Optimal Controlfor Hereditary Processes AVNER FRIEDMAN

Communicated by A. ERD~LYI Introduction Let x----(x1. . . . . x ~) be a variable point in n-dimensional Euclidean space R ~, and let u = (u 1. . . . . u ' ) be a variable point in a set U of R% Given n + t functions /~(t, x, u) (i----0, 1 . . . . . n) defined for - - o o < t < o o , x~R ~, uEU, and given real n u m b e r s t 0, t 1 and points xo, xl in R ", consider the following problem: L e t D be a set of functions u=u(t) which are defined on to<=t<=t1 with values in U, and let D o be the subset of D consisting of such functions for which the corresponding solution x (t) of

(0.t)

dxi --]i(t, x(t), u(t)) dt

(i

=

t,

"'"

n)

'

X(to)

=

xo

exists for to<_<_t<=t~ and satisfies x(tl)=x 1. Find functions u(t) in D o for which the functional tx

(0.2)

J = f /o(v, x(z), u(v)) dv to

takes its smallest value. The functions u (t) of D are called (admissible) control ]unctions (or steering ]unctions), U is called the control set, the solution x(t) of (0.1) is the traiectory corresponding to u (t), and J is the cost junctional. A pair u (t), x (t) which minimizes (0.2) is called an optimal solution; u (t) is an optimal control, and x (t) is an optimal traiectory. The above p r o b l e m is a typical one in the t h e o r y of optimal control. Other problems are obtained when (i) one or b o t h end-points xo, xl are not fixed b u t free to m o v e on given manifolds; (ii) t o or t~ are free; (iii) x is restricted to a region X 4 : RL The problems of optimal control fall outside the realm of the classical Calculus of Variations for the reason t h a t U (and also X) need not be open sets. The need to consider closed control sets arises n a t u r a l l y from physical experience and, moreover, in m a n y i m p o r t a n t cases the values of the o p t i m a l control lie, in fact, on the b o u n d a r y of U. The following f u n d a m e n t a l results have been p r o v e d in recent years: (a) PONTRYAClN'S m a x i m u m principle [6]; (b) existence theorems b y direct m e t h o d s (see [4] and the references given there), and (c) for linear systems: uniqueness and the " b a n g b a n g " principle.

Optimal Control for Hereditary Processes

397

In this paper we derive results analogous to (a), (b), (c) for hereditary processes of the form t

(0.3)

x'(t)=x~o+fh'(t--z)/'(z,x(z),u(~:))d3

( i = 1 . . . . ,n).

to

The cost functional may also assume the more general form tt

(0.4)

J = f h ~ (t1 - 3)/o (3, x (3), u (3)) d 3. to

Observe that in contrast to the process (0A), for hereditary processes an optimal solution in the interval (to, tl) is not, in general, an optimal solution in any subinterval. If h i ( t ) ~ l , then (0.3)reduces to (0.I). In w1 we prove a maximum principle which generalizes (a). In w2 we take the f (t, x, u) to be linear in x, u and prove uniqueness, the "bang bang" principle and the finiteness of the "switching set". Finally, in w we consider the question of existence. The results of this paper immediately extend, wittl minor modifications in the proofs, to the more general processes in which hi,/i are vector functions and also to processes where he(t--3) is replaced b y he(t--3, 3). The methods and results can undoubtedly be generalized also to hereditary processes of the form t

(0.5)

x~(t)=x~o+fhi(t--z)l'(z,x(,),u(3))dz t--~X

( i = t . . . . . n)

for t>=to, where ~ > 0 and xr162 for to--C~_<__t
For simplicity of notation we shall usually take t o = 0 in (0.3), (0.4); this clearly has no effect on the results for t o fixed. We shall first consider the optimal problem for autonomous systems, i.e., when the ]~ do not depend explicitly on t, and we shall also consider t 1 to be positive and free and the endpoints x0, x 1 to be fixed. We introduce the function x~~-x ~ (t) b y t

(1.1)

(t) -- f h0 ( t - 3)/o (x

(3)) dr

0

and denote b y ~ the straight line {(x~ x); - - o o < x ~ x---x1) where xa is the fixed right end-point. Taking X = (x~ x) as a variable point in R ~+x and setting X(t)=(x~ x(t)), we can restate the optimal problem as follows: Problem 1. Find a pair u (t), x (t) ((satisfying (0.3)) with intersects ~ at the smallest coordinate x ~

ucD

suctl that

X(t)

398

AVNER

FRIEDMAN

:

We proceed to define precisely the classes D of controls. We shall henceforth denote b y D a n y class of functions u (t) having the following properties: (i) u (t) is defined and is measurable and bounded in some interval 0--< t=< t1, t1 depending on u; (ii) u(t)EU for any O<=t~tl; (iii) if uED and its interval of definition is (0, tl), then for any O < t ' < t " < t l and for any v E U the function

u.(t)=lu(t)

if O<_t<_t' or if t'
Iv

t"
also belongs to D. We shall denote the class b y D* if, in addition to (i), (ii), (iii), the following condition holds: (iv) each u (t) is continuous at the right end-point t 1. We shall naturally assume t h a t every function u (t) satisfying (i), (ii) ((iv)) which is equal almost everywhere to a function in D (D*) also belongs to D (D*). The process we consider is (t.2)

xi(t) = X~o+ fh'(t -- z) f(x(z), u(z)) dz

(i = O, 1 ..... n).

0

Recall t h a t x0=(Xox . . . . . x~), x i = (xl . . . . . x~) are fixed, t h a t U is an arbitrary set in R '~, and that t ~ > 0 is free. Assume t h a t

hi(g), dhi(t)/dt, ]i(x, u), ~f(x, u)/Ox i are continuous functions for 0 ~ l < oo, x~R ~, uEU and O<~i<=n, l<=j<--_n. Given a pair u(t), x(t) (where uED and X(t) satisfies (t.2)), we set (A)

[~(t) =/~(x(t), u(t))

Op(t) '

Oxi

Op(x(t), u(t)) --

cOxi

For a n y vector ~oI let ~0(t) be the solution in O<=t~t 1 of the linear system tl

(1.3)

d~i(t) dt

0~(t) h~(o) ~(t) + ~w-

~0~(~) d~ d~ (i = o, 1 . . . . .

n)

with ~o(tl)----~o1. The existence and uniqueness of ~o(t) follow as for ordinary differential equations (see, for instance, [51). Thus, ~v(t) is absolutely continuous, and (1.3) holds almost everywhere. Setting t~

