Time Lag Control Systems (Mathematics in Science & Engineering, Vol. 24)

for t E Z[to , to

< t, <

< t,

*..

+ h], =to

G u R.

(2.7.6)

where x, , x2 E R and

i

Let T = {ti}$ be a subdivision of the interval I [ t o , to T : to < t,

=

+ h.

=

1 , 2.

(2.7.7)

+ h] of the form (2.7.8)

Put xo = +(to)and define xj = xi-,

+/

ti

V(t,

*

)) dt,

j = 1,2, ..., N .

(2.7.9)

ti-1

Thus

(2.7.10) Suppose that the inequality (2.7.11)

32

11. DELAY-DIFFERENTIAL

EQUATIONS

is true for k = 1, 2, ...,j - 1. Then applying the inequalities (2.7.10) and (2.7.1 1) we find

II xi

-

xo /I

< I/ xi

- xi-1

11

< M(tj - t j - 1 ) = M(ti

+ I/ xo I/ + M(tj-1 - to) xi-1 -

(2.7.12)

to).

-

Since the inequality (2.7.12) is true for k = 1, because of the inequality (2.7.10) with j = 1, the inequality (2.7.12) is thus true for all j = 1, 2, ..., N . From (2.7.12) we see that

I1 xi

- xo II

< M(tj - t o ) < M h d b,

(2.7.13)

i . e . , x j E R , j = 1 , 2,..., N . Now we define (2.7.14) E I[t,-l , t i ] , j = 1, 2, ..., N . Clearly the function +(t) i s continuous in the interval I [ t o , to h], and

for t

+

x r ( t i ) = xj ,

j

=

1 , 2,

..., N .

(2.7.15)

Furthermore it has a derivative =

*T(t)

. ))

V(t,

(2.7.16)

for t E I ( t j p l, t j ) , and has one-sided derivatives V(ti , and V ( t j ,x i ( . ) ) at the points t = ti. This derivative is continuous in I [ t o , to h] with the exception of the points t = t i ,j = 1, 2, ..., N , and bounded (2.7.17) II kT(t) I/ < M ,

+

+

for all t E I [ t o , to h ] . Moreover, i T ( t ) is continuous at t = to and i T ( t O= ) V ( t o ,xo(.)). Furthermore, for t E I [ ~ ti] ~ we -have ~ ,

I1 X T ( 4

-

xo /I

< II

< M(t i.e., x ( t ) E R for all t

E

- xi-1

-

to)

I [ t o , to

II

+ II

xi-1 -

< M h < b,

+ h].

xo I1

(2.7.18)

2.7. I . Euler's Polygonal Method Let

33 (2.7.19)

< M(tj Let TT

= max(t,

-

- tj-1).

t o , t , - t, , ..., t , - t&.

(2.7.20) (2.7.21)

Thus,

+

W )=

w, w( )) + (5(MTT)j, *

(2.7.22)

for t E I [ t o , to h], t # ti , j = 1, 2, ..., N , where ( 5 ) denotes a quantity whose norm is less than 5 : 11(5)11 5. I t can be easily seen that the relation (2.7.22) is also valid for the two derivatives at t = t i . Integrating Eq. (2.7.22) between t and t A t we find

< +

Now let r be an arbitrary positive number and consider two subdivisions Tl and T, of the interval I [ t o , to h] of the form (2.7.8) such that T ~ , ,T~~ < 7 . Let (2.7.24) 2(t) = X T l ( t ) - X&).

+

Thus

II q t ) II d 2Mk

Making use of Eq. (2.7.23) for T we obtain

t =

I [ t , , to

Tl and T

+4 =

(2.7.25)

T , , and subtracting,

34

11. DELAY-DIFFERENTIAL EQUATIONS

We introduce once more the function m(r) =

sup

0
11 2(t) 11,

Y

< h.

(2.7.27)

As in Section 2.5, we have

Writing

+MT,

(2.7.28)

dp < 3h.

(2.7.29)

p = m(r)

we find, as in Section 2.5, that SMilmlh)

MT

Now consider the integral for p

5(P)

> 0, (2.7.30)

where h is a given positive number. Clearly J(p) is continuous for > 0, J(0) = 0 and is monotonic increasing with p. Since by the property (i) in Section 2.5, c ( p ) = 5(26) for p 3 26, we have

p

lim J ( p ) Y-303

=

+a.

(2.7.31)

2.7. I . Euler's Polygonal Method Thus, there exists a positive number p = p(h) for h

35

> 0 such that (2.7.32)

It can be easily seen from the graphical representation of ( = ( ( p ) that the function p(h) is continuous for h > 0 and is monotonic increasing with A. Thus the limit of p(h) for h + 0 exists. Denote this limit by po . Then po > 0 if the Osgood integral (2.7.33)

converges, and po = 0 if the Osgood integral diverges. Put MT = h and consider the inequality (2.7.29). Then

Thus m(h)

<

(2.7.34)

= P(M'C).

Hence, if the Osgood integral (2.7.33) diverges, we have lim ~(MT =)0. 7-0

(2.7.35)

By the above discussion

+

for t E I[to, to h]. Thus, by Cauchy's criterion of convergence, the approximating function x T ( t ) converges uniformly in I[to , to h] to a limit function x ( t ) for T --+ 0. This limiting function satisfies the initial condition x ( t ) = d ( t ) for t E I t 0, since, by construction, all the approximating functions x T ( t ) have this property. Further, as a limit of a uniformly convergent sequence of continuous functions in I [ t o , to-+ h], x ( t ) is continuous in the same interval. And furthermore, for an infinitesimal refinement of the subdivision T (i.e., for T 40) in (2.7.23) we find, for t = to and A t = t - t o , x ( t ) = 4(to)

+

+J

V ( t ,x( . )) dt,

t

E I [ t o ,to

+ h].

(2.7.37)

t0

Since x ( t ) is continuous for t E I[to , to

+ h] and the functional V ( t ,x(-))

36

11. DELAY-DIFFERENTIAL

EQUATIONS

is assumed to be continuous in t for t e I [ t o ,to + h], differentiation is allowed in (2.7.37). Hence k(t) = V(t,x (

. ))

for t

E ] [ t o , to

+ h].

Thus x ( t ) is a solution of (2.7.1) which satisfies the initial condition x ( t ) = +(t)for t E I f 0. It is also unique by Theorem 2.5, since the Osgood integral is assumed to be divergent. Remark. The inequality (2.7.36) can be used to measure the rapidity of the convergence of the approximation process. In the case where a Lipschitz condition is satisfied, i.e., <(p) = Lp, L being the Lipschitz constant, we have

~ ( M T ) 3LhM7.

(2.7.38)

N

And for ((p) = Lp log( l/p), we find p(MT) 1(MT)expc-3Lh) - MT.

(2.7.39)

2.8. Construction of Solutions: II. Picard's Successive Approximations

We shall shortly present in this section another method of construction of solutions of a delay-differential equation based upon Picard's successive approximations. Consider an equation of the form =

w ,(. . )),

t E I[t, > to

+ 4,

(2.8.1)

where the functional is assumed to be continuous in t and locally Lipschitzian with respect to x with Lipschitz constant L, on I l O f ax D, D being a domain and a > 0. Let G be a compact subset of D and #(t) E C ( I t o ,G). Define (2.8.2)

By Lemma 1 of Section 2.6, there exist constants b such that

II whenever t E I [ t o 5,

+ a],

> 0 and M > 0

w ,4 . )>/I <

x E W+, D), and (I x - 6 (Itotn

(2.8.3)

< b.

