Phase-plane solutions to some singular perturbation problems

JOURNAL

OF JL%THEMhTICAL

ANALYSIS

AKD

54, 449-466 (1976)

APPLICATIONS

Phase-Plane Solutions to Some Singular Perturbation Problems* R. E. O'mLLEY, Department

of Mathematics,

University

JR.

of Arizona,

Tucson, Arizona

857-71

Submitted by h’enneth I.. Cooke

1Ve seek asymptotic solutions of the two-point problem &S +f(s) = 0, 0 < t S: 1, x(O), x(l) prescribed, as the small parameter c tends to zero. In the (s, G) phase plane, solutions generally tend to remain at rest points (x, 0) corresponding to maxima of the potential energy V(s) = j’.f(s) ds. As E -t 0. we (roughly) find that (i) no solutions of bounded variation exist if either esceeds all finite maximum values of I,’ (if any); (ii) if I,’ has a maximum at b with I’(x) max(.s(Ol. -v(I)) < 6, a solution tending to b within

I-(x(O)) or V(x(1))

I T-(b) for .I .-I b and (0, I) exists;

(iii) if T’has a finite number of maxima between x(O) and s(lj, there is a unique monotonic solution which remains at these maximum values; (ix-) if s(0) and ~(1) are contained between successive maximum points d, and d2 of Iv with I-(d,) == Lw(d?), there are denumerably many solutions xvhich switch between the values d, and d2 ; (v) if X(O) and x(l) are contained between successive points e, and e, with Ia := Ir(c,) where e, , but not e3, is a maximum point of V, there are denumerably many solutions with limit c1 almost every-where in (0, 1).

I. INTRODUCTIONAND SUMMARY OF RESULTS II’e shall consider the two-point 2.e c f(s)

problem = 0,

x(O), x( 1)

O,(t<-C, prescribed,

(1)

where the dot represents differentiation with respect to t, E is a small, positive parameter, and f is a differentiable function for all s. We can think of this autonomous equation as the dynamical equation of motion of a nonlinear * This work was supported in part by the Office of Naval Research under Contract Number NOOO14-67A-0209-0022.

449 Copyright I? I976 by Academic Press, Inc. .\I1 rights of reproduction in any form reserved.

450

R. E. O’MALLEY,

JR.

spring with spring constant large compared to the mass. Alternatively, McLaughlin [12, 131 relates such two-point problems to critical paths of Feynman path integrals. It is a singular perturbation problem because the order of the differential equation drops when l becomes zero (cf., e.g., [19,

1511, We note that every root off(x) = 0 is a solution of the differential equation of (1) and therefore might expect the limiting solution within (0, 1) to tend to certain zeros off, possibly jumping from one to another from time to time. We must expect some problems to have no solutions, while others have nonunique (possibly an infinity of) solutions. For example, it is easy to verify that the linear problem with f(x) = x has no solution for all E sufficiently small unless x(0) = X( 1) = 0. Likewise, the problem has the unique constant solution when x(O) = x(1) and f(x(0)) = 0. Let us introduce the new variable

and study problem (I) in the X-Z phase plane. There the trajectories are parameterized by their constant energy E while conservation of energy requires that

(s/2) + V(x) = E

(3)

for potential energy V given by

We note that it is simple to obtain the trajectory with energy E through any point of the phase plane (cf. [14, Chap. 21). One first draws the x - V(x) profile and then graphically draws the trajectory (x, Z) with

z = &(2(E - V(x)))l",

(5)

extending it as long as z remains real (i.e., until V(x) esceeds E). Such trajectories can be infinite in both x directions, semi-infinite, or bounded. Along any solution path of (l), then, the energy E must be no less than the maximum of the potential energy V(x) between x(0) and x(l); i.e., there is a minimum energy E, required to cross the potential barrier. Also note that zeros off are critical points of V and that the maxima (minima) of V correspond to rest points (x, 0) which are saddle points (centers) in the phase plane. We will find that solutions of (1) tend to remain at x values corresponding to such saddle points. Finally, we observe that

SOLUTIONS

TO PERTURBATION

PROBLEMS

451

E = F(d) along any trajectory through (2, 0) with k a maximum point of I’ so z = +(2( V(5) - V(x)))‘/2 = * (2 ITSf(s) ds)li2

(6)

there. Our results are summarized in the theorem which fohows. The five parts of the theorem are separately discussed (and illustrative examples are given) below. Readers might then find the presentation easier to comprehend by separately studying these statements and the related discussion. THEOREM.

