m: q$(a) = a+ (bl)
‘j’(&,
s0
t20.
i)ds,
For toE (0, co), 01= a, i a forward phase: z;,,(a) = a +
lf(~4,,~,i) d7, s 10
t > to.
(b2) For tog (0, oo), (Y= b, i a backward phase: q#4
= b+
s
i:f(+,bl i) d7,
tat,.
Note that as a result of the time invariance of the velocity profiles, the characteristic curves of phase i are all parallel to one another, for each i. We shall denote the time of arrival of a trajectory at boundary b (a), for i a forward (backward) phase, as follows: T;(a) = mi~{t:
z&,(cY)= b}
min (t : &(Ix) = a} . >
l&l
Lemma 2 Consider any two characteristic curves, one of phase i, the other of phase j, i # j. Suppose that the set of points in time at which these two curves intersect one another is of Lebesgue measure zero. Then for any dynamic phase 1, any to> 0, j # 1 and ~!,~(a) a characteristic curve of phase 1, w,x = &&4, Y =jl x0, k) = 0, i.e. with respect to Lebesgue measure for all t such that to < t < T;(a), for k = 1, . . . . m. Remark. The condition in the lemma implies the existence of no more than one rest phase. Note that this lemma eliminates systems such as that of Example 2. Lemma 3 Let i be a Suppose that continuously respectively.
forward or backward phase. X:,&(Y) is a trajectory in phase i. the velocity profiles of each forward and backward phase are differentiable, with bounded derivative, on [a, b) and (a, b], Suppose that the Lebesgue measure of the set (x: f(z, i) = f&j),
x E [a,
al, i
# j}
is zero for all pairs i # j. Then for all k, P(s, x = x$~(cx), y = i 1x0, k) = 0, for all s E [to, TiO(ol)),when (1) i # k, for any OL E [a, b] and to > 0, i.e. for any characteristic curve of phase i, or when (2) i = k, for (01,to) not lying on the characteristic curve x~,(x,), t 2 0.
Vol. 300, No. 1, July 1975
69
David Rappaport and Joshua Dayan The proofs of Lemmas 2 and 3 are summarized Ref. (5) for the complete proofs.)
in the Appendix.
(See
Theorem II (identi$ability) Let F(x) be known and continuously differentiable with bounded derivative on (a, b). Let I’(x) be an analytic function on (a, b). Suppose X~EI. Let there be no more than one rest phase I, and for which h,(x,) # 0. Assume that the set {x: x E [a, b], f(~, i) = f(z,j), i f j} has Lebesgue measure zero for all i # j. (a) Then for the case of accessible outputs, l?(x) is uniquely identifiable from the output distributions. (b) Suppose for the case of inaccessible outputs that Ti(x,,) # Tg(x,,), for each pair i # j where either both are forward or backward phases. Then l?(x) is uniquely identifiable from the output distributions. Remarks. The analyticity assumptions are motivated by Example 1 in which is illustrated that in the absence of a functional relationship between the parameters of the input area and those of the rest of the system the system is not necessarily identifiable. The assumption that X,(x,) # 0 for 1 the rest phase ensures that phase 1 “communicates” with other phases at x0. The arrival time assumption in part (b) of the theorem prevents the build-up of parallel discontinuities in the arrival time distributions for the different phases as occurs in Example 2. Note that the input region I c (a, b) is arbitrary except for the conditions on the rest phase and the arrival times for the inaccessible case. Proof. For the purposes of the proof we shall assume the existence of two non-identical analytic functions on (a, b), I’,(X) and l?,(x). We shall assume that for the associated systems with F,(X) = F,(x) that all the corresponding output distributions are identical. We first tackle the case of accessible outputs. Differencing Eq. (8) with f,, with Eq. (8) with f,, by the definition of the experiment, we obtain that for t 2 0 [fl(%)
-
fk%)l~t (91x0,Yo) =
0.
Since this holds for each g as defined in Section IV and as ri(xo) = f,(x,), i = 1,2, it follows that for t>O [r,(z,)
- I?&,)] g(t, II:= a or b, y = k 1zo)=
0t
(9)
for each forward or backward phase k. (Note that for k a forward (backward) phase P(t, x = a, y = k 1zo) =0 (P(t,x=b,y=k~z,)=O)fort>O.Seecase(2) in proof of Lemma 3 (Appendix) for detailed explanation.) Suppose l?l(xo) # l?Z(~o), i.e. there exists at least one pair i, j, i # j, such that hi;)(z,) # h$‘(~,). Now consider the sum corresponding to the ith component of Eq. (8), i.e. for k either a forward or backward phase
f) [A$)(x,) -X$)(x,)]
P(t, z = a or b, y = k) x0,j) = 0.
(10)
J=l
is t-spose ofthe vector
t P( -, . , -1%) the
70
[P( . , . , . 1x0,l)r. . .
p( . , ., .lx,,
m)~.
Journal of The
Franklin Institute
Tracer IdentiJiability of Multiphuse Transport Systems Let a, = h$‘(zo) -A$)@,), j = 1, . . . . m, and partition these coefficients positive (a+) and negative (a-) subsets. For k either a forward or backward phase we rewrite (10) as za,fP(t,x
into
= a or b, y = kjx,,,j) = Ca;P(t,x a-
t>O,
= a or b,y = k/x,&,
(11)
where a7 = - aj for j E a7 and a: = aj for j Ea+. Subcase (1) : one side of (11) empty (say a-) Suppose a: # 0, where (Y is a dynamic phase (say forward). Hence, P(t, z = b, y = CY 1x0,CX)= 0, all t 2 0. By Eq. (3a) we obtain an immediate contradiction. Thus a+ = 0. Let ai # 0 where aris the rest phase. Let 1 be a dynamic phase (say forward) for which h,(z,) # 0. By (11) P(t, x = b, y = I) x0, I) = 0 for all t > 0. By (3~) P(one jump only to I on [O,At]] z,,, IX)> 0, i.e. P(At, y = Zlx,,, a) > 0. Let &,=o, be the identity function of the set [y = I]. By the Markov property for T > (b - o)/f,i, P(T + At, 2 = b, y = I) z,,, a) > EAt(&,=rl P(no jump on [At, T] Ix, y) Ix,,, CL) > e=p ( -Lax Contradiction!
T)P(At,y
= Z~x,,,or)>O.
Hence a+ = 0.
Subcase 2 : neither side of (11) empty Clearly there exists a., # 0 where Z is a dynamic phase (say a,Ea+ and Z without loss of generality is a forward phase). Consider the system trajectory z&(x,) for t 2 0. Tb(x,) is the arrival time at x = b of the characteristic curve 4,&J [i.e. ~+c~~o,,&~) = bl. Note that by (3b) the probability P(u, x = b, y = Z(x,,, a) has a discontinuity at u = Tb(x,,). Therefore, by (11) this means that there exists an 01~a-, a # 1 for which P(u, 2 = a or b, y = Zlz,,, a) has a discontinuity at u = Th(x,). Designate this latter discontinuity by the notation AI. Recall that by Lemma 3, part (l), for every As > 0, P(T;(x,,) -As,
x = x”TO’(zOo)-*s,O(~O)~ Y = ZI5>4
= 0.