2.8. II. Picard’s Successive Approximations

37

Let h = min {a, b / M ) . We have seen in Section 2.6 that, under the above conditions, there is a unique solution of (2.8.1) for t E I[to , to h] which satisfies the initial condition x(t) = +(t) for t E Il0. Now define

+

v = 1, 2,

..., with xow = +(t).

(2.8.4)

(2.8.5)

From (2.8.5) and (2.8.6), we obtain (2.8.7)

+

for t E I[to , to h], v = 0, 1,2, ... . Hence { ~ , ( t is ) }a ~Cauchy ~ sequence in I [ t o ,to h], and the convergence is uniform. Thus this sequence approaches uniformly to a limit function x(t). Since each x,(t) is continuous in I [ t o , to h] and the convergence is uniform, the limiting function x ( t ) is continuous in the same interval. Furthermore

+

+

which converges to zero as v

--+ 00.

Thus,

TX = X,

38

11. DELAY-DIFFERENTIAL EQUATIONS

i.e.,

It follows, from the continuity of V(t,x(*)), that k ( t ) = V(t,x( . ))

for

t

E

I [ t o , to

+ h].

Thus, x ( t ) is a solution of (2.8.1) for t e I [ t O to , initial condition x ( t ) = +(t)for t E l t o .

+ h] satisfying

the

2.9. Dependence of Solutions on the Initial Data Consider the delay-differential equation

where V is continuous in t and locally Lipschitzian with respect to x with the Lipschitz constant L , for t E It,+r*and x E C(If,+a,D), D being a domain. Let G be a compact subset of D, and choose two functions +(t),$ ( t ) E C(I,,, G). By Theorem 2.7, there is a number h > 0 (h a), such that for each initial function belonging to C ( l t o ,G) there corresponds a unique solution of (2.9.1) valid for t E I [ t o , to h]. T h e n by (2.4.41), we have

<

+

II x(t, to 4) I

where t

E

I [ t o , to

+ h] and

E

4)II <

yoeL't-to',

(2.9.2) I

yo

Given

- q t , to,

= II 4

- B Ilrt0*

(2.9.3)

> 0, define 6 = Ee-hL.

(2.9.4)

(2.9.6)

i.e., a solution x ( t , t o , +) depends continuously upon the initial function +(t) E C(I, , G), to being fixed.

2.9. Dependence of Solutions on the Initial Data

39

Assuming to < to*, consider two solutions x(t) = x(t, t o ,+) and x*(t) = x ( t , to*, +*) of (2.9.1), where E C ( I l 0 ,G) and +* E ( I l o * ,G). Thus

+

and

+

I'

V(s, x*( .)) ds

for t E Ito* for t € Z [ t O * , to +) h].

(2.9.8)

to'

to*,

Now we shall establish an upper bound for II x(t, t o , (6) - x*(t, .I!)*+ For this purpose, consider the Picard's iterants for the solutions x(t) and x * ( t ) : for t € I t o xv+l(t) = + V(s,x,( .)) ds for t E Z[to , to h] (2.9.9)

):[:I

f

+

t,

and x,*+l(t) =

d*(t) /+*(to*)

for t €Ito*

+ t O*

V(s,x,*(

*

)) ds

for t E I[&-,*,to -I- h],

(2.9.10)

with

and

We have seen, in Section 2.8, that lim x,(t) = x ( t ) , V+W

uniformly in the intervals I [ t o , to Let

lim x,*(t) = x * ( t ) V-tW

(2.9.13)

+ h] and Ifto*, to + h] respectively.

r = II+*W - +(t> IlltOfh *

(2.9.14)

Since the functional V ( t , is assumed to be continuous in t and x, by Lemma 1 of Section 2.6, there are constants b > 0 and M > 0, such that (2.9.15) II 4 . )) II d M .(a))

w,

40

11. DELAY-DIFFERENTIAL EQUATIONS

whenever t € I [ t O, to

+ a ] , x ( t ) E C ( I l O t aD), , and I1 x

-

6 llIto+m< b.

(2.9.16)

Note that, h = min{a, b / M } .

A straightforward computation shows that

By construction, we have

It can be easily seen by induction that

Ir llx,*,l(t)

- %+#)ii

for t E I,” for t E ~[t,,to*]

< +

i=O

Thus, passing to the limit v

il x * ( t ) Let

E

- x ( t ) I1

--+ 00,

(v

+ l)! )-

for t E I [ ~ , * , to

+ h3.

(2.9.20)

we obtain

i‘

for t e l t 0 for t E I [ t o , to*] Kexpl(t-to*) for t € I [ t O * ,to

d K

+ h].

(2.9.21)

> 0 be given, and define (2.9.22)

Then, for r = 116*w - 6(t)IlltO+h<

and

0 < to* - to < 6,

*

(2.9.23) (2.9.24)

2.10. Dependence of SoZutions upon Parameters we find

II x * ( t ,

to*, +*) - x(t, to Y 4) II

41

<

(2.9.25)

This proves the following. Theorem 2.9 (Continuous Dependence on Initial Data). Let the functional V ( t , x(*)) be continuous in t and locally Lipschitxian with and x ( t ) E C(lto+a ,D), respect to x with the Lipschitx constant L for t E D being a domain. Let G be a compact subset of D. Let +(t)E C ( l t 0 ,G ) and +*(t)E C(Ilo*:G). Consider the solutions x(t, t o , +) and x*(t, to*,+*) of Eq. (2.9.1) valzd for t E I[to, to h] and t E ][to*, to h],respectively, satisfying the initial conditions x(t) = +(t) for t € I t , and x * ( t ) = +*(t) for t E It,* . Let E > 0 be given. Then we have

+

+

/I x*(t, whenever

to*, +*) - x(t, to 9 4) /I < € 9

IIB*(t) - B(t> II < 6 and I to*

- to

I < 6,

where the number 6 is defned by (2.9.22); i.e., the solution x(t, t o ,+) of Eq. (2.9.1) depends continuously upon the initial data {to,4). 2.10. Dependence of Solutions upon Parameters

Consider a functional V ( t ,x(-), A) where x ( t ) is, as usual, an n-dimensional vector function with components (xl, ..., xn), t is time, and h is a k-dimensional parameter vector. The functional V itself is, as we have always assumed, an n-dimensional vector-valued functional with components ( V , , ..., Vn). Let P be the space of parameter vectors h = (h,, ..., hk), and define J,(Ao) = {A

where A, = (Xlo , ..., )A', of ( t , x, h) space, QA

= {t

E

E

p : II A

- A0

P is fixed and 7

Ita+= Y

II < ?>,

> 0. Let SZA be

x E C(Zt0+,, 4, A

J,(Ao);.