Consider the problem e2.eif(x)

= 0,

0 < t -< 1

4% x( 1)

prescribed.

(1) If V(x(0)) OY V(x( I)) exceeds V(a) for allfinite nzaxinzumpoints a of V, then (*) has no solution of bounded x variation for c tending to zero.

(2) If b is a finite maximum point of V such that I’(x) < V’(b)for x < b and max(r(O), x(l)) < b (or for b < x and b < min(x(O), x(l))), then (*) has a solution x(t, 6) such that x(t, 6) + b as t ---f 0 within 0 < t < 1. (3) If the set max C = (c : V(c) = E, = r(o)s+sr(l) 64~ 40) G c d X(l)? f(c) = 01 is finite and nonempty, then there is a unique monotonic solution x(t, 6) of (*) whose limit as l - 0 is a monotonic step function with values in C. If f’ < 0 on C, the amount of time asymptotically spent at any c E C is inoersecvproportional to (-f’(c))‘/” (with this time halved if c == x(O) or x(l)). (4) Suppose d, and d2 are successiveJinite maximum points of V(x) such that V(d,) = V(d,) and dI < x(O), x(l) < d, . Then, for every integer n 3 0, there are two solutions of (*) with limiting solutions X,,(t) which switch n-times betweenthe values dI and d, . The total time with X&t) = d, is

(-f

(-f ‘(dW”

‘(dW2 + (-f

‘(dW2

while that with X0(t) = d, is

(--f’&W2 (-f’(4W2 + (-f’WY2 provided f ‘(di) < 0, i = 1, 2.

452

R. E. O’MALLEY,

JR.

(5) Suppose e, is a jinite maximum point of V(x) and that e2 > e, is the next finite x value such that V(e,) = V(e3. If e2 is not a maximum point of V and e, < x(O), x(1) < e2 , then for every nonnegative integer n, there are four solutions of (*) with a limiting solution X,,(t) such that

within (0, 1) except that

X&/n)

= e2,

m = 1, 2 ,..., n -

1.

The opposite conclusion follows ;f e2 is the makmum point instead of e, .

Remark. In the theorem, we treat extreme righthand (lefthand) values of x on trajectories as maxima of V if they are inflection points where IT is increasing (decreasing).

_----

EXAMPLE 1. Suppose the potential is as pictured in Fig. 1. Here, the second statement implies a solution with limit b in (0, 1); the third statement implies a unique monotonic limiting solution with steps at ci , and c,; the fourth statement implies denumerably many solutions switching back and forth between dl and d2; and the last statement again implies an infinity of solutions with limit e, except for “jumps” to ez . Analogous problems for partial differential equations would have to be studied by entirely different techniques. They would be of considerable interest, however, both mathematically and physically. Limited results are reported in [7, lo], for example.

SOLUTIONS TO PERTURBATION

2. STATEMENT

PROBLEMS

453

I

For any solution path T of (1) through x(0) and X( 1) of bounded x variation, we must have 1 = E 1”“’ (&Z(S)) do)

(7)

for is” = E - V(r) (E a constant) since dt = c(dx/z). For E-+ 0, s must become small along T in a manner which makes the integral J’(d.r/z(s)) unbounded. Thus, while passing from x(O) to x(l), z” must asymptotically have at least a second order zero while remaining nonnegative. This would require E to be nearly V(a) for some maximum point a of IJ-.This trajectory could not, however, extend from x(O) to x(l) since E > (V(x(O)), V(x( 1))) > ?;(a) by assumption. Nonexistence for small E follows if V has no finite maxima. -4 solution could exist if x(1) were an inflection point where I’ is increasing.

V(x) t

E,= I

FIGURE 2 EXAMPLE 2. Consider e2~-/- x = 0, X(O) = 0, x(1) = 1. The potential energy is plotted in Fig. 2. We note that the minimum energy required to move from x(0) to a(l) is E, = 1, which features the “time of flight” (on a monotonic path)

T=r

’ I o (2(1 -f&l’2

E =-+
For ci = 2/77(2j + l), note that there is no solution. In general, it is clearly necessary that the solution oscillate indefinitely and path length be 0(1/e) in order to have a solution.

454

R. E. O'MALLEY, JR.

3. STATEMENT 2 EXAMPLE 3.

Statement 2 is illustrated by the linear problem &

x(0) = 1,

- .t” = 0,

X(1) = 2.