By the sample path properties, the Markov property and Eqs. (3(a)-(c)) A, G W,(x,)
-As, x = ~&,~~zOo)--s,o(~o), Y = ZIx0, 4 + m&,,
As +- @As),
where the last two terms on the right side represent bounds on the probability of jumping on the time interval As. As As is arbitrary it follows that AI = 0. Contradiction! Thus, a, = 0 and for the accessible case rI(xo) = I’.Jx~). We turn now to the inaccessible outputs case. By Eq. (8) and the definition of the inaccessible case we obtain that for t > 0 [I’,(x,) - l?,(x,)] P(t, x = a or b 1zo) = 0.
Vol.300,No.1,
July 1975
(12)
71
David Rappaport and Joshua Dayan Using the same reasoning and notation conclude that
used to obtain Eq. (ll),
by (12) we
a; aif P(t, x = a or b, j xO,j) = x a7 P(t, z = a or b j xo,j). a-
(13)
Subcase (1): one side of (13) empty (say a-) Suppose a: = 0. This means that WL CP(t,z
= b, y =j(x,,~~)+P(t,x
= a, y =jlq,,ci)
= 0
j=l for all t > 0, i.e. each term is zero. Using the same reasoning as in the proof of subcase (1) in the accessible case we conclude that a: must equal zero.
Subcase (2); neither side of (13) empty This implies the existence of an 1 such that a, # 0 (say ale a+), where b, without loss of generality, denotes a forward phase. Hence, P(u, x = b 1x0, a) has a discontinuity at u = Tb(x,). Consequently, on the right side of (13) there exists an a,Ea-, a # I for which P(u, x = a or b j x0, a) also has a discontinuity at u = Tb(x,). It then follows that for some j, P(u, x=a or b, y = j 1x,,, a) has a discontinuity at u = Tb(s,), which we shall denote by Ai. If j = 1we reason as in subcase (2) for the accessible case that a, must be zero, If j # 1 we assume without loss of generality that j is a forward phase [i.e. P(u,x
=a,y
=j[xO,a)
= 0
for u > 01. From the trajectory properties it follows that there exists a unique DE [a, b] and a unique toe [0, Tb(x,)], defining the characteristic curve x,{,~(v), s > t,, for which x&(Z) t,,(v) = b [i.e. the unique characteristic curve of phase j for which the arrival ‘time at b is TA(x,)]. Reasoning as in subcase (2) of the accessible case we obtain that for any As > 0 Aj < P(Tb(x,) -As,
X: = ~~o~~so~-~s~o(~), Y =j 1x0,a) +mX,,,As
+ o(h).
For j # 01we conclude immediately by Lemma 3, case (l), that the probability on the right side of the inequality is zero. Forj = 01we note that by hypothesis !!‘;(~a) # Tb(x,) for all I # j and consequently (v, to) is not on the characteristic curve x$(x0). Thus by Lemma 3, case (2), the probability on the right side of the inequality is zero. As As is arbitrary we conclude that Aj = 0. Contradiction! Hence, a, = 0 and we conclude that rl(xo) = r&x,,). Continuing we note that the output distributions are available for all xOeI. Thus, for the accessible case I’(Z) is uniquely identifiable, for each x0 E 1. By analyticity we conclude that r(x) for x E (a, b) is uniquely specified by the output distributions. Secondly, T:(x) is continuous in XE (a, b) for each i. Hence, for the inaccessible case there exists a closed neighborhood I* of x,, in which T:(x) # T;(x) all i f j either both forward or backward phases. Consequently, I’(x) is obtainable from the output distributions for x ~ln I*. By analyticity we conclude that r(x) for XE (a, b) is uniquely specified by the output distributions.
72
Journalof
The Franklin Institute
Trucer Iclenti$ability of Multiphase Transport Xystems Discussion
It is quite reasonable to suggest that the conditions imposed in Theorem II are not the weakest possible under which identifiability is possible for some other class of transport systems. However, the problem is to find a common factor describing a class such as the restrictions on the velocity profiles as imposed in Theorem II. From a practical point of view these restrictions are not unreasonable and are applicable t,o a large number of systems. It can be similarly shown that for the case in which some of the phase boundaries are accessible and some inaccessible that the system is identifiable under the conditions imposed on J’(x) and I’(x) in Theorem II plus the additional requirement that Tk(x,) # T&(x,) for i # j, both either forward or backward phases with inaccessible boundaries. Note that the “solution” of the problem posed herein requires that an infinite number of experiments be conducted. Work is continuing in the means for the actual calculation of the direction of finding a “practical” unknown exchange coefficients. We once again reiterate that the purpose of this work is to provide a mathematically logical framework to indicate how to conduct tracer identification experiments of multi-phase plug-flow systems and under what conditions the results obtained can be meaningfully used. References (1) T. Mamuro and H. Hattori, “Flow pattern of fluid in spouted beds”, J. Chem. Engng Jap., Vol. 1, No. 1, pp. l-5, 1968. (2) H. Kabayashi, F. Arai and T. Chibe, “Behavior of bubbles in a gas-solid fluidized bed”, Kag. Kog. (J. Chem. Engng Jup.) (English Ed.), Vol. 4, No. 1, p. 147, 1966. (3) D. Rappaport and J. Dayen, “A probabilistic model for tracer distribution in multiphase spatially inhomogeneous transport systems”, J. Stat. Phys., Vol. II, No. 6, 1974. (4) E. B. Dynkin, “Markov Processes”, Vol. I, Springer, Berlin, 1965. (5) D. Rappaport and J. Daysn, “Tracer identifiability of spatially inhomogeneous multiphme tranxport systems”, EE Pub. No. 205, Technion,;SI\srael Institute \ iof Technology, Haifa, Israel, Aug. 1973. (6) R. Courant and D. Hilbert, “Methods of Mathematical Physics”, Vol. II, Interscience, New York, 1962.
Appendix (A.l)
Proof
Let 6>0.
of Lemma 2 Let B = (8: P(s,x
= &(a),
y = j Ixo,k)d6,j
# I, to
Without loss of generality we shall assume 1 to be a forward phase. From each pair of points in the plane [s, x&&x)] w h ere 8 E B, consider the plane curve generated by the points (t, zf) where XT = ~!,~[~:,~,(a)1 and t > s. These curves are segments of characteristic curves of phase j. It can be shown by the Markov property and Eq. (3) that all the pairs [a, z&(a)], where 8 E B lie on a finite number of distinct curves. Let these curves be numbered one through n. Each curve commences from some point [8i,z:,,tr(~)]
Vol.500, No. 1.July 1975
73
David Rappaport and Joshua Dayan n. On the ith curve there lie points {a, ~f~[~&,(a)]} where a E B But by the hypothesis the set of points s~!l’:~(a) at which &,b&,t.(41 = 4,,,[4&, (a)] has Lebesgue measure zero. Since the number of such curves is finite and B is in the union of such points on the n distinct curves, it follows that B is a subset of a set of Lebesgue measure zero. As 6 is arbitrary the result follows. wheresiEB,i=l,..., and {s, z$,[z:~,+(cx)]}.