Consider now the delay-differential equation *(t) = V(t,x( . ), A)

(2.10.1)

for t E I[to , to

+ a].

the domain (2.10.2)

(2.10.3)

Let G be a compact subset of D and choose any +(t)E C(lt,+a, G). Suppose that the functional V is continuous on SZA and satisfies a

42

I I. DELAY -D IFFERENTI AL EQUATIONS

Lipschitz condition in x uniformly on a,.Let x ~ ( t= ) x(t, t o ,4, A) be a solution of (2.10.3) satisfying the initial condition xA(t) = +(t)for t E I , , . T h e existence and uniqueness of xA(t) for each h E ],(Ao) is assured by Theorem 2.7. Now we shall prove the following Theorem 2.10 (Continuous Dependence upon Parameters). Let the above hypotheses be satisjied. Let x"t) be a solution of (2.10.3) in an a. Then, given E > 0, there exists a interval t E I [ t o , to 163, 0 < 16 6 > 0 such that

<

+

/I %(t) whenever X

E Ja(Xo);

- XA,(t)

I1 <

+ 81,

for t E I [ t O to 9

(2.10.4)

i.e., the solution xA(t) is continuous in A.

Consider the system of delay-differential equations

Proof.

j ( t ) = W(t,y( . ))

where y

=

(xl

, ..., x, , A, , ..., A,)

for t E q t o , t o

and

W

= (V,,

+ a],

(2.10.5)

..., V, , 0, ..., 0).

Obviously the functional W is continuous in t and Lipschitzian with Define respect to y on a,. @A

= (#l

***, #n

Y

A

+*.v

J,(AO)*

Clearly @ A ( t ) is continuous and bounded for t € I L , . Thus, by Theorem 2.7, the system (2.10.5) has a unique solution y ( t , t o , @A) which satisfies the initial condition y ( t ) = @,(t) for t E I . Moreover, !. by Theorem 2.9, the solution y(t, t o , @ A ) is continuous with respect to the initial function @ A ( t ) . Thus, given E > 0, there exists a number 8 > 0 (6 7)such that

<

II A t , to whenever i.e.,

11 h

-

1

@A)

/I @ A A,

- Y ( t , to -

9

@A,)

II < E

@loll < 8;

11 < 6. This proves the theorem.

2.11. Dependence of Solutions on the Right-Hand Side of the Equation

Consider an equation of the form

44 = q t , (. . )) + q t , 4,

t E4

t o , to

+ 4,

(2.1 1.1)

43

2.12. Gronwell’s Lemma

where 6(t, x) is, like x and V, an n-dimensional vector function, piecewise continuous for t E Ito+aand X E D ,D being a closed domain in En. Suppose that the functional V ( t ,x ( . ) ) is continuous in t and locally Lipschitzian with respect to x with the Lipschitz constant L for t E and x E C ( I f O f D a ,) and that

I1

w,x) I/ <

(2.11.2)

E

for t E l f o + a and x E C(Ito+a, D ) . Let Z ( t , t o ,+) be a solution of (2.1 1.1) and x(t, t o ,+) be a solution of the equation (2.11.3)

*(t) = V(t, x( * )).

Obviously x(t, t o , +) is an +approximate solution of Eq. (2.11.3). Thus, in view of the inequality (2.4.25), we have

I1 x”(4to 4) - x ( t , t o ,4) I/ < (+)(eL“-”) 9

+

- 1)

(2.1 1.4)

for t € I [ t o ,to h]. Clearly, we may make the right-hand side of (2.1 1.4) arbitrarily small by choosing e sufficiently small. 2.12. Gronwell’s Lemma

I n the sequel we shall require the following. Gronwell’s Lemma. I f f ( t ) ,g(t), and h(t) are nonnegative continuous functions in an interval I[to , to a] satisfying the inequality

+

(2.12.2) for t

EI[tO

Proof.

, to + a].

T h e function (2.12.3)

is continuous and continuously differentiable in I [ t o , to y(to) = 0. Moreover

W )=m + r(t)

+ .I] and (2.12.4)

44

11.

DELAY-DIFFERENTIAL EQUATIONS

(2.12.5)

Consider now the function I

rt

(2.12.6)

In view of the inequality (2.12.5), we have

Obviously z(to)= 0. Integrating both sides of the inequality (2.12.7) between to and t, t E I[to , to a],we get

+

(2.1 2.8)

It follows from (2.12.6) and (2.12.8) that (2.12.9)

The relations (2.12.4) and (2.12.9) give immediately the required inequality (2.12.2). Note that Gronwall’s lemma is true for more general functions f,g, and h, which are, for example, piecewise continuous in I[to , to 011.

+

2.13. Differentiability Properties of Functionals

In this section we shall brieffy sketch the differentiability properties of functionals. For a detailed study we refer to V. Volterra [l-31, E. Hille and R. S. Phillips [l], and L. A. Ljusternik and W. I. Sobolev [I]. Let F = F[x(t)] be a functional defined for an ordinary function x ( t ) E C(l[a,b]). F[x(t)] depends on all the values taken by x ( t ) when t ranges over the interval I[a,b]. The functional F may also depend upon certain parameters A,, ..., A, , i.e., F = F [ x ( t ) ;A], A = (Al, ..., A,). I n this case F, for each A, is a functional of x ( t ) , and is an ordinary function of A when x ( t ) is fixed.

2.13. Dzzerentiability Properties of Functionals

45

Let us give to x ( t ) an increment 8 x ( t ) = ~ ( tand ) consider the corresponding increment AF of the functional F [ x ( t ) ] : AF = F [ x ( t )

+ x(t)] - F[x(t)l.

(2.13.1)

Suppose that X ( t ) does not change its sign and that 1 x(t)l < c and ~ ( t= ) 0 outside an interval I(r, s) of I [ a , b] of length h = s - r, containing the point T in its interior; then T E I(r, s) C I [ a , b]. Let u =

s’

~ ( tdt. )

(2.13.2)

7

Suppose that:

<

(1) the ratio I AF/Eh 1 M , where M is a finite number; ( 2 ) limr+O,h+O AF/o = F’[x(t);T ] uniformly with respect to all possible functions x(t) and all points T ; (3) F’[x(t);T ] is finite and continuous in x and T . Then: Definition 1. Under the above conditions F‘[x(t);T ] is called the Jirst derivative of the functioiral F[x(t)]with respect to the function x ( t ) at the point T .

Obviously F’[x(t);T ] is a functional of x ( t ) and an ordinary function of the scalar parameter T E I[., b ] . Definition 2.

The quantity (2.13.3)

is called the dzfferential or first variation of the functional F [ x ( t ) ] . Let us give an increment ~ # ( t to ) the function x ( t ) where # ( t ) is a continuous function in the interval I [ a , b ] . Let AF be the corresponding increment of the functional F [ x ( t ) ] i.e., , AF = F [ x ( t )

+ 4 ( t ) ] - F[x(t)].

(2.13.4)

Fixing x(t) and #(t), let us vary the parameter E. Then AF may be considered as a function of E. Let us determine the limit

46

11. DELAY-DIFFERENTIAL

EQUATIONS

V. Volterra [3] has shown that F'[x(t);TI#(.)

dT.

(2.13.5)

+ y,

(2.13.6)

Thus AF

=E

S b F ' [ r ( t ) T; ] # ( T ) dr a

where y is an infinitesimal quantity of higher order than (2.13.3) and (2.13.6) we see that

E.