(1.4)

zi(t) = hi(0) ~~

+ f dhi(a--t) d~ ~vi(e)d a

(i = 0 , t . . . . . n)

t

and introducing the function (t.5)

H(t, x, % v) = ~. ]i(x(t), v) zi(t), i=0

we can now state the m a x i m u m principle for Problem 1 as follows:

Optimal Control for Hereditary Processes

399

T h e o r e m 1. Consider Problem I with a set D o/control/unctions, and assume that (A) holds. I f u (t), x (t) is an optimal solution and X ( t ) = (x ~ (t), x (t)) intersects z~ at t = t1, then there exists a non-zero vector ~1 such that the/ollowing property holds, with ~v(t) being the solution o/ (1.3) with ~v(tl) ~-~vl : max H(t, x, ~v, v ) = H ( t , x, ~v, u(t))

(t.6)

v~U

/or almost all t, 0 <=t <=t 1. Furthermore, ~o (tl) <=O. Remark. If we change the definition of u (t) on a set of measure zero, then the new control is again an optimal control with X(t) being unchanged. Thus if U is a compact set (so that the left side of (t.6) exists), then we m a y redefine u (t) on a set of measure zero in such a way that (t.6) holds everywhere. If u (t) was originally piecewise continuous and if at the points of discontinuity we define u (t) = u (t -- 0), then (t.6) holds everywhere b y continuity, since the left side varies continuously with t. Definition. An optimal control u(t) (or an optimal solution u(t), x(t)) is said to be normalized if (1.6) holds everywhere for O<_t<_tl, and if, when u(t) is piecewise continuous, u (t)----u (t--0) at all the points of discontinuity. Thus, if U is a compact set, then every optimal control is equivalent to a normalized one. In applications we may therefore consider only normalized optimal controls. Proof of T h e o r e m 1. For any measurable function v (t) (a--
f g ( t , v(t)) d t = e ( q - - p ) g ( , , u(,)) + o ( e ) .

z+p,

If g depends continuously on a parameter ~ which varies in a compact set, then o (e)/e-~0 uniformly with respect to ,~. To prove Theorem I we begin b y changing the optimal control u (t) into a control u* (t) as follows: Let 0 < ~t < ~a < " " < ~s < tl be arbitrary regular points of u(t), and denote b y Ij the interval T j - - e ~ t i < t ~ T j where 6tj are some nonnegative numbers and e > 0 is sufficiently small so that the intervals Ij are disjoint intervals and ~ - - e 6 6 > 0. Let

(1.8)

u*(t)

=fu(t) (v s

if

tr

if

t~I i

for all i

where vj are some points in U. u* belongs to D. Denote b y X*(t) = (x *~ (t), x*(t)) the solution of (t .2) with u----u*. Its existence for 0 ~ t =
400

AVNER

FRIEDMAN

:

We shall establish the variational formula

0.9)

x*~(tJ=x~(tJ +,A~x+

o(,)

( i = o , t . . . . . n)

where i=i

and the B~ (t) are defined as follows: Let

(~.11)

g'(t) = X h ' ( t - 3~) [f(x(3j, v~) - f(x(3~), ~ (3~))] ~t~, ~k
and let T be the linear operator defined b y t

(Tw)' (t) = "~, --]"h i (t -- z) 8/' (z) w ~ (3) d3

(1.12)

a=o o

~x~

"

Define Tkw inductively b y T k-1 (Tw). Thus, if we introduce the matrix F ( t ) : (Ol~(t)/Sx i) and the diagonal matrix H(t)----(~ii hi(t)), then we can write

T w (t) : f n(t -- 3) F(z)w (3)d z,

(t.t3)

0

and, as is easily verified,

(1.14)

T k w (t) : f Hk (t, 3) F(z) w (3) d 3, 0

where Hi(t, 3 ) = H ( t - - z) and t

(1. t 5)

n k (t, 3) : f H k_l (t, a) F((~) n ( a -- 3) d a .

The B~ (t) are now defined b y

0.~6)

~ ~(t),~t;:~'(t)+ ~ (~g)'(t). ~=1

i=1

To justify this definition note first that the series on the right-hand side is convergent, as can be verified b y majorizing its terms in a standard fashion. Furthermore, since the gi(t) are linear in 6t 1 . . . . . 6t~, it follows t h a t the righthand side of (t.t6) is also linear in the (~ti. Thus the B~(t) in (t.16) are well defined as functions independent of ~t~ . . . . . 6ts; they depend, however, on the parameters 3~. . . . . %, v1. . . . . vs. The proof of (t.9), (1.10) is direct:

~,'(t) - ~' ( t ) : f h' (t - 3) If (~,(3), ~*(3)) - f (x (3), ~*(3))] a 3 + 0 t

+ f h'(t - 3) [/'(x (3), ~*(3)) - f (~ (3), ~ (3))] a3 _-- ~' (t) + j'(t). 0

By the Mean Value Theorem, f

~ ; oi,(.(~),~,*(.)) F-~ ~'(t)= 0r~,fh'(t- -, C~.~(3)_.~(3)?d3+o(.)

Optimal Control for Hereditary Processes since Ix*(t)--x(t)l =-O(E), as can be seen from the fact t h a t set of measure 0 (e). F r o m this last fact it also follows t h a t

(t.t7)

40t

u*&u

only on a

n

~'(t/= ~ [ h'/t- ~) ~~l'/~/ Ix* ~ (,)

-

x ~ (~)~ d ~ + o

(,).

cX=0

Next, since zl . . . . . z, are regular points of u (t), we have (recall (1.7))

Ji(t)=eg(t)+o(e)

(1.t8)

if

tr

for a l l j .

Since J~ (t) = 0 (e), using (t .t 8) we obtain tt

f [ J ' (t) - , g (t) l d t = o

(1.t9)

(,).

0

Setting

z(t)= x*(t)- x(t)

z'(tl--

~, [

and using (t.17), we have

t

h'lt- ~1 ~l'l.l z'l,l~, + J'ltl +ol~l,

i.e., z(t)=(Tz)(t)+J(t)+o(e). (1.19), we get

Applying this relation successively and using oo

(t.20)

z(t) - - J ( t )

+ e~

k=l

(Tk g) (t) + o(s).