Its unique solution

x(t, E) =

(

1 - 2e-11’- &-*I( ) e-tiE + ( ; 1:;;;: 1 - e-2/c

) e-‘l-t’;E

tends to zero as E-+ 0 within (0, 1). W e ob serve, however, that zero is the unique maximum point of the potential energy V(x) = - fX s as = -(X2/2) + *. '1

No proof of Statement 2 will be given, since it follows more easily than the other statements which will be proved. We note that unique solutions occur under appropriate restrictions (cf. [II, 31). I n p art’KU1ar, uniqueness holds iff(b) = 0, f’(x) < m < 0 for all N while existence would follow from Statements 2 or 3.

FIGURE 3

4. STATEMENT 3 EXAMPLE 4. We consider the problem with the potential energy profile as in Fig. 3. Here, x(0) < - 1, x(I) = 3, C = {-I, 2, 31, V(- I) = V(2) = V(3) = Et, , whilef’(1) = -4, f’(2) = - 1, andf’(3) = -9. Statement 3 implies that there is a unique monotonic trajectory between x(0) and 3 with the time spent at - 1,2, and 3 being asymptotically in the proportion l/41/2: l/11& l/2: (l/9)‘/“.

SOLUTIONS

TO

PERTURBATION

PROBLEMS

455

Thus, this solution satisfies

0 < t < 3;‘10,

s(t, 6) -+ - I)

- 2,

3/10 < t < 9,110,

- 3,

9,‘lO < t : [ I,

XI

32-

f

I l/2

I

*t

-I X(O)FIGURE

4

as E -+ 0. The trajectory is pictured in Fig. 4 for some small 6 > 0. There is an initial boundary layer jump from x(O) to - 1 at t = 0 and interior nonuniformities (transition layers) at t = 3/10 and t = 9/10. M:e note that the energy along this trajectory is asymptotically equal to the minimal energy E,, required to go from x(O) to x(l). Further, relatively more time is spent at the flatter maxima (x = 2) that at the more peaked ones (s =:- - 1, s = 3). As Fig. 1 indicates, other limiting solutions which are not monotonic are often possible. Thus iff(6) = O,f’(6) < 0, and V(6) > E,, in Fig. 3, there is also a solution with only endpoint nonuniformities (cf. Fig. 5 and Statement 2). Proof.

Since C is nonempty,

the trajector!

passes through (c, 0) for some c E C and extends at least from x(0) to x( 1). Moreover, since T, passes through rest point(s), transit time from x(0) to S( 1) along To will be infinite. For trajectories above T,, with energy E > E, L with increasing E (since f,‘(c), transit time decreases monotonically

456

R. E. O’MALLEY,

JR.

X 6

4

i

X(I) 2

I

I

l/2

I

*

t

I X(O) FIGURE

5

dt = E(&/z) for &zz = E - V(X)). Since z must tend to zero on the solution path for (I), its energy E, must tend to the maximum value E,, of V in this interval. Specifically, on any trajectory from x(O),

so to meet the terminal condition, we need

Since E, - V(x) > E, - E,, , then, 1 (

’

441)

-

4-N

(2(E, - E,,))1/2 ’

and E, = E, + O@).

SOLUTIONS

TO PERTURBATION

457

PROBLEMS

By monotonicity, there is a unique trajectory near T,, with unit transit time for each sufficiently small E. Moreover, since r(l) 1 =

z:iz

I‘&)

(E” -

the large contributions to this integral where I-(c) == E. . Further, since

E. - b’(s) = V(c) -

r.ysPs-t O(E*))l:*

’

occur near the points c of the interval

L’(s) s -(f’(c)/2)

(s - 0’

locally (provided f’(c) # 0), the proportionate contribution to the time integral of each c E C is asymptotically l/( -Sf’(c))llz (with this halved for endto this time points c). For points of [X(O), x(l)] not in C, the contribution integral is asymptotically negligible. Note that C may include appropriate inflection points at the ends. Re~rark. If we consider the situation where f’(c) is relatively small for some c E C, the proportion of time which the monotonic limiting solution spends at c will be relatively large compared to that spent elsewhere in C. This makes it reasonable to espect that the limiting solution will generall! tend to those points of C which are higher-order zeros off. Further, our argument shows that if all points of C are zeros off of order 2m -+ 1, the proportionate amount of time spent at each c E C is asymptoticall! 1/( -.f(Z’rS~l)(~))l 2 (half this at endpoints). EXA~IPLE

5.