(A.2)
Proof
of Lemma 3
Without loss of generality we shall assume i to be a forward phase. It can be demonstrated by the Markov property and Eqs. 3(a)-(c) that the probability p(a, x
y = i
120, k)
is absolutely continuous in s on [to, TtO(a)- AT] for arbitrary AT > 0. Hence its derivative with respect to s exists, ae., and is integrable on [to, T&u) - AT], i.e. for AT> 0 (neglecting the OLnotation) : et
X2$.
d X < x:+A~,ta~
!,
=
i 120,
k)-P(to,ol~x
=
i Izo,
k)
=
=i 1X0,k) d8. J~$P(8,xf,,,~x
(Al)
0
It can further be shown that the integrand is bounded uniformly, liIIl 38 AT&O d8
, Xi#,tO< X < Xf+A7,to>
!,
=
i 1 xO~
k)
G
mhmXj~zp~8~
x:,to’
a.e., and that Y
=
j
1 xO?
ffl)
GW
(Zi
for a E [to, Tb(ar)], a.e. By the lemma in the Appendix, haragraph (A.3), the hypothesis of Lemma 3 implies that of Lemma 2. Therefore, by Lemma 2, the right side of (A2) is zero for 8 e [to, Tj@)], a.e. Applying the Lebesgue dominated convergence theorem and stochastic continuity to (Al) as AT + 0 we obtain: P[u, 2 = x;,&),
y = i
j 20, k] - P(to, z = a, y = i Izo, Ii) s 0
(A3)
for all u E [to, TQ(a)). Two cases: (1) to = 0, aE[a,b]: For i # k we conclude by stochastic continuity that P(0, x = 01,y = i 1x0, k) = 0. For i = k we recall from the hypothesis of Lemma 3 that for to = 0 cy. # x0. We thereby arrive at the same conclusion. (2) t,>o, a= a: Recall that z. E [a, b]. Recall that i is a forward phase. From the structure of the sample space it is evident that the only paths that arrive at time to at state x = a, y = i, must first reach z = a, y = j, where j is a backward phase. But x = CL,y = j, is an absorbing state. It then immediately follows from the strong Markov property that P(t, x = a, y = i 1x0, i) = 0 for all t > 0. Hence, P(u, xf,,,(a), y = i 1x0, k) = 0 for all u E [to, TQ(ol)). (A3) Lemma Let f (2, 1) and f (x,2) represent two forward phase velocities. continuously differentiable with bounded derivatives on [a, b). Let A = {x: 2 E [a, 61, f
Suppose
each is
(x,1) = f (x,2)}.
Let xi and x,“, respectively, denote two arbitrary characteristic curves, x&,(ol,) phase 1 and &a(Z) (CGJof phase 2. Let B = {s: xi = z,” < b}. If the Lebesgue measure of A is zero, then the Lebesgue measure of B is zero. Proof:
74
of
Standard arguments.
Journal of The Franklin
Institute
of Multiphase Transport
Systems b’
DAVID
RAPPAPORT
Faculty of Electrical Engineering Technion-Israel Institute of Technology, Haifa,
and JOSHUA
DAYAN
Faculty of Mechanical Technion-Israel
Israel
Engineering
Institute of Technology, Ha;fa,
Israel
Using the two-compolzent Markov procese of Ref. (3) to de-scribe tracer$ow in a multiphase spacidly inhomogeneoua transport (plu.g@w) system a mathematical framework i8 established for the conduction of tracer experiment8 for 8y8tem identification. of the Transport velocities are ass-unzed known. The problem of unique identij2ation
ABSTRACT:
exdange coefficients between various phases ting boundary rnemuremmte is studied. f3~e.8 of different output-boundary con$gurations are considered (acceasibb and inaocesm’ble outputa). Su$?cient condition for identifiability are given.
I. Introduction We consider a conservative mass flow system in a steady state defined on the subinterval of the real line [u,b]. We shall assume the existence of m different interwoven parallel flows, called phases, each of which at any cross-section, i.e. x E [a, b], is characterized by a velocity f (x, i) m/see for i=l , . . . , m. That is, all particles belonging to a particular phase at crosssection x are moving with the same velocity. In chemical engineering this is known as “infinite radial diffusion” (i.e. a “white noise” type assumption which implies that particles are uniformly distributed within each phase and that present location within the phase cross-section is independent of the past). The phases will be restricted to one of the following three categories: forward, backward and rest. Forward and backward phases shall be called dynamic phases. Rest phases are those for which f(~, i) = 0 for x E [a, b]. If i is a forward (backward) phase f(x, i) > 0 ( > 0) for all XE [a, b) ((a, b]). For the case of forward (backward) phases new material enters the system from the outside at x = a (x = b) and exits at x = b (x = a). For the case of rest phases no new material enters from without. At exits the material is collected in some fashion. We also suppose that at any XE [a, b] there exists a mechanism (e.g. chemical, mechanical) by which particles or material move from phase to phase. This mechanism will be represented by a positive exchange coefficient and which accounts for the fraction of the &(x), whose units are se+
59
David Rappaport and Joshua Dayan material in phase i moving to phase j, i # j, at time t and cross-section xE[a,b]. A common method used to identify the spatial parameters of flow systems is by the introduction of marked tracer material into the system. The dynamics of the tracer material are presumed to be described by a firstorder variation about the steady state. Letting C(t, x, i) denote the concentration of tracer material per unit length of material in i at time t and position x by performing a mass balance we obtain the following set of linear hyperbolic partial differential equations (utilizing the obvious shortened notations) :
aCi
at-
-~(fici)+(j=~i;il,cj)-h,,ci~
hij
=
j_g.+ihij
for i = 1 , . . . , m. Systems described by Eq. (1) are commonly called transport or plug-flow systems. We assume that the following situat,ion prevails: (a) The steady-state velocity profiles f (x, i) for i = 1, . . . , m can be measured along the system. This is the case in many physical systems such as fluidized beds (l), in spouted beds (2) and many other flow systems. (b) Injection of tracer material can be made at a number of neighboring points along the system. At each such point injection is in the form of a “pulse” of high tracer concentration. (c) At boundaries (exits), x = b for forward phases, x = a for backward phases, tracer concentrations are measured (e.g. counts in the case of radioactive material). Phase boundaries are classified as follows: an accessible (inaccessible) phase boundary is one at which the concentration in that phase can (cannot) be separately measured. We treat the cases in which either all the phase boundaries are either accessible or inaccessible, i.e. in the latter only the overall concentration without regard to phase is available. Unless stated otherwise we shall refer to the (in)accessible case as being that for which all phase boundaries are (in)accessible. The case of inaccessible boundaries is by far the most common and most important. We assume that the exchange coefficients &(x) are not measurable and hence unknown. The problem of system identifiability will be defined as the ability to uniquely determine the exchange coefficients on the basis of the input and output measurements described above. Basically the idea is that given the phase velocities and certain concentration histories at the boundaries (i.e. outputs) for each of the various injection points, under what conditions does there exist a unique relation between the system described in Eq. (1) and the outputs. The purpose of this work is to provide a logical mathematical framework for the conduction of tracer experiments and to derive conditions under which results can be meaningfully interpreted given the underlying assumption that the idealized plug-flow transport model is a reasonable description of the particular system in question. By taking a probabilistic view of the tracer experiment rather than the deterministic presented above, in (3) it was shown that the behavior of a
60
Joumal of The Franklin Institute
Tracer Identi$ability
of Multiphase
Transport Xystems
typical tracer particle in the system described by (1) is described by a particular two-component Markov process with discrete second component. This model is specified in the next section along with a summary of pertinent system properties. In Section III a precise definition of the tracer experiment is given, Examples are cited of systems which are not uniquely identifiable from input and output measurements. We use these examples to motivate the class of inputs and conditions for identifiability which we use in the sequel. In the fourth section we imbed the system into a larger class of “extended” systems in order to obtain additional system properties. In Section V we prove a result which states sufficient conditions for identifiability of multiphase inhomogeneous plug-flow systems for both the cases of accessible and inaccessible outputs.