Finally, using (t.18), we h a v e the result t h a t for t>T~ (1.20) remains true if J(t) is replaced b y eg(t). Recalling (1.16), we complete the proof of (1.9) and

(1.1o). If 0 < T I ~ T ~ < - - ' . . ~
zJX+/IX

is the vector having a representation (z~ . . . . , z~,

vt . . . . . v~, 6tl + ~ . . . . . 8t~ + ~'t~). The p r o d u c t 2 A X is defined b y the representation (z~ . . . . . z~, v~ . . . . . v~, ~6t~ . . . . ,2~tv). We t a k e all the vectors A X to have their initial point at X ~ = (x ~ (t~), x~). Their t e r m i n a l points form a convex cone K with v e r t e x X~. K possesses the following p r o p e r t y : If L is a n y smooth curve which begins at X1 and whose t a n g e n t at X~ points into the interior of K, then there exists a control u* obtained from u b y the Arch. Rational Mech. Anal., VoL t5

28

402

AVNER FRIEDMAN :

process (t.8) such that X*(t) intersects L, for t = t 1, at some point of L other than X 1. The proof is based on (t.9) and is exactly the same as for the process (0A) (see [6; p. 94, Lemma 3])Since u(t), x(t) is an optimal solution, it thus follows that the ray L defined by x = q , x~ < x ~ (tl) does not point into the interior of K. Consequently, there exist non-zero vectors a = (a~. . . . . a ") such that (a, L0)>= 0 and

(a, dx)__
and also we have denoted by L 0 any tangent vector to L at the point X 1. Let ~o(t) be the solution of (1.3) with v/(tl)=a. Then

(W(6), A X ) G O

(I.2t)

for all A X in K .

Since we may take L o = ( - - t , 0 . . . . . 0), we also have %(tl) G0. It remains to prove (t.6). We shall use (1.21) with the special choice of s = 1, 8 tt = t and set B i (t) = B[ (t) From (1.16), (1.12) we find that t

B'(t) = ~2

f h'(t--z) O['(z) B~(T) d r + g ' ( t )

( i = 0 , t,

d a=0 0

,n)

B y (t.tl), g i ( / ) = 0 if 0 = < / < v l . Hence, b y uniqueness, B~(t)=0 if 0= zl, gi(t) =h~(t-- zl) C~ where (t .22)

C i ~ ~ ( x (~'1), Vl) - - /i (X (~1), " (~1)) "

It follows that, for h < t G q , t

(23)

B'(,/= #

~

cX=0q

Introducing the vector notation B ( t ) = (B~(t)), C = (C ~) and differentiating (t.23) once, we obtain t

dB (t) = H(O) F(t) B (t) + dt

f dH(t--3) F(z) B (z) d z 4 dH(t--5) C. dt dt

II we denote b y A* the transpose ol a matrix A, then we can write (1.3) in the form h

ddt~ - -- -- F*(t) H(O) W (t) -- f*(t) f dH(a--t)da W(a) as. t

As is easily verified,

./(/

I ;( ;

dH(t~Z)dt F(r) B(~)d*, w(t) dt~-

B(t),F*(t)

ft

t*

>

dH(a--t)d, ~o(a)da dr.

Optimal Control for Hereditary Processes

403

Therefore tl

tl

fY/

3[t

TI

TI

dt

'~

tl

=

.,

c.w(t))d,.

TI But the left-hand side is equal to (B(tl), V,(t,))--(B(~I+0), V'(~)), and B ( T I + O ) : H ( O ) C , by (t.23), and (B(t~),,p(t~))=(AX, v(t,) ) gO, by (t.2'I). I t thus follows that tl ( U ( 0 ) C, 'i/) ("gl) ) "-t-

((rill(t--T1) C,~(t))dt
d~

dt

=

Recalling the definitions (1.4), (t.5) and (t.22), we get H(zi, x, % vl) H(v~, x, % u (T~)). Since this inequality holds for any v~C U and for any regular point Ti of u(t), (1.6) follows for almost all t. This completes the proof of Theorem t. Counting the free parameters of the optimal problem, one can see that in order to obtain as many conditions as the number of parameters, so that (roughly speaking) there will be only a finite number of pairs u(t), x (t) satisfying the necessary conditions of optimality, we need one additional condition. We shall now derive this condition. The following assumption will be needed: (B)

For each i=O, 1 . . . . . n, either hi(0)=0 o r / i is a function independent of u.

Introducing the function t

(1.24)

K(t) =

r

h~(0) f(t) +

i=0

et

f(~) d~ ,

0

we can now state an extension of Theorem t, by giving one additional necessary condition. Theorem 1'. Consider Problem t either with a set D* o/ control /unctions and assume only (A), or with a set D o/ control /unctions and assume (A), (B). I / u (t), x (t) is an optimal solution, then all the assertions o/ Theorem t hold and, in addition, K (t~) = O. Proof. We proceed similarly to the proof of Theorem 1 but introduce (in addition to the Ti, vi, ($ti) a real parameter (it. Instead of (1.9), (1A0) we now derive the variational formula (t .25)

x* i(tl + e ~ t ) = xi (tl) + ~/~iX + o (e)

where

[

(1.26) -FiX = A i X + hi(O)/i(x(t~), u(ti) ) +

S

dt

0

28*

404

AVNER FRIEDMAN:

Note that X*(t) exists for O<_t<_tl+ ~ for some 7 > 0 independent of e, provided e is sufficiently small. We also decrease e, if necessary, so that edt<--<_~. In order to prove (t.26) we only need show that x * i ( t l + e d t ) - x*i(tl) is equal to the expression in brackets on the right side of (1.26) plus a term o (e) since then the proof is completed if we make use of (1.9), (1.t0). The evaluation of x * i ( t l + e dr)-- x*i(tl)is done directly, using the Mean Value Theorem if u is continuous at tl, and the assumption (B) if u is not continuous at tl. The convex cone K is next replaced by the convex cone K' consisting of all tile v e c t o r s / ' X . The existence of a non-zero vector ~o(tl) satisfying (~o(tl), FX) <=0 is proved as before, and the assertions of Theorem 1 then follow without any change in the arguments. Finally, taking / ' X which corresponds to s----O, we obtain from (v/(t~),FX)<=0 the inequality K(t~)~tGO. Since dt is any real number, K ( t l ) = 0, and the proof is completed. In the following remarks we give extensions of Theorems 1, t'. R e m a r k 1. If the points x 0, x~ are not fixed but are restricted to lie on some manifolds S o and S l, respectively, then the assertions of Theorems l, t ' clearly hold and, in addition, the following transversality conditions are satisfied: (~0~(0). . . . . ~o~(0)) is orthogonal to S O at x(0) and (~(tl) . . . . . ~o'(tl)) is orthogonal to S 1 at x(tl). The proof is based on Lemma t0 [6, Chap. 21, which remains true for the present hereditary process, and on the following modification of the variational formulas (t.25), (t.26): (1.25) holds w i t h / ' i X replaced by / ' i X . +. ~, E~(tl)~i