For the problem

2.2 + sin 77x =: 0, explicit

solutions

are obtainable

-3
< -1,

in terms of elliptic

sin rrs ds = -(l/sr)(cos

N(1) = 3, integrals.

zx - cos XV(O))

has its maximum value between x(0) and ~(1) at .v == -1, trajectory through the corresponding rest points is (2/2)

Here

I, and 3, and the

- (1 /?T) (cos 7r.X- cos %Y(O)) =: 0.

The familiar phase-plane portrait is shown through the rest points with energy E = 0. are ellipses (and thereby represent periodic while trajectories with E > 0 are monotonic tris.

in Fig. 6. There is a separatrix Trajectories with energy E < 0 motion) within the separatrix, in x and lie outside the separa-

458

R. E. O’MALLEY,

FIGURE

JR.

6

To go from X(O) < - 1 to X( 1) = 3 will require traversing a trajectory with

E > 0 and z > 0. Thus, z(t) = c"(t) = (z*(o) + (2/n)(cos ?-r&x - cosm(O)))l~~ and to solve the boundary value problem, we need

as usual. Since z must become small at the maxima of V in [x(O), x(l)], we must asymptotically have

z*(o) + (2/7r)(- 1 - cos TX(O)) - 0 there, so 2(O) N f

(+-

11.2 (1 + cos 7rr(O))) .

This implies that there is a boundary layer of thickness B at t = 0 unless cos ~$0) = - 1, (or x(0) = - 1. Then, other arguments imply that r(O) = O(CT’/~) for some y > 0.) For -3 < x(0) < - 1, x( 1) = 3, principal contributions to the t integral come from s = -1, 1, and 3 where f(s) = 0, and f’(s) = n cos n = --P. Thus, our third statement implies that the proportionate stay at these three values is 1 : 1 : 3 if x(0) < - 1 and 4 : 1 : $ if x(0) = - 1. Therefore, the limiting solution is X0(t) = -1,

O
Y=z1,

g
= 3,

4
if

x(0) < -1,

SOLUTIONS

TO PERTURBATION

PROBLEMS

459

and

X&t) = - 1)

o
= 1,

a
== 3,

$
if

r(0) = -1,

x

*t

FIGURE

7

We note that the nonuniform structure exhibited by these solutions for small l is determined by segments of trajectories of the form in Fig. 7. These Y-shaped paths are related to the solitons and wavetrains of the sineGordon equation (cf. [16]), as might be inferred from the differential equation. Such boundary value problems may be relevant to the study of the nonlinear pendulum and the elastic deformation of rods (cf. [4]). Finally, we note that Statement 4 shows that an infinite number of solutions will result if, instead, we consider the equation and boundary values such that - 1 < x(O), X(1) < 1.

5. STATEMENT

4

Proof. We can takef(d,) =f(d,) = 0, V(d,) = V(d,), and p-(s) < V(4) for dI < x < d, (alternatively, J:,~(s) ds < 0 for .r E (dl , d,)). The trajectory through (dl , 0) is given by

It has energy 6, = V(dl). We consider nearby trajectories with energy E < E, . Pictorially, the situation is shown in Fig. 8. Here, OL,(Y’, and /I, t3 denote the intersections of a trajectory with energy E < E, with the lines .r = X(O) and x = X( I), respectively. Such a solution path of (1) must traverse

460

R. E. O’MALLEY,

I

JR.

xi01 X(l) FIGURE

8

the trajectory clockwise, passing near 4 or d, (in order that z + 0). This rules out paths like c@, at least for E sufliciently small. We will number these trajectories by the number of times they near d, or d, . Solution paths approaching only one rest point are ~$3fl’ and OL’& We picture the corresponding solutions for small c > 0, in Figs. 9 and 10. These solutions feature only endpoint nonuniformities (boundary layers) as E-+ 0 and correspond to our statement with 1~= 0. XA

XI dz-

d2. 7

X(l)-

X(l)-

X(O)-

X(O)-

dl

dlI

0

l/2 FIGURE

I

-t

0

1 ,t

FIGURE 10

9

x4

X d2 X(I)

d2 X(I)

X(O)

X(O)

dl

l/2

dl

h-

I

01

FIG. 11.