ZZ. The Markov
Tracer Model for an m-Phase
Plug-flow
System
(A) We denote the space of sample functions of the process by R. The letter z denotes the vector [x, y], where x takes values in the closed interval of the real line [a, b], a< b and y takes values in the set of m points Y = (1, . . . . m>. Each point in Y is called a phase; H symbolizes the state space of the process which is the product space [a, b] x Y. Cl is the set of all the functions z(t, w) taking values in H, where t belongs to the semi-infinite interval [0, co) and where y(t, w) is a function in the space of all right continuous step functions on [O, 00) taking values in Y with a finite number of isolated jumps (i.e. changes in value) on every bounded subinterval [tl, tz], 0
w) = f [W, w), $4, w)l
(2)
with the initial condition ~(0, w) = x,, in [a, b]. We shall arbitrarily assume that the phase j for 1
Vol. 300, No. 1, July 1975
2
61
David Rappaport and Joshua Dayan For each w we shall interpret each z(t, w) for t 2 0 as one of the possible trajectories of a tracer particle with the jump times and the phases visited given by the function y(t, w). (B) We take as the u-field of events on Q, denoted 8, the one generated by all finite base cylinder sets of the type {w: z(t,, o) dI, . . ., z(t,, w) EA,) for every finite subset of times ti belonging to [0, co) and for arbitrary events A, belonging to the u-field in H, denoted H, generated by product sets of the form [c, d] x (‘}z w h ere a
$ h,(x) j=l, jsi
and
h,,,
= sup h,(x).
(D) Consider a time stationary strong Markov process defined on (a, /3, P) and state space H where P is characterized by the properties stated below : P(Y~+~ = yl each 7 E [0, CL]1x1 = x, yt = y) = exp
IS
1
t+CX &&,)ds
where 9’s = f(xs, y), s E [t, t + a}, X~= x. In view of the assumption If@, i) I>, fmin, equivalently, (backward) phase and for a
1, Pa)
for y a forward
x2 without jumping 1x1, y) = exp [ - /;Hdv].
(3b)
Further, P(Yt+, = j z i,
one jump on [t,t+a]lxl
= x, yt = i)
where 8, =f(x,G),
rE[t,Sl,
2* =f(x*,j),
pE[8,t+a-j,
x, = x, cc,*= 5,.
Consider the Banach space B of bounded measurable functions (in x) g(x, y) on H with the norm supH 1g(x, y) I. Let D be the subset of B in which g is continuous and has finite piecewise continuous derivatives with respect to
62
Jonmalof
The Franklin Institute
Tracer Identifiability of Multiphase Transport Systems x, with at most a finite number of discontinuities. These derivatives are right (left) continuous at the points of discontinuity for y a forward (backward) phase. For gE D define the operator A by MGY)
=f(l,Y)~(2.Y)-/\uu(Z)9(5.Y)+~_lfiiyn,(x)o(x.j).
(4)
In Eq. (4) for y a forward (backward) phase the derivative is the right (left) derivative. In Eq. (4) we invoke the notational convention that f(x, i) = 0 means that the first term on the right side (i.e. f(ag &r) is zero. A is the weak infinitesimal operator for the process. By Dynkin (4) paragraph 1.15, p. 40, for gE D the expectation E,(gIz,) is continuous from the right for every x,,EH, and in addition the time derivative from the right exists, is continuous for every .z,,EH, and satisfies
=
wherelb,&(g
%Ag Izo),
(5b)
Izo)= s(z,).
It will prove convenient in the sequel to represent the infinitesimal operator in vector form defined below. We shall use the scalar (4) and vector forms (6) of A interchangeably without much comment. Let
and
Therefore, Ag(x) = P(x)
‘2 +r(x)
g(x),
(6)
where we take Ag(x) to mean
A as defined in (4). We remark at this point that for i a forward (backward) phase the state (b, i) ((a, i)) is an absorbing state, i.e. Ag(b, i) = 0 (Ag(a, i) = 0).
Vol.300,No. 1,July 1975
63
David Rappaport and Joshua Dayan 111. Precise
Definition
of the Tracer
Experiment
We now precisely define the tracer experiment : (4 The tracer process is Markovian in nature and its description is given by the Markov process represented by the generator A with domain D. (b) The set of possible initial states (at time zero) for the process will consist of points (2, i) where i = 1, . . . , m and x belongs to a small closed set 1~ (a, b). The interpretation of this is that the tracer pulse may be injected at any of these initial states. In a control context this may be interpreted as the ability to exert a distributed control in an arbitrary small neighborhood of (01,i) where 01E I. The following arrival time distributions (outputs) are available (cl (measured) : (i) For the case of accessible boundaries all the distributions P&x = b, iIxO,j) (P(t,x = a,i lx&) for the case of i a forward (backward) phase i = I, + 1, . . ., I, (i = I, + 1, . . ., m), for all t > 0, all X~EI andj = 1, . . ..m. (ii) For the case of inaccessible boundaries the available distributions (outputs) are 2
P(t,x = b,ilxO,j)
i=t1+1
The problem of identifiability profiles under what conditions (i.e. the exchange coefficients) ? Example
and
$J P(t,x = a,i]z,,j)
for all t>O,
i=za+1
all ~,fl and j = 1, . . ..m. is paraphrased as follows : given the velocity do the outputs uniquely define the process
1
Consider two systems (A and B) defined on [0,5-5771, each composed of two phases. Both systems have the same velocity profiles while the exchange coefficients of I3 are a translation of those of A. Let f(~, 1) = 2 + sinx, and f(~, 2) = 2 - sin x for x E [0,5+57-r].Let X&(x) and vi(z) be as shown in Fig. 1.