Combining (2.13.7)

AF=e6F+y.

When the functional F'[x(t);T ] is differentiable, we may apply the formula (2.13.5) to it. Th u s we find the second-order derivative with respect to x(t) at the point T of the functional F [ x ( t ) ] :

Generally, under certain continuity conditions, we find that the nth order derivative is given by ($F[X(t)

f

=

E$(t)]) *=O

/:

.'' r F l n ) [ X ( t ) ;71, ..., T,]#(T1) ".$(Tn)d71"'dTn. (1

(2.13.9)

I t can be shown that F("'[x(t);T~ , ..., -i,] is a symmetric function of the n parameters T ~ ...,, 7,. Supposing x(t) and $(t) to be fixed; consider

Let us assume that the functionalF[x(t)] is differentiable u p to nth order. Th e n Taylor's theorem yields f(1)-f(0)

df 1 dn-lf (---) + (z ) + ..' + ___ (n I)! '

S=O

where 8 lies between 0 and I .

-

C=O

1 dnf +-n! ( den

2.13. Dzyerentiability Properties of Functionals

47

Therefore

(2.13.10)

which is an extension of the ordinary Taylor’s formula to differentiable functionals. Note that differentiable functionals considered above do not exhaust the class of all functionals F[x(t)] having a differential, because the differential 6F, in the preceding cases, was a regular linear functional of Sx(t). But in many cases the variation 6 F depends specia2Zy on Sx(t) and on its derivatives at some points of the interval. I n general, for every differentiable functional F[x(t)] we have the following relation:

+ x(t)l

@(t)

=

Wt)l

+ Wx(t), x(t)l + +(x),

(2.13.11)

where L(x) denotes the norm of x,7 is a quantity which tends to zero with L(x),GF[x(t),~ ( t ) is ] the first variation of F[x(t)] which corresponds to the increment Sx(t) = ~ ( tof) x. Note also that GF[x(t), ~ ( t ) is] a functional of two functions x ( t ) and ~ ( t )linear , with respect to ~ ( t ) . Putting X(t) = E$(t) in (2.13.11) we see that d

( Z W )+ &)I)

F=O

=W ( t ) *

#(t)l.

(2.13.12)

If the function x depends on a scalar parameter h as well as on t, assuming the differentiability of the functionalF[x(t; A)] for < h < /3, a < t < b, we obtain (Y

a

- F [ x ( t ; A)] = GF[x(t;A), x,’(t; A)] ah = I l F ’ [ X ( t ; A);

T]Xh‘(T;

A) dT.

(2.13.1 3)

If the functional F[x(t)] has a relative maximum (or minimum) at x o ( t ) = x ( t ; ho), i.e.,

@O(t)l

if always

> F[x(t)l

(or F[xrJ(t)l < &”

48

11. DELAY-DIFFERENTIAL EQUATIONS

for x ( t ) sufficiently near to x,(t), then GF[x(t;A), x'(t; A)] = 0

for A = A,, x'(t; A,) being an arbitrary function of t. T h u s the variation W [ x ( t ) ,#(t)]of a functional is zero for those functions xo(t)which make the functional itself a maximum or minimum. For a differentiable functional, whatever T may be, F'[x(t);T ] = 0 for those functions x,(t) for which the functional is a maximum or a minimum. Finally, we shall state the following definitions which we shall need in the sequel. Definition 3.

for all x, y

E

Let F[x(t)]be a functional dejned in a function space x. If

+ PA41 = @4t)l + PF[Y(f)l,

x and for all scalars a,p, then F [ x ( t ) ]is said to be linear.

Definition 4.

A linear functional F[x(t)]is said to be bounded in x, i f

!W 4 l ! for all x

E ,Y,

(2.1 3.14)

<

ll x Il

(2.13.15)

where M is a finite number.

Definition 5.

The quantity

IIF I/ = SUPIl F[x(t)lI; I1 x ( t ) I1 < I >

(2.13.16)

is called the norm of the linear functional F[x(t)]. Remark. Suppose that the functional F[x(t)] has derivatives of all orders and that the limit of the last term in (2.13.10) is zero for n + co. Then we have

W t )

+ I)@

+2

d f i !

=

Wt)l

1: 1;.'.

rF'.'[X(t); a

TI

, ..., T ~ ] $ ( T ~...) $(T~)dT1 - * * dr,, . (2.13.17)

2.14. Differentiability Properties of Solutions with Respect t o Parameters

Consider an equation of the form (2.14.1)

2.14. Dgerentiability Properties of Solutions

49

where A, as in Section 2.10, is a K-dimensional parameter vector, and t , x, and V have the usual meaning. I n this section we shall discuss differentiability properties of solutions x ( t , A) = x(t, to , +, A) with respect to A,, ..., A, , components of the vector A. We shall denote by C r ( H )the space of all functions having continuous T derivatives (ordinary or functional) in the domain H . Define Q = {t E

Ito+. > x E D, A

E

JdAn)},

(2.14.2)

where D is a domain in En,A, is a fixed element in the parameter space P , 01 and 6 are positive numbers, and JXA,) is defined by (2.10.1). Suppose that V E C1(Q). Let G be a compact subset of D,and consider any + ( t ) E C 1 ( l t 0G , ). Let h = (A,, ..., A), and AX = (0, ..., 0, A h , , 0, ..., 0). Set dx(t, A) = x ( t , A

and dV

=

V(t,x( . , A)

+ Ah)

-

x ( t , A)

+ dx(t, A), A + Ah)

-

V ( t ,x( . , A), A).

By Theorem 2.10, the solution x(t, A) is continuous in A, i.e., lim dx(t, A)

AA+O

= 0,

and for sufficiently small 11 Ah 11 the point (x in Q. Moreover, by Section 2.13, we have d V = J [ t ,A; AX(

+ A x , t , A + Ah) remains

+ yv(t,A)dA, + ?L(dx(t,A), AX,)

. , A)]

(2.14.3)

where

At, A; 4 .

=

s v t , (.

. , A),

A;

4 . ,A)],

(2.14.4)

i.e., the first variation of V[t,x( -,A), A] with respect to the vector x(t, A) which corresponds to the increment d x ( t , A), and (2.14.5)

Note that J [ t , A; Ax(t, A)] is a linear functional of d x ( t , A) and that J ( t , A; d x ( t , A)] and y V ( t ,A) are n vectors. Moreover, in (2.14.3) L(Ax(t, A), Ah,) is an n-dimensional vector whose norm is an infinitesimal of higher order than that of I( Ax(t, A)ll and that of I/ A h , 11, and 7 is an n x n matrix such that lim

AAv+O

II 77 Ilft,+a

i.e., 7 + 0 with Ah, uniformly in t for t

= 0, E

I[to,t ,

+ a].