I/2 For f’(&)

I = f’(r&).

*t

1I 01

FIG: 12.

I

l/2 For f’(dJ=

I

I

f’&).

*t

SOLUTIONS TO PERTURBATION PROBLEMS

461

Paths nearing a rest point twice are &3’~~‘0ip and OI’@$!?‘.The energy required to achieve unit time of flight is again asymptotic to E, from below, but it is somewhat larger than for n = 0. The corresponding solutions are shown in Figs. 11 and 12. These solutions feature one interior nonuniformit! as well as endpoint boundary layers. In general, n + I approaches to a rest point correspond to n switchings in the limiting solution. For each n, there is one solution with 2(O) > 0 and one with k(O) < 0. As for Statement 3, the time spent at s = dl and x =: d, is asymptotically inversely proportional to ( -f’(dl))lF2 and (--j“(dJ)‘? provided eachf’(d,) < 0. Moreover for rz ;C>I, this determines the asymptotic location of the switching points. (Higher-order zeros off would be treated as in Statement 3.) EXAMPLE 6. Suppose the potential b’(x) has the profile pictured in Fig. 13. The phase-plane protrait is given by Fig. 14. Here, the trajectories with energies V(d,) and V(d,) are drawn, while the dashed paths within the separatrices represent solution paths for the boundary value problem (I) for V(x) 4

/

x;--; X X(O) X(I) FIGURE 13

X(O)

X(I) FIGURE 14

462

R. E. O’MALLEY,

JR.

x(O) and x( 1) as pictured. The preceding discussion implies that there will be countably many solutions with limits X’A’) tending to dI or d, within (0, l), and another countably many solutions with limits Xi3’ tending to d3 or d4 there. Remarks. (1) Solutions of optimal control problems (cf. [2]) often feature discontinuous limiting behavior. Considerations as in the discussion of Statement 4 may be relevant to such problems. (2)

Fife [9] treats boundary value problems of the form c”(p(t, x, c) 2). +f(t, 4Oh

p>o,

x, 6) = 0,

o
prescribed,

41)

when the reduced problem f(h x, 0) = 0 has the distinct solutions x = d,(t)

and

x = d,(t).

Under appropriate hypotheses, he finds that the limiting solution will be d1,

0

d2%

t,
<

I,

where

JM = 0

and

./C&J# 0

for

JO)= (;I:’ f( t, x, 0) p(t, 1

x, 0) dx.

The t dependence is crucial in Fife’s work. However, it is still interesting to interpret his hypotheses in the simple autonomous case where p(t, x, .E)= 1 and f(t, x, c) = f(x). They are that (i) dl and d2 are successive equal maxima of the potential energy V(x), i.e., f(di) = 0, f’(dJ < 0, j:YJ(s) a%= 0, and j:,j(s) ds < 0 for K between d, and d, . (ii) The potential energy V(x) is less than V(d,) between s(O) and dl (i.e., Ji,f(s) ds < 0 for I b et ween dl and x(O)) and V(x) is less than V(d,) (i.e., s,“,f(~) ds < 0) between d, and x(l)). Our statements 3 and 4 for the autonomous problem suggest these conclusions in the two cases that (i) dl and d, lie between x(O) and x(l), and (ii) x(O) and x(l) lie between dl and d2 .

SOLUTIONS TO PERTURBATION PROBLEMS

463

We also note that periodic solutions for certain related autonomous systems are discussed by Vasil’eva and Tupciev [18], while some nonautonomous problems are considered by Fife [20], Vasil’eva [17], and Boglaev [3].

6. STATEMENT 5 EXAMPLE

7. Now consider the autonomous problem &i! + x2 - 1 = 0,

-1
x(l) < 2.

Here, the potential energy b’(x)

=

f

-

x

-

(q!L

-

x(q)

has its only maximum point at x = - 1 and V(z) > L’( - 1) for x > 2. The trajectory with energy E = L’( - 1) is pictured in Fig. 15. Unlike the situation with Example 5, the point (2, 0) is not a rest point. We note that Statement 1 implies that there is no asymptotic solution of the two-point problem if either x(0) or x(1) exceeds 2; Statement 2 implies that there is one solution (staying near x = - 1 within 0 < t < 1 for E- 0) provided - 1 > x(O), X(I); Statement 3 implies that there is a unique monotonic solution if x(0) <. - I 6 x(l) < 2 or x(1) < - 1 < x(O) < 2; and, finally, Statement 5, implies that there are denumerably many solutions when - 1 < x(O), x(l) < 2. The limiting behavior in the last case will become clear from the following discussion.