FIG. 1.
Ley A;(X) = X#X - 2n) for 47-r6 5 d 5~r and zero otherwise on [0,5-5771. We can easily show that for each X~E [0,2r] that for both the cases of accessible outputs and inaccessible outputs the two systems are indistinguishable on the basis of the input-output measurements described above. However, if we were to allow for initial states in an arbitrarily small neighborhood of 01= 2x, for X~EI, x0 > 27r, we see that the tracer particle
64
Journal
of The Franklin
Institute
Tracer IdentiJiability of Multiphase Transport Systems entering system A no longer sees on his right exchange coefficients which are simply a translation of those of system B. This is, in fact, the ciux of the matter ; for a system to be identifiable the exchange coefficients in the region I on which the inputs are distributed must have a functional relationship t’o the exchange coefficients in other parts of the system; in particular, we shall further on prescribe analyticity. In the next example it is demonstrated that some restriction must be placed on the relation between the velocity profiles of a given system in order to ensure identifiability for the case of inaccessible outputs. Example 2 Consider a two-phase
system with forward phases. Let f(x) = f(G 1) = f(G 2),
all x E [a, b]. Let &(x) and &(z) be arbitrary. structure of the sample space implies that
Then for any x0 E [a, b] the
P(t,x~b?lxo,j)= p(P[x,,+/;.j-(x&ds]), where
% =f(x8),
x,I,,~ =
x0
and
&?) =
1, pa0 0,
B
Obviously in the event of inaccessible outputs no information about the exchange coefficient is contained in the outputs. IV.
The Extended
whatsoever
System
Defined in this section are systems on the real line subinterval [c, d] where c ,< a < b < a?, the “projection” of which on [a, b] is the original system which will enable us to explore further properties of the original system. Consider the m phase Markov system on [c, d] processing all of the properties given in Section II. The parameters of this system will be identified by the symbol “. The system on [a, b] will be distinguished by the notation used until now. We shall assume that for: (a) Rest phases: &#) = h,,(x) for i,j pairs and for all XE [a, b]. i&(x) arbitrary, continuous on [~,a] and [b,d]. (b) Forward (backward) phases : (1) .h,i) =f(x,i) for x~Ca,b) (bbl); (2) f(b, 4 = lim,&(x, i) (fb, i) = lim,J(X, 9); (3) &i(X) = &j( x ) f or x E [a, b], for both forward and backward phases ; (4) Aii(x) = 0 on (b,d] and [~,a). Summing up, on the interval [a, b] the velocity profiles and the transfer coefficients are identical for both systems. On the intervals (c, a] and [b, d) the velocity profiles of the extended system are continuous extensions of those of the original. On theee latter two intervals the transfer coefficients for
Vol. 300, No. 1, July 1975
65
David Rappaport and Joshua Dayan dynamic phases of the extended system are zero whereas those for rest phases are continuous extensions of those of the original. Thus, exchange is prevented outside of the interval [a, b]. Note that the transfer coefficients of the extended system as chosen are right and left continuous, respectively, for the cases of forward and backward phases. Thus the extended system possesses all of the required system properties, and one identifies with it the generator whose form is given in Eqs. (4) and (6). We remark that the imbedding of the original system in the above class of extended systems is possible by virtue of the fact that the probabilistic plug-flow model allows for the existence of internal discontinuities in the transfer coefficients. This was verified in Ref. (3). the Consider the following two results which relate probabilistically extended system to the original. Lemma 1 Let (x,, y,,) be a state of the original system, then for i a forward (backward) or rest phase, for each a, p such that a < 01< /3 < b (a < 01< /3c b) and for all t>o:
P(t,a
= ilxo,yJ =P(t,a
= ilzg,yJ, = P(t,ol
= ilx,,y,)).
Theorem 1 For all t > 0, for each i, and (x,,, y,,) a state of the original system P(t,
x 2
b, y = i j x,,,
ZJ,,) =
P(t,
x =
b, y = i 1x,,, yo)
(74
and B(t,z
= il2,,y,)
= P(t,x
= a,y = ilx,,y,).
W)
Rigorous proofs for these intuitively obvious results are available in Ref. (5). The basic idea is that for every initial starting point (x0, y,), a state of the original system, there is a one-to-one correspondence between the trajectories of the original and the extended systems. The trajectories are identical until an absorbing boundary of the original system is reached. In the original system the trajectory remains fixed at the barrier thereafter. In the extended system the trajectory leaves the confines of the original system never again to return. Note that by Theorem I and the path properties for t 2 0, b
= b,y = ilx,,y,),
where t,, 1 = 1,2, are the unique times satisfying xl
=
b+
s
:If^(x,,i)ds.
If t - t, < 0 the corresponding probability is taken to be zero. A similar result exists on the subinterval [c, a]. Hence, there is a one-to-one relation between
66
Joomal
of The Franklin
Institute
Tracer IdenliJiability of Multiphase Transport Systems the boundary arriva,l time distributions associated with the original system and with “spatial” probabilities, for a fixed time (t), on the subintervals [c, a] and [b,d]. In particular “spatial” discontinuities on the subintervals [b,d] and [c, a] corresponds to “time” discontinuities in the boundary distributions of the original system. Therefore, by using the arrival time distributions associated with the original system we can evaluate expectations associated with the extended system of the form, &(gIz,, yO) where g(x, y) = 0 for x E [a, b]. In the sequel it will be convenient to use a particular form of the extended system. Below we shall classify the various cases and state their properties. (1) Case of accessible outputs If i is a forward (backward) or a rest phase” let f”(z, i) = limzraf (2, i) (lim ,,,f(z,i)) for all x~[b,d) ((~,a]). Also let f&i) = f&i) (f(b,i)) for x E [c, a] ([c, d]). The extended system exchange coefficients are given at the beginning of this section. (2) Case of inaccessible outputs We extend the velocit,ies of the system in two stages. Choose an arbitrary velocity v> 0. Using elementary arguments one can show that on the subinterval [b, b + p] ([a-p, a]), where p > 0 is arbitrary, that it is possible t,o “accelerate” all forward (backward) phases to velocity v (-v) on the subinterval [b,b+p] ([a-p, a 1) such that the time At (also arbitrary) it takes to traverse this subinterval is the same for each phase. In addition, for i a forward (backward) phase we choose f(x, i) constant and continuous on [a - p, a] ([b, b + p]). Thus after the first stage we have a multiphase system for which the output velocities are the same for all forward (backward) phases. We then extend the system to [c, d] from [a-p, b + p] exactly as in the case of accessible outputs. Notice that here this means that for all forward (backward) phases for 2 E [b + p, d] ([ c, a - p]) the phase velocity is v ( - v). We extend the exchange coefficients as described at the beginning of the section. Clearly from the above description, if i is a forward (backward) phase, p and At as above, for x0 E [a, b] by the strong Markov property :
= ilx,,y,)
P(t-At,x>b,y
= &,z2b+P,y
(fi(t-At,x
= ilz,,y,)
= il x,,,y,,), = p(t,x
= ijx,,y,)).