( i = 0 , t , . . . , n),

where X*(t) is the solution of (t.2) corresponding to the control u*(t) and with x*i(0)=x~(0) + e $ i + o ( e ) , i = 0 , 1 . . . . . n, ~ ~ Here the E~(t) are defined by (!.27)

~ E~(t) ~i = $, + ~ (Tk~)i(t)

/=1

k=l

(i = 0 , t . . . . . n),

where T is defined in (1.t2). The convex cone K (K') is now being replaced b y the convex cone generated by K (K') and the vectors ~. E~(tl)~i where ~ = ]=l (~ . . . . . f") is any vector tangent to S o at x (0). The arguments for the process n

(0A) (in [6; p. t t 3]) extend to the present situation, thereby yielding ~ or (t~)~i= 0 i=i for any vector ~/ tangent to S 1 at x(tl), i.e., (~o1(ti),..., ~ (tl)) is orthogonal to S 1 at x (tl). Also,

(~ .28)

i=l

1=1

for any ~ tangent to S o at . (0). Setting ( i = o, t . . . . . j=l

n),

Optimal Control for Hereditary Processes

405

we have t

dgi(t) 2 h i " " O/i(t) 2/ dt -(0)~ ~e(t) + otto

dh'(t--z) " dt

ottO

O]'(z) dx ~ cPi(z)d~:"

It follows, as in the proof of Theorem t, t h a t

Z ~' (t~) ~ (tl) - Z ~' (o) ~(o) = o. i=0

i=O

Since, b y (1.27), 9 ~(0) = ~i and since 20= 0, using (1.28), we find t h a t ~, 8i~v~(0) = 0, i.e. (~o1 (0) . . . . . ~o"(0)) is orthogonal to S o at x (0). ,=1 R e m a r k 2. Consider the optimal problem for a non-autonomous system t

(1.29)

x'(t)=x~o+fh'(t--z)]~(%x(r),u(v))dz

( i = 0 , 1 . . . . . n)

o

with fixed end-points x0, Xl. The system can be reduced to an autonomous one b y introducing the variable x ~ + l = t and the additional equation t

(1.3o)

x "+1 (t) = f d T. 0

Thus, h ~+1 ----t, ff+l ~ 1. Tile ~vi satisfy, in addition to (1.3), the equation

dv,.+~(t) (t.31)

at

~T-

--

(o) r

0t=0

and

+

de t

X~+1 (t) ------~o"+1 (t). The notation

0n(t) _ on(t, x(t),u(,))

l (t) an(t) _

ot

an(t, x(t), u(t)) ot

is now being used in (t.3), (t.31). Set

H*(t, x, % v)----~o~+1(t) + H(t, x, % v) where

(1.32)

n(t,

v/-i=0

f (t, It/, v) x'Ct/.

Since the m a x i m u m principle holds for H* and since H * - H is independent of v, the m a x i m u m principle (t.6) remains true for the system (t.29) with H defined b y (1.32). Furthermore, as x~+l----t is flee, the terminal point varies on the line x~=xi(tl), i = 0 , t . . . . . n, -- o o < x ~ + X < o% and the transversality condition becomes

(t .33)

~,+1 (tl)=o.

Finally, since the function K(t) corresponding to (t.29), (1.30) is equal to ~o'~+l(t)+K(t) where K(t) is defined b y (1.24) (with ]~(t)=[~(t, x(O, u(t))), the

406

AVNER FRIEIJMAN:

last assertion of Theorem 1', in conjunction with (t.33), yields (1.34) In s u m m a r y we have:

K(tl) = 0 .

For non-autonomous systems (1.29) all the assertions o/ Theorems 1, 1' remain true, and, in addition, (t.33) holds. Note t h a t instead of (A) we now have to assume t h a t hi, dhi/dt, [i, 8f/Ox i and 8f]St are continuous functions. R e m a r k 3. A particularly i m p o r t a n t class of optimal problems are tile timeoptimal problems, i.e., problems where h ~ ~ 1, [0 ~_ 1. Since/~176is now independent of v, the m a x i m u m principle is valid with H replaced b y

Ho(t, x, ~o, v)= ~ /i(t, x(t), v) gi(t).

(I.35)

For convenience, when dealing with time-optimal problems we shall always denote b y ~v and Z the n-dimensional vectors (~o1. . . . . ~") and (Z1. . . . . Z~), respectively. Similarly, the matrices n ( t ) = (diihi(t)) and F ( t ) = (Sli(t)/Sxi) will be taken to be n • matrices (i, j" = 1 . . . . . n), and (SF/Oui) will be an n • matrix (i = t . . . . . n; i = 1 . . . . . m).

2. Linear Hereditary Processes We specialize in this section to time-optimal problems (see R e m a r k 3, w 1) and ]i linear in x, u, i.e., (2.t)

f(t,x,u) =~a~(t) xJ+~,b~(t)ui+c'(t) i;1

(i = 1, . . . , n ) .

j=l

I t is assumed once and for all, throughout this section, t h a t a~(t), b~(t), ci(t), hi(t), dhi(t)/dt are continuous functions for 0--
A(t) : (a~(t)),

The family of control functions is a set

B(t) : (b~(t)),

H(t) : ((Siihi(t))

and the vector c (t) = (c l(t) . . . . . c~ (t)) and noting t h a t H 0 (t, x, ~, v) differs from (B (t) v, g (t)) b y a function independent of v, we can write the m a x i m u m principle in the equivalent form: m a x (B(t) v, Z(t)) = (B(t) u(t), z(t))

(2.2)

vEU

for all 0 <~t <=t 1 (u (t) is always taken to be a normalized control; see the remark following Theorem I). Here, tz

(2.3)

e~(t) _

dt

[

A*(t) H(O)~(t)+

i

d~

t

tl

(2.4)

z(t)=H(O)~~

+ . f1 - dH(~--t) d~ W(a) da" t

~(~)d~

]'