50

11. DELAY-DIFFERENTIAL EQUATIONS

On account of the continuity of x(t, A ) in t and in A, t, A E Q, J[t, A; dx(-,A)] with x(t, A ) and y.(t, A ) are continuous in Q. With the above notation we have

where L*

=

7

L ( A x ( t , 4, 4

4

)

which is continuous, and L* 4 0 for Ah, ---f 0, and h is as usual. Consider now the following delay-differential equation

with the initial condition z V ( t )= 0 for t E l t 0 Since . this equation is linear in x and continuous in t, it satisfies a Lipschitz condition in Q. Hence, there exists a unique solution zv(t)with the initial condition z,(t) = 0 for t E 1," . Clearly the solution x,(t) is continuous for t €1[tO , to h]. Consequently /I z Y ( t ) / /is bounded in I [ t o , to h]. Furthermore

+

+

Set Ax(t A) v(t) = A - z,(t). A A"

It follows from (2.14.6) and (2.14.8) that

Whence, writing

(2.14.101

2.15. Analyticity Properties of Solutions

51

and making use of the linearity of J in v(t), we get

0 < u(t) <

s’

[Mu(s)

ta

+ I/L* 111 ds,

t € I [ t 0 , to ,

Clearly the function u(t) is continuous in I [ t o ,to (2.14.12) Gronwall’s lemma, we find

<

~ ( t ) h 11 L* /I ehM,

Since 11 L* 1) + 0 with Ah, to (2.14.10),

-+ 0,

t E I[to , to

+ h].

(2.14.12)

+ h]. Applying

+ h].

to

(214.13)

u(t)-+0 for Ah, -+ 0. Thus, according (2.14.14)

Hence

ax(t’ to

’”

exists, is continuous in t for I[to, to

+ h], and is

ah” the solution of the delay-differential equation (2.14.7) which satisfies

the initial condition z ( t ) = 0 for t € I t o . Taking into account the continuity of J[t, A; x(-, A)] and yv(t,A) in I[to, to a ] , we can easily see

+

that dx(t’ “ ’” ’) exists throughout 52 and satisfies (2.14.7). ax, We can proceed from class C’ to class C2 by replacing the given equation (2.14.1) by (2.14.7), etc., down to class C’. Furthermore, we see, by ordinary or functional differentiation, that the solutions x(t, t o ,4, A) of (2.14.1) with V E C‘(L2) are of class C“l(I[t, 9 to 013, D). Thus we have the following.

+

Theorem 2.11 (Differentiability with Respect to Parameters). Let the functional V ( t ,x(-), A) E CT(52),where Q is dejined by (2.14.2). Let G be a compact subset of the domain D. Choose any +(t)E C r ( I t 0 G). , Then, the solution x(t, to , 4, A) of the equation i ( t ) = V ( t ,x(*), A), where ( t ,4, A) E 52, is of class Cr in A and of class Cr+l in t.

2.15. Analyticity Properties of Solutions

Consider a delay-differential equation qt)

=

V ( t , x(

. ), A),

t

E I [ t o , to

+ a].

(2.15.1)

Let the functional V = V(t,x(*), A) be analytic with respect to all its arguments; i.e., let the components V , , V 2, ..., V , of V be analytic, in

52

11. DELAY-DIFFERENTIAL

EQUATIONS

the ordinary sense, in t and in A,, A,, ..., A,, the components of the parameter vector A, and have a Taylor series expansion of the form (2.13.17) in terms of functional derivatives with respect to the vector , A E Js(Ao), and x ( t ) E Cm(Ilo+a , D).Define function x(t), for t E (2.15.2)

Let G be a compact subset of the domain D.Choose an analytic initial function +(t)E C r n ( I i 0G). , By existence and uniqueness theorems there exists a unique solution x(t, t o ,+, A) o f (2.15.1) in I [ t o ,to h] for each A E J6(ho)such that x(t, t o ,4, A ) = +(t) for t E I l 0 ,h being defined by (2.6.8): h = min{a, b / M } , where the constant M is such that

+

for all ( t , x, A) E Q* (note that, since the functional V is supposed to be analytic in A, it is bounded there). By Theorems 2.10 and 2. I 1, the solution x(t, to ,4, A) is continuous and differentiable in t and A,, A, , ..., A, . Consider now Picard’s iterants for x(t, to ,+, A):

with

It can be easily verified that each iterant x.(t, t o ,+, A) is analytic in t and in A,, A 2 , ..., A, in I [ t o ,to h] and J6(Ao) respectively. Owing to the analyticity of V(t,x(-), A) in Q*, the inequality (2.15.3) is satisfied uniformly in Q*. Thus, by the process in Section 2.8, the sequence {xv(t,to,4, A)} approaches its limit x(t, to,+, A) uniformly for t EI[to,to h] and h E J6(Ao). By extending the interval I [ t o ,to + h] as we did in Section 2.6, we see that x(t, t o ,+, A) is analytic for t E I [ t o ,to a ) and A E JdAo). Thus we have the following extension of PoincarC’s expansion theorem.

+

+

+

Theorem 2.12. If the functionaE V ( t ,x(*), A) is analytic with respect to all its arguments in the region Q* and if the initial function +(t)is chosen to be analytic ir,( I I o, G),where 8*is defned by (2.15.2) and G is a compact subset of the domain D,then the solution x(t, to , A) of the delay-dtgerential

+,

2.16. First Variation of a Solution equation $ t ) E J6(ho)-

=

53

V ( t , x( .), A) is analytic in t and h f o t~E I[to , to

+ a) and

Remark. If the functional V ( t ,x(-), A) is analytic in x and h and continuous in t, by the same argument as above, the solution x ( t , t o ,+, A) is analytic in h and continuously differentiable in t , h E J6(h0) and t E q t o , to a).

+

2.16. First Variation of a Solution with Respect t o an Initial Function

Let the functional V = V(t, x(-)) be continuous in t and differentiable with respect to the components of the vector function x ( t ) ; i.e., let the components Vl , ..., V , of V be continuous in t and differentiable, in the functional sense, with respect to xl, ..., x, , the components of x, for t E I,,, and X E C1(lt,+or, D),D being a domain in En. Let G be a compact subset of D . Consider the delay-differential equation k ( t ) = V(t, x(

. )),

t

E

q t o , to

+ a].

(2.16.1)

By the existence and uniqueness theorems, for each initial function +(t)E C 1 ( l t oG), , there corresponds a unique solution x(t, t o , 4) of Eq. (2.16.1) for t € I [ t O to , h], where h is as usual. Now we give an increment

+

€ A + = (0, ...)0,E A + # , 0, ...)0)

to the initial function

d + E Ad

+

= (+1

=

(dl

9

.*a,

, ..., 4,). Consider the function dj-1 ,dj E 4, +$+I ...,d,).

+

9

Assume that A+ E C1(Ilo,G) and that E > 0 is sufficiently small such that EA+ E C 1 ( I t oG). , Consider the solution x ( t , to , €A+) and set (2.16.2) 9) = x ( t , to ,d E 4) - x(t, to 9 $1.

++

++

+

By Theorem 2.9, we have lim A x ( t , 4) r-10

uniformly in I [ t o , to

+ h].

Let

=0

(2.16.3)

54

11. DELAY-DIFFERENTIAL

EQUATIONS

I n view of the continuity of V in x, we have lim A V = 0. r-to

With this notation we may write

Let us denote by J = J [ t ,x ( t , t o ,4);A x ( t , 4)] the first variation 8V of the functional V corresponding to the increment dx(t, 4)of x ( t , t, ,4). As we have seen before, J is a functional of two functions x and A x , linear with respect to A x , and an ordinary function of the variable t. Furthermore, by Section 2.13, we have Al'

=

J

+

y

II AXII,

(2.16.6)

where y is an n vector whose norm is an infinitesimal quantity of higher order than 11 A x 11, i.e., lim 11 y

AX+O

Ij

(2.16.7)

= 0.