In the light of Theorem I and the above chosen extended system we shall interpret the description of the experiment as stated in Section II as follows : (1) For the case of accessible outputs availability of the distributions: P(t,z
= 6 y = ilz,, yO) (P(t,x = a,y = il~O,yO))
for each x0 E.Z and each t > 0, for i a forward (backward) phase, will be understood to mean that for each integrable g(x, y) for which g(x, y) = 0 for
Vol. 300, KGO.1, July 1075
67
David Rappaport and Joshua Dayan x E [a, 61, the expectation $,(g[ x0, y,,) can be calculated for t 2 0, X~EI. If further g(x, y) E Da then also the partial derivatives of the expectation with respect to t and x,,, x,, E I, are calculable. (2) For the case of inaccessible outputs the availability of the dist,ributions P(t, x = b 1x,,, y,,) and P(t, x = a 1x,,, y,,) for each x,, EI and each t > 0 shall be understood to mean that for each integrable g(x, y) for which g(x, y) = 0 for XE [a- p, b + p] and for which g(x, i) = g(x,j), for all i # j, the expectation 8,(g 1x0, go) can be computed for t >, 0, X,,E I. If in addition g(x, y) E Dg then also the partial derivatives of the expectation with respect to t and x,,, z,,EI are computable, The essence of the interpretation stated above is that the availability of the output distributions implies the availability of the solution to the partial differential equations generated by the weak infinitesimal operator.
where t > 0, x~EI, g EDa_, g(x, y) = 0, for x E [a, b]. In the event of an inaccessible output system we further require that g(x,j) = g(x, i) all i,j. As &(g] x,,, y,,) is a functional of the arrival time distribution the identifiability problem can be paraphrased as follows: “Given a set of solutions on the boundary of a plug-flow system under what conditions does this set uniquely specify the unknown system parameters”. V. Identifiability
In this section we present the main result, Theorem II, the identifiability theorem, which gives conditions under which the transfer coefficients are uniquely related to the output distributions. Establishment of this result utilizes two lemmas which are stated prior to Theorem II. Lemma 2 is used to prove Lemma 3, while Lemma 3 is needed in the proof of Theorem II. Essentially, the lemmas maintain that under certain conditions the only discontinuity in the process probabilities, i.e. P(t, z < CL, y = i ]x0, y,,), is that which propagates from initial point (x,, yO) through phase y,, in accordance with the respective phase velocity. That is to say, these conditions preclude the transference of discontinuities from one phase to another, or to another trajectory in the same phase which is a time shift of the one above commencing from (.x0,y,,). Physically, these conditions mean that if two particles enter the system at the same time, one in phase yl, the other in phase yz Z yl, and each propagates through its respective phase without jumping to another, then the t’otal time that these two particles spend abreast of one another is zero. For each t,>O, cuE[a,b] and i~[l, . . . . m], we shall denote the trajectory starting from position a, at time t,, generated by the dynamics of phase i by x$,(4
ilf(x:,r.(4, i) d7, t>t,. s The time t, should not be confused with the initial state (x,, y,,) at time zero.
68
= a+
Journal of The Franklin Institute
Tracer IdentiJiability of Multiphuse Transport Systems For the original system we define the field of characteristic curves (6) of phase i on the rectangle [a, b] x [O,co) in euclidean two space, to be the collection of the following curves, each of which is specified by an initial point on the boundary of the rectangle: (a) For t, = 0, olE [a, b], and i = 1, . . . . m: q$(a) = a+ (bl)
‘j’(&,
s0
t20.
i)ds,
For toE (0, co), 01= a, i a forward phase: z;,,(a) = a +
lf(~4,,~,i) d7, s 10
t > to.
(b2) For tog (0, oo), (Y= b, i a backward phase: q#4
= b+
s
i:f(+,bl i) d7,
tat,.
Note that as a result of the time invariance of the velocity profiles, the characteristic curves of phase i are all parallel to one another, for each i. We shall denote the time of arrival of a trajectory at boundary b (a), for i a forward (backward) phase, as follows: T;(a) = mi~{t:
z&,(cY)= b}
min (t : &(Ix) = a} . >
l&l
Lemma 2 Consider any two characteristic curves, one of phase i, the other of phase j, i # j. Suppose that the set of points in time at which these two curves intersect one another is of Lebesgue measure zero. Then for any dynamic phase 1, any to> 0, j # 1 and ~!,~(a) a characteristic curve of phase 1, w,x = &&4, Y =jl x0, k) = 0, i.e. with respect to Lebesgue measure for all t such that to < t < T;(a), for k = 1, . . . . m. Remark. The condition in the lemma implies the existence of no more than one rest phase. Note that this lemma eliminates systems such as that of Example 2. Lemma 3 Let i be a Suppose that continuously respectively.
forward or backward phase. X:,&(Y) is a trajectory in phase i. the velocity profiles of each forward and backward phase are differentiable, with bounded derivative, on [a, b) and (a, b], Suppose that the Lebesgue measure of the set (x: f(z, i) = f&j),
x E [a,
al, i
# j}
is zero for all pairs i # j. Then for all k, P(s, x = x$~(cx), y = i 1x0, k) = 0, for all s E [to, TiO(ol)),when (1) i # k, for any OL E [a, b] and to > 0, i.e. for any characteristic curve of phase i, or when (2) i = k, for (01,to) not lying on the characteristic curve x~,(x,), t 2 0.
Vol. 300, No. 1, July 1975
69
David Rappaport and Joshua Dayan The proofs of Lemmas 2 and 3 are summarized Ref. (5) for the complete proofs.)
in the Appendix.
(See
Theorem II (identi$ability) Let F(x) be known and continuously differentiable with bounded derivative on (a, b). Let I’(x) be an analytic function on (a, b). Suppose X~EI. Let there be no more than one rest phase I, and for which h,(x,) # 0. Assume that the set {x: x E [a, b], f(~, i) = f(z,j), i f j} has Lebesgue measure zero for all i # j. (a) Then for the case of accessible outputs, l?(x) is uniquely identifiable from the output distributions. (b) Suppose for the case of inaccessible outputs that Ti(x,,) # Tg(x,,), for each pair i # j where either both are forward or backward phases. Then l?(x) is uniquely identifiable from the output distributions. Remarks. The analyticity assumptions are motivated by Example 1 in which is illustrated that in the absence of a functional relationship between the parameters of the input area and those of the rest of the system the system is not necessarily identifiable. The assumption that X,(x,) # 0 for 1 the rest phase ensures that phase 1 “communicates” with other phases at x0. The arrival time assumption in part (b) of the theorem prevents the build-up of parallel discontinuities in the arrival time distributions for the different phases as occurs in Example 2. Note that the input region I c (a, b) is arbitrary except for the conditions on the rest phase and the arrival times for the inaccessible case. Proof. For the purposes of the proof we shall assume the existence of two non-identical analytic functions on (a, b), I’,(X) and l?,(x). We shall assume that for the associated systems with F,(X) = F,(x) that all the corresponding output distributions are identical. We first tackle the case of accessible outputs. Differencing Eq. (8) with f,, with Eq. (8) with f,, by the definition of the experiment, we obtain that for t 2 0 [fl(%)
-
fk%)l~t (91x0,Yo) =
0.