Optimal Control for Hereditary Processes

407

The maximum of (B (t)v, z(t)) =(v, B * ( t ) Z (t)) must obviously occur on the boundary of U. This fact can be considered as the bang bang principle for our linear hereditary process. This principle is well known for processes of the form (0.t) with fi linear (see [31 and the references given there; also [6; Chap. III]), and it can be stated in the following words: If one wishes to change a system from one state to another in a minimum time, one should utilize at each moment all the power available. Let v = ~ (t) be a value for which max (B (t) v, Z (t)) = (B (t) ~ (t), Z (t)). vEU

Either 77(t) is uniquely determined, for a given t, and is then a vertex of U, or ~ (t) is not unique and can then be taken to be any point on a whole face of U. In the second case we call t a switching point of the optimal control (or of the optimal solution). The switching set is the set consisting of all the switching points; denote it by S and its complement in O<_t<=t1 b y S. From the continuity of B*(t) Z (t) it follows that S is an open set in [0, tl] and that 77(t) is a continuous function of t in any interval lying in S. Since for tc S, ~ (t) is a vertex of U, it must then be a constant function in every interval lying in S. Finally, by the maximum principle, u(t)~-~ (t) for t~ S. In summary, we have T h e o r e m 2. Let u(t), x(t) (O<--_t<=tl) be an optimal solution, and let S be the corresponding switching set and S its complement in O<--_t<=t1. Then the values of u (t) belong to the boundary of U and, restricted to S, u (t) is a continuous piecewise-constant function whose values are vertices o/ U. The next theorem establishes the uniqueness of the optimal solution. T h e o r e m 3. Let u 1 (t), xl (t) and u S (t), x2 (t) be two (normalized) optimal solutions which transfer a point x o into a point x 1 in a m i n i m u m time t 1. Let S i (i ----t, 2) be the switching set corresponding to u i (t), xi(t) (i-=t, 2), and denote by Si the complement o/ S i in O<--t<--t1. Then ul(t ) =--us(t ) in SireS 2. Corollary.

xl (t)

I] the switching sets S 1, S 2 are finite, then ul(t ) =--us(t ), and

=_ x~ (t).

Indeed, u 1 (t)------uS(t) follows b y continuity since, b y Theorem 2, both controls are piecewise continuous functions for 0--< t--
(2.5)

x i (t) = x o + c (t) + Yi (t) + f n ( t -- z) A (z) xi (z) d*, 0

where t

(2.6)

Yi (t) = f H(t -- z) B (z) u i (z) d'~. 0

Defining H 1(t, z) = H(t -- z) and t

(2.7)

Hk+l (t, z) = f H k (t, a) A (a) H(a -- z ) d a ,

408

AVNER

FRIEDMAN

:

we can represent xi(t) in the form t

t

xi (t) = [ 1 + d R (t, ~) A (~) d~] x o + [ c ( t ) + of R (t, ~) A (~) c (~) dT] +

(2.8) + [yi(t)+foR(t,')A(z)yi(z)dz

},

where

R (t, ,) ---- E Hi(t, ~).

(2.9)

i=l

Since xl (tl) = x2 (tl), we conclude that tt

(2.t0)

tt

yl(tl)+fR(tl,*)A(z)yl(z)d*=y2(tl)+fR(t,z)A(z)y2(*)d*. 0

0

We shall need the following lemma. L e m m a 1. For any n-dimensional vector tol and/or any t l > 0, let to (t) be the solution o/ (2.3) which satisfies to(tl)=to 1, and let z(t) be de/ined by (2.4). Then, /or any measurable and bounded m-vector u (t) (0<_<_t<= tl) h

tt

(2.1t) where t

r (t) = f xc(t- ,) B (z) u (~) d,. 0

Proof. From the definition of R (t, ~:) we obtain the relation t

f R (t, g) A (a) H(a -- z) A (z) d a = R (t, z) A (~) -- H(t -- z) A (z). Setting t = tl, S (~) = R (tl, z) A (z), we get tt

S (T) = H ( t 1 -- 7) A (~) + f S (a) H ( a - - ~)A (~)da. Consequently, the vector 9~(t)=S*(t)tol satisfies the integral equation

q~(t) = A*(t) H(t 1 -- ~) tol + f H(z -- t) q~(~) d z . t

Next, integrating b y parts on the right-hand side of (2.3), we find t h a t -- dry (t)/dt satisfies the same integral equation as 9 (t). B y uniqueness, therefore, 9 (t)= -- dto (t)/dt, i.e., (2.t2) d~(t) _ A*(t) R*(t 1,t) tol. dt Integrating b y parts on the right-hand side of (2.4) and then substituting

dto/dt from (2.12), we get tl

(2.13)

Z (t) = H(t 1 -- t) tol + f H(a -- t) A* (a) R* (tl, o') ~Pldo'. t

Optimal Control for Hereditary Processes

409

Using the definition of y (t), we then find tt

~x

f (B (t) (t), Z (t)) dt = f (H(tl -- t) B (t). (t), 0

0

ct +

$

dt

and since the inner integral on the right-hand side is equal to y (a), the proof is completed. We proceed with the proof of Theorem 3. Denote by ~v1 (t), Z1(t) the functions ~, Z associated with the optimal solution u~(t), x1(t), and let ~vl----~v~(tl). Taking the scalar product of each side of (2.t0) by ~vl and using Lemma I, we get tX

~l

(2.14)

f (B (3) u~ (3), Zl (3)) d 3 ----f (B (3) u S (3), gl (3)) d 3. 0

0

Since by the maximum principle (B (3) ul (3), Zl (3)) >= (B (3) u3 (~), Zl (~)) for all z, it follows from (2.14) that, for almost all t,

(2.15)

(B (t) u S (t), Zl (t)) = (B (t) u 1 (t), Zl (t)).