Hence, using the linearity property of J in Ax, we may write

for t € I [ t o ,to

+ h].

(2.16.8)

Consider now the equation

with the initial condition z ( t ) = A+(t) for t E I l 0 . Since the functional J is linear with respect to z and continuous in t, it satisfies a Lipschitz condition with respect to z for all t E I [ t o , to h ] , z E D. Thus, Eq. (2.16.9) has a unique solution z ( t ) for t E I [ t o , to h], satisfying the initial condition z(t) = A+(t) for t E I t o fObviously z(t) is continuous in l [ t o ,to h ] , and therefore bounded there. Putting

+

+

+

(2.16.10)

55

2.16. First Variation of a Solution we obtain

[

0

for t € I t ,

w jl0m, =

X(SI

+ r ( l l 4 s ) + 4 s ) ID> ds

t o 7 $4;fJ(41

for

+

t EI[tO9 to 4. (2.16.1 1)

Let us denote by M the norm of the functional J (M is defined by (2.14.7)). Let u(t>= II

fJ

1 1 1 ,*

Then, making use of the linearity of J in o, (2.16.10) yields u(t)

< N + ( M + K ) j t u(s) ds, to

t EI[to,to

+ h],

(2.16.12)

where (2.16.13)

Whence, by Gronwall’s lemma, we have u(t)

< Neh(M+K),

t € I [ t Oto ,

+ h].

(2.16.14)

+

Since z ( t ) is continuous and bounded in I[to, to h] and K -+ 0 with d x by (2.16.7), combining (2.16.3), (2.16.7), (2.16.13), and (2.16.14) we see that lim,,,u(t) = 0, i.e., (2.16.15)

+

uniformly in I [ t o ,to h]. By definition, the left-hand side of this equality is the first variation of x(t, to , 4) with respect to 4 corresponding to the increment E A 4 of 4. Therefore w t , to I

$1

=4t),

+

t E I [ t , to 1

+ 4.

(2.16.16)

By extending the interval I [ t o ,to h ] , as we did in Section 2.6, we see that (2.16.16) is valid for t e I [ t O to , a). We formulate the above result as follows.

+

Theorem 2.13 (First Variation with Respect to the Initial Function). Let the functional V ( t , x ( - ) )be continuous in t and daxerentiable, in the functional sense, in x for t E and x E C1(I,p+a, D), D being a domain in En. Let G be a compact subset of D. Then the solution x(t, t o ,4) of

56

11. DELAY-DIFFERENTIAL EQUATIONS

(2.16.1), where +(t)E C 1 ( I t oG), , is dzyerentiable, in the functional sense, with respect to the initial function +(t). The $first variation Sx(t, to ,4) of x(t, to ,4) with respect to 4 corresponding to an increment E A 4 is equal to the solution z ( t ) of the linear delay-dzflerential equation (2.16.9) satisfying the initial condition z ( t ) = A+ for t E I f o. 2.17. Differentiation with Respect to the Initial Moment to

Consider an equation

q t ) = V(t,x( . )),

t E I [ t O , to

+ 4,

(2.17.1)

where V is assumed to be continuous in t and differentiable, in the functional sense, with respect to x for t E and x E C1(Ilo+a, D), D being a domain in En.Let G be any compact subset of D and choose any 4 ( t ) E C1(Ifo,G). Consider the solution x(t, t o ,4) of (2.17.1). I t satisfies the functional equation ((t) x(t' to ")

= !((to)

+

V(s,x( . , t o , $)) ds

for

t E Ito

for

t E I [ t o , to

+ h], (2.17.2)

t0

h being as usual. Now let us give an increment At, to the initial moment t o , and consider the solution x(t, to At, ,$), where

+

(2.17.3)

2.17. Initial Moment to

57

We know, by Theorem 2.9, that lim Ax(t) = 0

Ato+,

+

uniformly in I[to , to h], and because of the continuity of the functional V in x (by hypothesis V is supposed to be differentiable with respect to x), lim A V ( t ) = 0.

Ato+,

Moreover, with the previous notation, we have

where y is an n vector such that lim

Ato*,

11 y 11

(2.17.8)

= 0.

From (2.17.3) and (2.17.4), we find 0 -- -

At,

for t

1 _At,

st

J

t"

E It"

V(s, x( . , t o ,d)) ds

for

t E I [ t o , to

for

t E Z[to

+ At,]

+ A t , , to + h]. (2.17.9)

Consider now the following linear functional differential equation

Clearly, it has a unique solution z(t)which satisfies the initial condition z(t) = - V ( t o , d(.)) for t € I I o . Note that z(t) is continuous, and therefore bounded in ] [ t o , to h]. Furthermore

+

Put (2.17.12)

58

11. DELAY-DIFFERENTIAL

EQUATIONS

A straightforward computation shows that

+

Let t be any number in the interval I ( t o ,to h ] . Keeping this t fixed let us choose any A t , such that At,, < t - t o . Hence, making use of the last equality of (2.17.13), we find u(t)

< K + (111 + N ) j t u(s) ds,

t e I ( t o , to

t,

+ h],

(2.17.14)

where M is the norm of the functional J defined by (2.14.7) and N

=

/I Ylll,,,

K

=

hN

+ (M + N ) j

to+Atn

/I 4 s ) II ds

to

(2.17.15) Now, let us apply Gronwall’s lemma to the inequality (2.17.14). We get u(t)

< Keh(M+N),

t

E

I ( t o ,to

+ h].

(2.17.16)

Making use of the relation (2.17.8) we see that (2.17.17) uniformly in ] ( t o ,to

+ h ] . Hence

dx(t, to ’ #) = Z ( t ) ,

at,

Thi s proves the following.

t

€

Z(to , to

+ h].

(2.17.18)

2.18. Measurable Right-Hand Sides

59

Theorem 2.14 (Differentiation with Respect to the Initial Moment). Let the functional V ( t , x(-)) be continuous in t and dzfferentiable, in the functional sense, with respect to x, for t E and x E C1(Zfo+a, D), D being a domain in En.Let G be a compact subset of D. Then the solution x(t, to , +), where +(t)E C1(Ifo,G), is dzfferentiable with respect to the ax(”to”) is equal to the initial moment to and the partial derivative ---at0 solution z(t) of Eq. (2.17.10) which satisfies the initial condition x(t) = - V ( t , , for t E I f o. +(a))

2.18. Delay-Differential Equations with Measurable Right-Hand Sides

Consider the delay-differential equation V(t,x( . )),

A(t) =

t E I [ & ,to

+ a].

(2.18.1)

Frequently, especially in many control problems, the functional V ( t , x( *)) is Lebesgue-integrable, rather than continuous, in t. In this section we shall briefly investigate this kind of equation. First, we shall give some definitions. Definition 1. Let x ( t ) be an n vector defined for I[a, b]. Suppose that I(tl , t , hl), Z(t2 , t, h2), ..., I ( t , , t, h,) are nonoverlapping intervals in Z(a, 6 ) . If,for each given E > 0,we can find a 6 > 0, such that

+

+

+

for all choices of intervals with (2.18.3)

x(t) is said to be absolutely continuous in I[a, b]. I t is well known that: (i) an absolutely continuous function is continuous; (ii) an absolutely continuous function on I[a, b] has a derivative almost everywhere (i.e., except on a set of Lebesgue measure zero) on Z[a, b].