Since this holds for each g as defined in Section IV and as ri(xo) = f,(x,), i = 1,2, it follows that for t>O [r,(z,)
- I?&,)] g(t, II:= a or b, y = k 1zo)=
0t
(9)
for each forward or backward phase k. (Note that for k a forward (backward) phase P(t, x = a, y = k 1zo) =0 (P(t,x=b,y=k~z,)=O)fort>O.Seecase(2) in proof of Lemma 3 (Appendix) for detailed explanation.) Suppose l?l(xo) # l?Z(~o), i.e. there exists at least one pair i, j, i # j, such that hi;)(z,) # h$‘(~,). Now consider the sum corresponding to the ith component of Eq. (8), i.e. for k either a forward or backward phase
f) [A$)(x,) -X$)(x,)]
P(t, z = a or b, y = k) x0,j) = 0.
(10)
J=l
is t-spose ofthe vector
t P( -, . , -1%) the
70
[P( . , . , . 1x0,l)r. . .
p( . , ., .lx,,
m)~.
Journal of The
Franklin Institute
Tracer IdentiJiability of Multiphuse Transport Systems Let a, = h$‘(zo) -A$)@,), j = 1, . . . . m, and partition these coefficients positive (a+) and negative (a-) subsets. For k either a forward or backward phase we rewrite (10) as za,fP(t,x
into
= a or b, y = kjx,,,j) = Ca;P(t,x a-
t>O,
= a or b,y = k/x,&,
(11)
where a7 = - aj for j E a7 and a: = aj for j Ea+. Subcase (1) : one side of (11) empty (say a-) Suppose a: # 0, where (Y is a dynamic phase (say forward). Hence, P(t, z = b, y = CY 1x0,CX)= 0, all t 2 0. By Eq. (3a) we obtain an immediate contradiction. Thus a+ = 0. Let ai # 0 where aris the rest phase. Let 1 be a dynamic phase (say forward) for which h,(z,) # 0. By (11) P(t, x = b, y = I) x0, I) = 0 for all t > 0. By (3~) P(one jump only to I on [O,At]] z,,, IX)> 0, i.e. P(At, y = Zlx,,, a) > 0. Let &,=o, be the identity function of the set [y = I]. By the Markov property for T > (b - o)/f,i, P(T + At, 2 = b, y = I) z,,, a) > EAt(&,=rl P(no jump on [At, T] Ix, y) Ix,,, CL) > e=p ( -Lax Contradiction!
T)P(At,y
= Z~x,,,or)>O.
Hence a+ = 0.
Subcase 2 : neither side of (11) empty Clearly there exists a., # 0 where Z is a dynamic phase (say a,Ea+ and Z without loss of generality is a forward phase). Consider the system trajectory z&(x,) for t 2 0. Tb(x,) is the arrival time at x = b of the characteristic curve 4,&J [i.e. ~+c~~o,,&~) = bl. Note that by (3b) the probability P(u, x = b, y = Z(x,,, a) has a discontinuity at u = Tb(x,,). Therefore, by (11) this means that there exists an 01~a-, a # 1 for which P(u, 2 = a or b, y = Zlz,,, a) has a discontinuity at u = Th(x,). Designate this latter discontinuity by the notation AI. Recall that by Lemma 3, part (l), for every As > 0, P(T;(x,,) -As,
x = x”TO’(zOo)-*s,O(~O)~ Y = ZI5>4
= 0.
By the sample path properties, the Markov property and Eqs. (3(a)-(c)) A, G W,(x,)
-As, x = ~&,~~zOo)--s,o(~o), Y = ZIx0, 4 + m&,,
As +- @As),
where the last two terms on the right side represent bounds on the probability of jumping on the time interval As. As As is arbitrary it follows that AI = 0. Contradiction! Thus, a, = 0 and for the accessible case rI(xo) = I’.Jx~). We turn now to the inaccessible outputs case. By Eq. (8) and the definition of the inaccessible case we obtain that for t > 0 [I’,(x,) - l?,(x,)] P(t, x = a or b 1zo) = 0.
Vol.300,No.1,
July 1975
(12)
71
David Rappaport and Joshua Dayan Using the same reasoning and notation conclude that
used to obtain Eq. (ll),
by (12) we
a; aif P(t, x = a or b, j xO,j) = x a7 P(t, z = a or b j xo,j). a-
(13)
Subcase (1): one side of (13) empty (say a-) Suppose a: = 0. This means that WL CP(t,z
= b, y =j(x,,~~)+P(t,x
= a, y =jlq,,ci)
= 0
j=l for all t > 0, i.e. each term is zero. Using the same reasoning as in the proof of subcase (1) in the accessible case we conclude that a: must equal zero.
Subcase (2); neither side of (13) empty This implies the existence of an 1 such that a, # 0 (say ale a+), where b, without loss of generality, denotes a forward phase. Hence, P(u, x = b 1x0, a) has a discontinuity at u = Tb(x,). Consequently, on the right side of (13) there exists an a,Ea-, a # I for which P(u, x = a or b j x0, a) also has a discontinuity at u = Tb(x,). It then follows that for some j, P(u, x=a or b, y = j 1x,,, a) has a discontinuity at u = Tb(s,), which we shall denote by Ai. If j = 1we reason as in subcase (2) for the accessible case that a, must be zero, If j # 1 we assume without loss of generality that j is a forward phase [i.e. P(u,x
=a,y
=j[xO,a)
= 0
for u > 01. From the trajectory properties it follows that there exists a unique DE [a, b] and a unique toe [0, Tb(x,)], defining the characteristic curve x,{,~(v), s > t,, for which x&(Z) t,,(v) = b [i.e. the unique characteristic curve of phase j for which the arrival ‘time at b is TA(x,)]. Reasoning as in subcase (2) of the accessible case we obtain that for any As > 0 Aj < P(Tb(x,) -As,
X: = ~~o~~so~-~s~o(~), Y =j 1x0,a) +mX,,,As
+ o(h).