Since both sides are continuous functions on the set SI~S~, (2.t5) holds for all tcSlc~S 2. Finally, since the common value of each side of (2.15) is equal to max (B (t) v, Zl (t)) vEU

and since for t E ~ the maximum is obtained for only one v, namely, v=ul(t), it follows that u s (t) = ua (t) for t CSlc~ S 2. Theorems 2, 3 indicate the importance of studying the nature of the switching set S and, in particular, of establishing conditions under which S is a finite set. We shall consider this question in the remaining part of this section. For the differential process (0.1) very simple sufficient conditions on A, B, U can be given to ensure the finiteness of S. For hereditary processes the conditions are quite involved. We first make a general useful observation: If S is not finite, then there exists a sequence {t]}, ti-+t* ( 0 ~ t * ~ t l ) such that ticS. Now if tcS, then there exist two different points, say v~ and v2, lying on some edge w of U, such that the maximum of (B (t) v, Z (t)) is obtained at vl and at v2. Hence (B (t) w, Z (t)) -----0. Since U has only a finite number of edges, we may assume that, for some edge w, (B (t]) w, Z (t])) = 0 for all/'. Using Rolle's theorem, we show that each derivative of (B(t)w, Z(t)) vanishes on a sequence of points converging to t*, provided B (t) and Z (t) are infinitely differentiable functions. Hence, (2.16)

d-~-k(B(t)w,Z(t))=Oat

dtk

t=t*

for k----0, t,2,

....

4t 0

AVNER FRIEDMAN :

The problem thus reduces to finding conditions on A, B, H and U under which (2A6) cannot hold. We shall solve this problem under the additional assumption that A, B, H are analytic functions since in the non-analytic case the conditions which can be obtained by the same method depend upon solving a sequence of integral equations and, therefore, are difficult to verify. b

We shall use the convention Y, M k = 0 if b
For ]>i=>0, set i--1

(2.17)

di-k-i H(o) dk-iA*(t) dti-k-~ dt~-i '

;~=i

and for/" => 1

(2.~8) (2.19)

Bi(t) =C]o(t)H(O ) + (--1) i

p, (t) = Bj (t) + y,

~,

dill(O) dtJ

Cio6 (t) Ci,i,(t) ... Ci,_llk(t) B~,(t)

k = I l
where i0=i- Finally, set

(2.20)

Po(t) =H(o),

(2.211

i=o i J~

(t)

....

dt~-J

/

D e f i n i t i o n . Let A, B, H be analytic functions for 0 G t < ~ . We say that A, B, H and U are in general position if for any edge w of the polyhedron U and for any 0 G t < oo, the infinite set of vectors Qk(t)w (k=0, 1, 2 . . . . ) contains n linearly independent vectors.

4. Let A, B, H be analytic/unctions/or O<=t< ~ . I[ A, B, H and U are in general position, then the switching set o / a n y optimal solution is finite. Theorem

By the corollary to Theorem 3, the (normalized) optimal solution is then unique. Proof. If the switching set is not finite, then (2.t6) holds. Because of the analyticity assumptions, z(t) is analytic, and therefore the left-hand side of (2.16) vanishes identically. In particular, (2.22)

dk (B(t) w , x ( t ) ) = O d#

at

t = t 1 for

k = 0 , t,2,

....

We proceed to calculate the left-hand side of (2.22). From (2.4)

(2.23)

dJz(tl) J di-~H(o) dk~(6 ) dtJ - ~, ( - 1)j-k dtJ-k dtk

i f = 0 , ~, 2 , . . ) .

k=0

Since, by (2.3), (2.4), & p / d t = - - A * ( t ) z , we also have (2.24)

da+l~v(h' -dt~+l

2 (~/~)d~-t'A*(h' dvz(tl, /*:0

d#-v

dry

( 2 : 0 , t,2, ..). "

Optimal Control for Hereditary Processes

41t

Combining (2.23), (2.24) and using the definitions of C]i, B~. and • (tl) ----H(0) ~v(tl), we obtain a recursive formula for diz(tl)/dti: (2.25)

dig(t1)

dti

i-1 --

~' Ci' (tl) ~diz(tl) + Bj(tl) ~ (tl)

(i = t, 2 .... )

i=l

and setting Cik----C~,(t~), B i : B i ( t l ) ,

Applying (2.25) k times, l ~ k ~ y ' - - l , we obtain d i z (tl)

dti

: B i ~ ( t l ) + ~' Cji'B'1~(tl) + l~_it< j

+ "'" +

+ For k : y ' -

X

Cii, C,,i,B,~v(t~) +

l<~i~
~.

Ci i, Cilia... Ci,_~~-1 B~,_,~v(tl) +

l
Y.

Cj~, Ci,i ... C~_xi, d~*z(t')

l
d ti*

1 the last sum consists of only one term with i a : 1"--#, and, therefore, di~z

dz

dt i*

dt -- B1 ~ (t~) = Bi, ~v(t~).

Using the definitions (2A9), (2.20), we conclude t h a t

(2.26)

dJz (tl) _ pj(tl) ~ (t~) dr]

(i = 0, t, 2,

""

.).

Expanding the left-hand side of (2.22) and substituting (2.26), then using the definition of Qk in (2.2t), we find t h a t the Qk (tl)w ( k = 0, 1, 2 . . . . ) are orthogonal to ~ (t~). Since A, B, H and U are in general position, ~o(tl) ----0. But, b y Theorem I, ~o(tl) ~ 0, a contradiction. We next specialize to the case n----1 and give conditions which ensure t h a t the switching set is empty. We need the following assumptions" (C1) Set ~ = a~ (t), b] (t) = b~ (t), h (t) = h 1 (t) and assume t h a t c~= const. =4=0 and b] (t) 4:0 for all t ~ 0, 1"= 1 . . . . . m. (C2) Either a h ( t ) > 0 for all t>--0 or --o~h(t) is ~ 0 and is completely monotonic in 0 G t < o o , i.e., (--t)~d~[--o~h(t)~/dt~>~O for all t>__0 and k = 0 , 1 . . . . . (C~) U is the m-dimensional cube - - t G u j G l (1"~--1. . . . . m). T h e o r e m 5. Under the assumptions (C~)--(C~) the switching set o] any optimal

solution is empty. m

Proof.