FIGURE 1.5

464

R. E. O’MALLEY,

JR.

Proof. The proof of Statement 5 follows like that of Statement 4. It relies on the trajectories of Fig. 8 where d, is replaced by e, and d, by es . Here, zz will only have a simple zero at x = e2 on the separatrix, so x = e2 will not contribute substantially to the time integral for E- 0. Indeed, any solution path must lie within the separatrix and spend most of its time near x =: e, . This implies that it will generally pass the rest point (e, , 0) quickly, and that the paths o$ and 0$$3’are ruled out since their time of flight tends to zero with E. The four paths nearing x = e, once are 01’~$3,01’01&3’,a/3/3’or’$, and &3’01’+3fi’. The corresponding solutions for small E look like those pictured in Figs. 16-19. In all four cases, the limiting solution is X0(t) = e, , 0 < t < 1. The last three solutions, however, feature initial and/or terminal spikes of height e, - e, . The four paths passing near x = e, twice are c/c&Y~‘~~, a’~#3’ol’&!I’, $?~~‘oL’&~‘~‘cx~,and o&~‘o~‘cx/#~‘cc’c$?~. The first and simplest looks like that shown in Fig. 20 for small E > 0. Thus, there is a limiting discontinuity at t = + where &,(a) = e2. Elsewhere in (0, l), the limiting solution satisfies So(t) = eI . The three other solution paths, in addition, feature initial and/or terminal spikes of height e2- e, . Proceeding, one analogously finds that there are four solutions which pass near (e, (0) n + 1 times (for each n >, 0), and they satisfy the stated conclusions as E-+ 0. X

X(I) X(O) 7L---J e2 I

I 01

X

e2 X(I)

9 X(I)

X(01

X(O) e,

e, bi.

! *I I

l/2

FIGURE

X

0

FIGURE

16

X e2 X(I)

X(O)

i

I 01

I wt I

I l/2 FIGURE

19

t

112

17

0 !‘!!=A

I

l/2

FIGURE

18

SOLUTIONS TO PERTURBATION

PROBLEMS

465

Remarks. (1) We recall that Carrier and Pearson [S] present a fascinating discussion of spurious solutions to singular perturbation problems. They consider Example 7 where e, = - 1 and es = 2 and note that a second solution to the boundary value problem can be obtained from the simplest solution (with trajectory OI’C@)by adding to it the function

4(l) = 12ec/(l + &j*, where [ is the stretched variable

for any t,, within (0, 1). We note that 4 behaves like a narrow spiked function of height 3 for E sufficiently small. (It looks like a solution of the related Korteweg-de 1’ries equation.) Indeed, all the solutions obtained through Statement 5 are obtained by adding such peaked functions at regularI> spaced points in (0, 1). One usually has faith that formally obtained asymptotic approximations correspond to actual solutions. Our results (and Carrier’s) tend to imply, however, that these spikes cannot be added at arbitrary points in (0, l), but only simultaneously, at regular mesh points. (This seems to correspond to the observed periodic spacing of solitons.) Thus, our faith in formal asymptotics may be misplaced, while the oscillatory possibilities allowed here may be significant in some applications (cf. [6]). (2) For E small, but nonzero, the number of solutions may actually be limited, though the preceding asymptotic conclusions for E --, 0 indicate an infinite number of asymptotic solutions. il problem related to Example 7 with E large and f quadratic was discussed by Boa [I]. His results can be interpreted as multiple steadv-state solutions to a biochemical reaction with large diffusion. (3) IVe note the closeness of the techniques method for asymptotic expansions of integrals.

used here and the Laplace

.‘kKNOWLEDGRIENTS iVe gratefully acknowledge with thanks many valuable by H. Flaschka, P. Fife, and D. McLaughlin.

comments

and calculations

REFERENCES Doctoral Dissertation, California I. J. .I. Boa. “.I &Iodel Biochemical Reaction,” Institute of Technology, 1974. 2. Ya. P. BOGLAEV, Smoothing of the solution in a control problem, Z’SSR Cornput. ~l/lnt/z. Moth. P~JT. 10 (1970), 226-230.

466 3. 4. 5. 6. 7.

8. 9. 10. 11. 12.

R. E. O’MALLEY,

JR.

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