60

11. DELAY-DIFFERENTIAL EQUATIONS

Definition 2. The functional V(t,x ( * ) )is called measurable in t , if V ( i ,x(.)) is a measurable function in t for t E I ( t o , to a] for all x(t) E c(40+ol 9 D).

+

Definition 3. An absolutely continuousfunction x“(t),whose range is D, is said to be a solution of Eq. (2.18.1), in the extended sense on I[to, to a ] , i f i ( t ) = V ( t , x“(.)) for all t E I [ t o ,to a ] , except on a set of Lebesgue measure zero.

+

+

+

Note that, if V is continuous on I[to, to a] and Z(t) is a solution of (2.18.1) in the extended sense, then i ( t ) E C(I[to, to a ] , D ) , and

+

therefore x”(t)is also a solution in the ordinary sense. Let D be a domain in En,and G be a compact subset of D. Let us , ) and define choose any 4 ( t ) E C ( l f oG (2.18.4)

In this section we shall consider the set R = { t E Ito+u>

II x

-

B /I <

(2.18.5)

4 - a

Now we shall prove the following. Theorem 2.15 (Extension of Caratheodory’s Theorem). Let V(t,x( .)) be a functional defined on the set R, and suppose that it is measurable in t for each fixed x , continuous in x for each fixed t. If there exists a Lebesgueintegrable function m(t)for t e I [ t o ,to a] such that

+

1I v t , 4 . )) II

e W)?

( t , 4E R ,

(2.18.6)

then there exists a solution of (2.18.1) in the extended sense on some interval q t o , to 81, 8 > 0, satisfring

+

x ( t , to

1

4)

=4(t)

for t E Ito

1

(2.18.7)

where +(t)E C ( I t o ,G). Proof.

As in the case of ordinary differential equations, consider the

function

Clearly the function M ( t ) is nondecreasing and uniformly continuous in I f o + a .Let e be the n vector whose components are equal to 1. From

2.18. Measurable Right-Hand Sides

61

the properties of M ( t ) we see at once that there exists a constant /3, 0 -==c /I < a, such that ( t ,4 + M(t)e)E R for t E ] [ t o , to + /I]. Let the approximations xY(t), Y

=

1, 2,

...,be

defined by

Clearly x l ( t ) is defined on Ito+Bwhere x l ( t ) = $(t);for any Y 2 2 the first expression in (2.18.9) defines x Y ( t ) on Ilo+B,v, and since (t,r$) E R for t ~ l ~ , the + ~second / ~ expression of (2.18.9) defines x,(t) as a continuous function on the interval I(to (/3/v),to (2/3/u)]. Moreover, making use of the relations (2.18.6) and (2.18.9), we find

+

II x,(t)

- B(t)

II < M ( t - (PI.)

for t

EI(t0

+

+

to

PIY,

+ (2Piv)I.

(2.18.10)

Let us assume that x Y ( t ) is defined on IjS/”for 1 < j < Y . Then, j/?/u, the second expression of (2.18.10) defines x Y ( t ) for I(to to + ( j + 1)/3/u], since, for this interval, we only need the measurability of the integrand on I j S / .. It can be easily verified that the inequality (2.18.10) is valid on I [ t o + j / 3 / u , to ( j 1)/3/u]. Therefore, all the approximations x Y ( t ) are continuous functions for t E I [ t o , to 81, with the properties

+

+ +

for t

= B(t)

II ~ ” ( t-) &t) II

< M(t

-

PI.)

+

6 It0+Bl”

+ (Pi.), to + 81.

(2.18.1 1)

for t ~ I ( t o

Moreover, we have

+

for t, , t, d [ t o ,to 81. Making use of the uniform continuity of the function M ( t ) on I [ t o ,to /I] and the inequality (2.18.12), we see that the set (x.(t)} is equicontinuous on the interval I [ t o , to /3]. Hence, by Ascoli’s theorem, there exists a subsequence {xY,(t)} of {xY(t)}which converges uniformly on I [ t o , to /3] to a continuous limit function x ( t ) , as K -+ co.By the definition of the number /3, all the x y ( t ) belong to R for I [ t o , to 81, and on account of the hypothesis of (2.18.6),

+

+

+

+

II V(t,XYk( . N /I < 4 t h Since V(t,

))a(.

t

E

I[to t o ?

+ 81.

is supposed to be continuous in x for fixed t, we have lim V(l,XYk( . 1) k-m

=

q t , 4 . I),

62

I I . DELAY -DIFFERENTIAL EQUATIONS

+ /3].

for every fixed t in I [ t o , to of Lebesgue we may write lim

k-tm

By the dominated convergence theorem

j” V(s, xvk( . )) ds

(2.18.1 3)

=

4J

for every t in Z [ t o , to

+ /3].

T h e second equality of (2.18.9) yields

where, clearly, the latter term approaches zero as k + oc). Thus, passing to the limit for k = m in (2.18.14) and making use of the equality (2.18. I3), we find

which proves the theorem. We now deal with the question of uniqueness for solutions of Eq. (2.18.1). As in Section 2.5, consider the set S of all functions x ( t ) E C(l,o+a, D ) , which satisfy the following conditions: (1)

4 t ) = ?xt)

(2) I/ 44

-4(to)

I/

<6

where d ( t )E C(It0,G ) ;

for

t €Ito,

for

t EI[tO, to

+ .I;

and, supposing that V(t,x ( - ) ) is bounded and measurable in t and continuous in x , let us define

+

X(P> = SUP{Il

v(t> 4 . ))

for t € ] [ t o , t , a ] , /I x ( t ) - r(t)ll We can easily show that:

-

v(t> Y ( . 1) Ill

(2.18.16)

< p, x , y E s.

(i) x(p) >, 0 and x(p) is monotonic increasing with p ; x(0) = 0 and x(p) = ~ ( 2 6 for ) p >, 26; ( 4 x(p) is subadditive, i.e.9 x(p1 (iii) x(p) is Riemann-integrable.

+

P2)

< x(p1) + x(p2);

By the same procedure as in Section 2.5, we may prove the following. Theorem 2.16. Let the functional V ( t , x ( - ) ) be bounded and measurable in t and continuous in x f o r t E Zlo+ix , x E C(ZlO+, , D ) , where D is a domain

2.19. I . Auxiliary Theorems

63

in En.Let +(t)E C ( I t 0 ,G), G being a compact subset of D. Consider the function x(p) which is defined by (2.18.16). If the integral

soxo dP

(2.1 8.17)

is divergent, then Eq. (2.18.1) has at most one solution x(t, t o , 4) on I[to, to a],with the given initial function +(t).

+

2.19.