Since (B (t) v, X (t)) :- ~. bi(t) vj Z (t) (~ (t) a scalar function), the maxi]=1

m u m of (B(t)v,z(t)) is attained at the unique point vi=sgn~bi(t) z(t)l (1"= 1 . . . . . m) whenever z ( t ) ~ 0 . It thus remains to show t h a t z ( t ) ~ 0 for all t>=0. From (2.3) we have t1

Integrating and setting ~v(t) :~v (t~ -- t), we find t

(2.27)

9 (t) ~-~ (t~) + ~ f h (t -- a) 9 (a) d a. 0

4t 2

AVNER FRIEDMAN :

Assume t h a t ~o(tl)>0. If ~ h ( t ) > 0 for all t>--0; then, if we write the solution 9(t) of (2.27) in the expanded form co k=l

t

]

[0

where h 1 (t) = h (t) and t

hk+l (t) = f hk (t -- a) h (a) da, 0

it follows t h a t def(t)/dt>O. (2.28)

Since cf(t)=~(t~--t) d~(t)

dt

_

and

~z(t),

(t) :~ 0 for all t_-->O. If, on the other hand, --~ h (t) is completely monotonic, then, b y E2; Theorem 8], 9(t) is also completely monotonic, i.e., (--l)kdkg(t)/dtk>=O for t=>O, k~--O, 1 . . . . . We claim t h a t d q~(t)/dt < 0 for all t--> O. Indeed, in the contrary case d qJ (to)/d t-~ 0 for some t0=>0. Since dq~/dt<~O, d2qJ/dt2>=O, it follows t h a t d g / d t ~ O for t>=t o and, consequently, ~0(t) ~ const, for t--> t 0. q~(t) however is completely monotonic, and, b y a well known theorem, it is then also analytic. Hence 9 (t) ~ const.----~p (tl) for all t>_0. F r o m (2.27) we obtain t

f h(t--a)da=-O,

i.e.,

h(t)=--O,

o

which contradicts our assumption t h a t ~h (t) ~ O. We have proved t h a t d 9 (t)/dt O , and it follows from (2.28) t h a t Z ( t ) ~ 0 for t=>O. If ~(tl)
3. Existence of Optimal Solutions In this section we consider the question of existence of optimal solutions to the optimal problems associated with the hereditary process (0.3) and the cost functional (0.4). We first describe the assumptions which will be made. (D1) U is a compact convex set, and the set D of controls is the set of all measurable functions u (t) (to<--_t<~tl) with values in U. t 1 depends on u, but t o and the end-points x o, xl are fixed. Concerning t h e / i , we take them to be linear in u, i.e., (3.t)

f (t, x, u) ----g~(t, x) + ~, h~ (t, x) u i j=l

(i = 0, t . . . . . n)

and assume (D~) #(t, x), h~(t, x), ~g~(t, x)/3x k, ~h~(t, x)/~x ~ are continuous functions for to<=t
Optimal Control for Hereditary Processes

4t 3

T h e o r e m 6. Let (I31)--(D3) be satisfied, and assume that (a) there exists at least one control [unction in D such that the corresponding traiectory x (t) satisfies x ( t l ) = xl; (b) there exists a constant A such that /or any control u (t) in D with traiectory x (t) satis/ying x (tl) = xl, the inequality ]x (t) l <__A (0 <=t <=tl) holds. Then there exists an optimal solution o] the optimal problem associated with (0.3), (0.4). The proof is similar to the proof of the analogous result for the process (0.t) as given in [4~, and we therefore omit the details. T h e o r e m 5 can be extended to the case where x0, x 1 are not fixed b u t restricted to some manifolds and also to the case when t o is not fixed. Note t h a t the a s s u m p t i o n (b) is satisfied if the g~ and the h~ are linear functions in x, or, more generally, if

Ig'(t, x)] <-_M(t)(Ix I + t ) ,

Ih~(t, x)l <=M(t)(lx I + t )

where M(t) is a b o u n d e d function on b o u n d e d sets of [0, oo). The proof of this fact is obtained b y estimating directly the right-hand side of (0.3) and using the inequality obtained successively. Combining Theorems 2, 3, 5 and the last remark, we get the following Corollary. Consider the time-optimal problem /or the linear case (2.t) and assume that U is a compact convex polyhedron and that h~(t), dh ~(t)/dt, aj' (t), b (t) are continuous/unctions/or 0<=t < oo. I / t h e condition (a) o/ Theorem 5 is satisfied and i/ the switching set o] any optimal control is finite, then there exists a unique normalized optimal solution, and the corresponding control is a piecewise continuous piecewise-constant /unction whose values are vertices o/ U. In order to m a k e T h e o r e m 5 and its corollary more effective, one should establish direct conditions on the s y s t e m which ensure t h a t the a s s u m p t i o n (a) is satisfied, i.e., t h a t there exists a control u(t) (to<=t<~tl) for which the corresponding t r a j e c t o r y x (t) satisfies x (to) = t o, x (tx) ----x 1. The remaining p a r t of this section is d e v o t e d to this question. For simplicity we take the t e r m i n a l point x 1 to be the origin 0 of R ' , t 0 = 0 , a n d / i : - - / i ( x , u). We also assume (El) U is a c o m p a c t convex set containing 0 in its interior. Definition. A set F is called a set o/ controllability (with respect to 0) if for a n y Xo~F there exists a control u(t) in D (O<=t<=t*) which steers x o into 0, i.e., a control u (t) for which the corresponding t r a j e c t o r y x (t) satisfies x ( 0 ) = x 0, x (t*) ----0. Our p r o b l e m is then: (Po) Find sets X of controllability. F o r the process (0A) this p r o b l e m can be divided into two parts, each of which is t h e n t r e a t e d b y a different m e t h o d : (Px) Find conditions which ensure t h a t some neighborhood N of 0 is a set of controllability. (P~) Find conditions which ensure t h a t every point in some large set X can be steered into some point of N. I t then follows t h a t X is a set of controllability. For hereditary processes, of course, if we solve (P~), (P~), it does not y e t follow t h a t X is a set of controllability.

AVNER FRIEDMAN

4t4

:

In what follows we consider one case for each of the problems (Pi) ( i = 0 , t, 2). We shall take X = R ~. Case 1. Problem (P1) can be solved analogously to the case of the process (0.t) (see [4D. In the case where m>=n the situation is particularly simple, and we shall describe only this case in detail. It suffices to find, for any 9 * c N , a solution x(t; u*, 9*) of t

(3.2)

x(t; u*,

~*)=~*+fH(t--~) l(x(~; u*, ~*), ~*) d~ 0

such that x(t*; u*, 9*)----0 for some t* fixed. Here the control u is taken to be a constant u*, and / is the diagonal matrix Setting

((~i}/i).