For j # 01we conclude immediately by Lemma 3, case (l), that the probability on the right side of the inequality is zero. Forj = 01we note that by hypothesis !!‘;(~a) # Tb(x,) for all I # j and consequently (v, to) is not on the characteristic curve x$(x0). Thus by Lemma 3, case (2), the probability on the right side of the inequality is zero. As As is arbitrary we conclude that Aj = 0. Contradiction! Hence, a, = 0 and we conclude that rl(xo) = r&x,,). Continuing we note that the output distributions are available for all xOeI. Thus, for the accessible case I’(Z) is uniquely identifiable, for each x0 E 1. By analyticity we conclude that r(x) for x E (a, b) is uniquely specified by the output distributions. Secondly, T:(x) is continuous in XE (a, b) for each i. Hence, for the inaccessible case there exists a closed neighborhood I* of x,, in which T:(x) # T;(x) all i f j either both forward or backward phases. Consequently, I’(x) is obtainable from the output distributions for x ~ln I*. By analyticity we conclude that r(x) for XE (a, b) is uniquely specified by the output distributions.
72
Journalof
The Franklin Institute
Trucer Iclenti$ability of Multiphase Transport Xystems Discussion
It is quite reasonable to suggest that the conditions imposed in Theorem II are not the weakest possible under which identifiability is possible for some other class of transport systems. However, the problem is to find a common factor describing a class such as the restrictions on the velocity profiles as imposed in Theorem II. From a practical point of view these restrictions are not unreasonable and are applicable t,o a large number of systems. It can be similarly shown that for the case in which some of the phase boundaries are accessible and some inaccessible that the system is identifiable under the conditions imposed on J’(x) and I’(x) in Theorem II plus the additional requirement that Tk(x,) # T&(x,) for i # j, both either forward or backward phases with inaccessible boundaries. Note that the “solution” of the problem posed herein requires that an infinite number of experiments be conducted. Work is continuing in the means for the actual calculation of the direction of finding a “practical” unknown exchange coefficients. We once again reiterate that the purpose of this work is to provide a mathematically logical framework to indicate how to conduct tracer identification experiments of multi-phase plug-flow systems and under what conditions the results obtained can be meaningfully used. References (1) T. Mamuro and H. Hattori, “Flow pattern of fluid in spouted beds”, J. Chem. Engng Jap., Vol. 1, No. 1, pp. l-5, 1968. (2) H. Kabayashi, F. Arai and T. Chibe, “Behavior of bubbles in a gas-solid fluidized bed”, Kag. Kog. (J. Chem. Engng Jup.) (English Ed.), Vol. 4, No. 1, p. 147, 1966. (3) D. Rappaport and J. Dayen, “A probabilistic model for tracer distribution in multiphase spatially inhomogeneous transport systems”, J. Stat. Phys., Vol. II, No. 6, 1974. (4) E. B. Dynkin, “Markov Processes”, Vol. I, Springer, Berlin, 1965. (5) D. Rappaport and J. Daysn, “Tracer identifiability of spatially inhomogeneous multiphme tranxport systems”, EE Pub. No. 205, Technion,;SI\srael Institute \ iof Technology, Haifa, Israel, Aug. 1973. (6) R. Courant and D. Hilbert, “Methods of Mathematical Physics”, Vol. II, Interscience, New York, 1962.
Appendix (A.l)
Proof
Let 6>0.
of Lemma 2 Let B = (8: P(s,x
= &(a),
y = j Ixo,k)d6,j
# I, to
Without loss of generality we shall assume 1 to be a forward phase. From each pair of points in the plane [s, x&&x)] w h ere 8 E B, consider the plane curve generated by the points (t, zf) where XT = ~!,~[~:,~,(a)1 and t > s. These curves are segments of characteristic curves of phase j. It can be shown by the Markov property and Eq. (3) that all the pairs [a, z&(a)], where 8 E B lie on a finite number of distinct curves. Let these curves be numbered one through n. Each curve commences from some point [8i,z:,,tr(~)]
Vol.500, No. 1.July 1975
73
David Rappaport and Joshua Dayan n. On the ith curve there lie points {a, ~f~[~&,(a)]} where a E B But by the hypothesis the set of points s~!l’:~(a) at which &,b&,t.(41 = 4,,,[4&, (a)] has Lebesgue measure zero. Since the number of such curves is finite and B is in the union of such points on the n distinct curves, it follows that B is a subset of a set of Lebesgue measure zero. As 6 is arbitrary the result follows. wheresiEB,i=l,..., and {s, z$,[z:~,+(cx)]}.
(A.2)
Proof
of Lemma 3
Without loss of generality we shall assume i to be a forward phase. It can be demonstrated by the Markov property and Eqs. 3(a)-(c) that the probability p(a, x
y = i
120, k)
is absolutely continuous in s on [to, TtO(a)- AT] for arbitrary AT > 0. Hence its derivative with respect to s exists, ae., and is integrable on [to, T&u) - AT], i.e. for AT> 0 (neglecting the OLnotation) : et
X2$.
d X < x:+A~,ta~
!,
=
i 120,
k)-P(to,ol~x
=
i Izo,
k)
=
=i 1X0,k) d8. J~$P(8,xf,,,~x
(Al)
0
It can further be shown that the integrand is bounded uniformly, liIIl 38 AT&O d8
, Xi#,tO< X < Xf+A7,to>
!,
=
i 1 xO~
k)
G
mhmXj~zp~8~
x:,to’
a.e., and that Y
=
j
1 xO?
ffl)
GW
(Zi
for a E [to, Tb(ar)], a.e. By the lemma in the Appendix, haragraph (A.3), the hypothesis of Lemma 3 implies that of Lemma 2. Therefore, by Lemma 2, the right side of (A2) is zero for 8 e [to, Tj@)], a.e. Applying the Lebesgue dominated convergence theorem and stochastic continuity to (Al) as AT + 0 we obtain: P[u, 2 = x;,&),
y = i
j 20, k] - P(to, z = a, y = i Izo, Ii) s 0
(A3)
for all u E [to, TQ(a)). Two cases: (1) to = 0, aE[a,b]: For i # k we conclude by stochastic continuity that P(0, x = 01,y = i 1x0, k) = 0. For i = k we recall from the hypothesis of Lemma 3 that for to = 0 cy. # x0. We thereby arrive at the same conclusion. (2) t,>o, a= a: Recall that z. E [a, b]. Recall that i is a forward phase. From the structure of the sample space it is evident that the only paths that arrive at time to at state x = a, y = i, must first reach z = a, y = j, where j is a backward phase. But x = CL,y = j, is an absorbing state. It then immediately follows from the strong Markov property that P(t, x = a, y = i 1x0, i) = 0 for all t > 0. Hence, P(u, xf,,,(a), y = i 1x0, k) = 0 for all u E [to, TQ(ol)). (A3) Lemma Let f (2, 1) and f (x,2) represent two forward phase velocities. continuously differentiable with bounded derivatives on [a, b). Let A = {x: 2 E [a, 61, f
Suppose
each is
(x,1) = f (x,2)}.
Let xi and x,“, respectively, denote two arbitrary characteristic curves, x&,(ol,) phase 1 and &a(Z) (CGJof phase 2. Let B = {s: xi = z,” < b}. If the Lebesgue measure of A is zero, then the Lebesgue measure of B is zero. Proof:
74
of
Standard arguments.
Journal of The Franklin
Institute