A = ( O/i(~176)

B = ( O/i(~ ~ )

we assume (E~) ] (0, 0 ) = 0 and H(0)B is a matrix of rank n. T h e o r e m 7. Under the assumptions (D2), (Ds), (EI), (E~) there exists a neighborhood N o] 0 in R ~ which is a set o/controllability. Proof. Since [(0, 0)----0, if u * = 0 , ~0"=0, then x(t; u*, 9*)~-0, and, consequently, x(t*; 0, 0)----0. Hence if we prove that the matrix [ ~,iit; o, o) /

z (t) =

has rank n at t = t*, then by the Implicit Function Theorem it would follow that the equations x(t*; u*, q~*)=0 can be solved uniquely in the form u*=u*(q~*) for any ~0" in some neighborhood N of 0. Differentiating (3.2) with respect to u *i, for 9*----0, u * = 0 we get t

(3-3)

t

Z (t) = f n ( t -- z) A Z (z) d v + f H(t -- z) B d v. 0

0

It follows from (3.3) that Z(t)----O(t). Hence the first term on the right-hand side of (3.3) is 0 (t2). Take some n • submatrix A in H(0)B with determinant which does not vanish. The submatrix consisting of the same rows and columns on the right side of (3.3) is of the form tA +O(t2). Hence, the corresponding submatrix in Z(t*) is non-singular if t* is sufficiently small, i.e., the rank of

Z(t*) is n. If m < n, then sufficient conditions can also be stated to ensure the assertion of Theorem 7. They are derived similarly to the method for (0A), but they take a more complicated form.

Case 2. We now consider the problem (P2) for linear systems (2.1) with ci ~ 0, a~ ~ constants, hi(t)--h (t) and take u (t) ~ 0. We then want to show that for every y cR ~ the solution x(t) of t

(3.4)

x(t)---- y + f h(t-- z)A x(v)dv 0

comes arbitrarily close to the origin if t increases indefinitely.

Optimal Control for Hereditary Processes

4t 5

T h e o r e m 8. Assume that all the eigenvalues o[ A are negative numbers and that A can be diagonalized. I] h(t)>=O, h'(t)<--<_O, h"(t) h(t)>=(h'(t)) ~ [or all t>=O oo

and i / f h(t)dt=oo, then, /or any y~R n, x(t)---~O as t---~oo. 0

Proof. Let A = P - 1 A o P where A o is tile diagonal m a t r i x (~ii~i), h i being the negative eigenvalues of A. Set ~ = P x, ~ = - P y. Then t

~ (t) ----~ - - J~J f h (t -- z) ~i (z) d z

(i = t . . . . . n).

0

B y results of [~1 it follows t h a t if ~ > 0 , then ~ ( t ) ' x 0 as t T o o , and if ~ < 0 , then ~ ( t ) 7 0 as t / r oo. If ~ is a complex number, then R e ~ ~ and I m ~ ~ t e n d to zero as t---~ oo. Hence, for any y c R ~, x(t)-+O as t-+oo. Theorem 8 can be extended (with the same proof) to non-homogeneous systems (i.e., c i * 0 ) under some conditions on the c~. B y taking controls u ( t ) ~ 0 one can further obtain various different conditions on the c~. Different assumptions on h can also be made, for instance, h ( t ) > _ c o n s t . > 0 , h'(t)<--_O; or h(t) is co

completely monotonic and f h (t) dt = oo. 0

For some special choices of h (t) the conditions on A can be relaxed. Thus, if h(t)=t -~ ( - - t < c ~ < t ) , then the solution of (3.4) can be written in the form

x(t) =E~ (F(fl) t~A) y where E a ( z ) -

(fl = 1 --o~),

zk/F(flk + t) is the Mittag-Leffler function. (For its properties, /~=0

see [11.) Use of the asymptotic formula

Ea(z)--

F(t--fl)zl+O(~ r

as

]zJ-+oo,

Jarg(--z)J<(t--~)~,=

shows t h a t if A can be diagonalized and if its eigenvalues Jti satisfy (3.5)

~=0,

larg( - ~)l =< '+~2 ~ '

then Jx (t)! <=C/ta (C a constant). Thus Theorem 8 holds/or h (t) ----t-~ (-- 1 < ct< t) assuming (3.5) instead o/arg(--~i)----0. Case 3. We shall solve the problem (Po) for X----R ~ in case n = m = t, setting a~ = a, b~----b, h 1-- h and assuming

a(t)<=O,

(3.6)

h(t)>=O for t>--0,

t

(3.7)

f h ( t - - z ) Jb(z)J d z - + oo

as

t--~ oo.

0

Given any y~R 1 we shall find a control u(t) such t h a t Ju(t) I =<*o (where *0 is sufficiently small so t h a t u (t)C U) and such t h a t the solution x(t) of t

(3.8)

x(t) = y + f h ( t - - z ) b ( ' r ) u ( z ) d z + f h ( t - - z ) a ( ' r ) x ( z ) d z 0

vanishes at some time t* (depending on y).

0

4t6

AVNER FRIEDMAN: Optimal Control for Hereditary Processes

Suppose first t h a t y > 0 and take u (t)----- e0sgn b (t). If then x ( t ) > 0 for all t=>0, and, b y (3.8),

x(t)4~o for

all t>=O,

t

x(t)<--_y-- sof h(t-- ~) Ib(~) I d~-~--oo, 0

a contradiction. Suppose next t h a t y < 0 , a n d take u(t)=eosgnb(t ). If x ( t ) ~ 0 for all t=>0, then x(t)=0, and we obtain t

x(t)> y + eof h(t-- z ) Ib(z)l dz-+ oo, 0

a contradiction. We have thus p r o v e d t h a t under the assumptions (3.6), (3.7), the set o] con-

trollability is the whole space. This work was supported partly by a Sloan Fellowship and by Grant G i4876 of the U. S. National Science Foundation.

References [1] ERD]~LYI, A., et al, Higher Transcendental Functions. Bateman Manuscript Project, vol. 3. N e w Y o r k : McGraw-Hill t955. [2] FRIEDMAN, A., On integral equations of Volterra type. J. d'Analyse Math., vol. 11, 381--4t3 (1963). [3] LASALLE, J. P., The time optimal control problem. Contributions to the Theory of Nonlinear Oscillations, vol. 5, pp. 1--24. New Jersey: Princeton University Press 1960. [4] LEE, E. B., • L. MARKUS, Optimal control for nonlinear processes. Archive Rational Mech. Anal. 8, 36--58 (1961). [5] NOHEL, J.A., Some problems in nonlinear VoltelTa integral equations. Bull. Amer. Math. Soc. 68, 323--329 (t962). [6] PONTRYAGIN, L. S., V. G. BOLTYANSKI, R. V. GAMKRELIDGE& E. F. MISHCHENKO, The Mathematical Theory of Optimal Control. New York : Interscience Publishers 1962. Northwestern University Evanston, Illinois

(Received October 6, 1963)