Boundedness and Stability: 1. Auxiliary Theorems2

In this section and in the next two, we shall briefly discuss boundedness and stability properties of delay-differential equations. Let the systems * ( t ) = V(t,x( . ))

(2.19.1)

q t , 2( ))

(2.19.2)

and i(t) =

+

be given, where V and P are n vector functionals defined for t E I[to, m) and x , x” E C(I, Em),I being the real line. Let x(t, t o , 9) and x”(t, to , $) be solutions of (2.19.1) and (2.19.2) corresponding to the initial functions +(t)E C ( I t o ,G ) and $(t) E C ( I t 0 ,e) respectively, where G and e are compact sets in En. Let a function F(t, x , y ) 2 0 be defined and continuous on I x En x En,and suppose that it satisfies a Lipschitz condition with respect to x and y locally. Let Frtlbe the segment of the function F(t, x , y ) on I , whose values are defined by F I t I ( 4 = F(T, x ( 4 , Y(T)), 7 E It . (2.19.3) Let us denote the norm of Fltl by the symbol (2.19.4)

Let A = A ( t ) > 0 be a scalar function defined and continuous on I . We define the segment A I l lof the function A ( t ) as above and write (2.19.5)

* The author is particularly indebted to Professor V. Lakshmikantham for his kind permission to adopt his results here.

64

11. DELAY-DIFFERENTIAL EQUATIONS

In the sequel we shall need the following subspaces:

Theorem 2.17 (V. Lakshmikantham [2]). Let thefunction W(t,r ) 3 0 be defined and continuous for t 3 to and r 3 0. Let r ( t ; t o ,ro) be the maximal solution of the scalar dzfferential equation i =

W(t,r),

(2.19.8)

~ ( t , ;t o , ro) = T o ,

existing to the right of to . For each t >, to and x , y

E

C,, suppose

I f x(t, t o , C$) and a(t, t o ,6) are any solutions of (2.19.1) and (2.19.2) with the initial functions C$ E C ( I t 0 G , ) and $ E C ( I t 0 G , ) existing for all t 3 to such that (2.19.10) II F[to]ll < 10 Y

then

F(4 x ( t , t o , 41, z(t, t o , $>I

< r ( t ; t o , TO),

t

> to.

(2.19.11)

Theorem 2.18 (V. Lakshmikantham [2]). Let the function W(t,r ) 3 0 be de$ned and continuous f o r t >, to and r 3 0. Let r ( t ; t o ,ro) be the maximal solution of (2.19.8) existing f o r t 3 t o . For each t 3 to and x , y E C, , let

4)[ "y+yP

1

+ h, x(t) + h W , 4 . )I, ' q t ) + h q ,q .)I) - F(t, x"(t))l] + D+A(t)F(f,+), q t ) ) x(t),

where

< W t , F(t, 4%x"(t)))A(t), D*A(t)

=

1

lim sup - [A(t h+0+ h

+ h)

-

A(t)]

(2.19.12)

(2.19.13)

f o r each t 3 to . I f x(t, t o , 4) and a(t, t o ,6) are any solutions qf (2.19.1)

65

2.19. I . Auxiliary Theorems and (2.19.2) with the initial functions existing for all t > to such that

+

E

II F[t,lA[to]II < ro

C(It,,,

e) and C$

E

e)

C(ltU,

(2.19.14)

?

then for t

> to. Let

Proof of Theorem 2.17.

4 4 = w, 44 to $1, q t , t o ,$1). 9

(2.19.16)

Since mrto1= Frlo1 , we have 4to)

< I/ T t o l II < yo -

(2.19.17)

Now consider the differential equation i. =

W(t,Y )

+

r(to; t o , T o )

E,

=Yo,

(2.19.18)

which has solutions rE(t;to , yo) for all sufficiently small E > 0 existing to the right of to as far as r ( t ; t o , y o ) exists. It is known (see, for example, E. Kamke El]) that (2.19.19) lim rC(t;t o ,ro) = r(t; t o ,yo). C-10

It is therefore sufficient to show that m(t)

< r C ( t ;t o ,

yo)

for

t >, t o .

(2.19.20)

If this inequality does not hold, let t , be the greatest lower bound of numbers t 3 to for which (2.19.20) is not true. Since the functions m(t) and r6(t;t o , y o ) are continuoils, we have

I n view of (2.19.18) and (2.19.21), we may write

Since, by hypothesis, W(t,Y )

> 0,

the solutions of (2.19.18) are

66

11. DELAY-DIFFERENTIAL

EQUATIONS

nondecreasing as t increases. Hence, it follows from (2.19.17) and (2.19.21) that

/I

/I

=4 l ) ,

which implies, according to the definition of m(t), that x(t, to ,+), 2(t, to , E C, at t = t , . Therefore the condition (2.19.9) holds for the solutions x(t, t , , 4)and Z ( t , to , 6)at t = t, . Since F(t, x, y ) is assumed to be locally Lipschitzian with respect to x and y for each t, we have, for small h > 0,

4)

"(tl

+ h)

-

4tl)

+ h, to 4) "(tl , to 4) - h W 1 (. . , to ,4)) I1 + I1 ?(tl + h, to 6) G ( t l ? to 6) m.(t,Z( . t o , i))lll + F (t, + h, "(t, 4)+ hV(t1, (. . , t o ,419

< L[I!

X(t1

-

3

9

ti

-

~(1,

7

to

ti

9

7

-

9

9

9

-

20 >

,$1 + hP(ti

9

g( . 9 to

9

>

6)))

t o , +)* g(lo ti 7 $)), 9

where L is the Lipschitz constant of F. Making use of (2.19.1), (2.19.2), and (2.19.9), this inequality yields

which, in view of (2.19.22), is a contradiction. Therefore (2.19.20) is true, which is equivalent to the stated result. Proof of Theorem 2.18.

Define

K ( t , x, y ) = A(t)F(t,x, y).

(2.19.23)

Since F ( t , x,y ) is locally Lipschitzian with respect to x and y for each t, we have

2.20. II. Dejnitions

67

+ 0 as h -+ 0. Thus, in view of according to (2.19.12), where ~ ( h ) /-h the definition of K(t, x , y ) , (2.19.6), and (2.19.7), for each t 3 t o , x, y E C, where C, is taken with respect to K ( t , x,y ) , we have

2.20. Boundedness and Stability: II. Definitions

I n this section we shall introduce the following definitions, due t o V. Lakshmikantham, to unify the results on stability and boundedness of delay-differential equations. We assume that the solutions x ( t , t o , 4) and Z(t, to , 6)of (2.19.1) and (2.19.2) exist for all t 3 to . Definitions:

(i) T h e system (2.19.1) [or (2.19.2)] is said to be equi-norm-bounded with respect to the system (2.19.2) [or (2.19.1)] if, for each a > 0 and to 2 0, there exists a positive function. /3(to, a ) , continuous in to for each a, satisfying

It x(t9

for all

11 4 - 6 11

to

< a and

4) - Z ( 4 to > 6)/I < B ( t o 4 t 2 to .

9

7

(2.20.1)

(ii) If in (i) is independent of t o , the system (2.19.1) [or (2.19.2)] is said to be uniform-norm-bounded with respect to the system (2.19.2) [or (2.19.1)]. (iii) T h e system (2.19.1) [or (2.19.2)] is said to be equi-stable with respect to the system (2.19.2) [or (2.19.1)] if, for each E > 0 in to >, 0, there exists a positive function 6 ( t o , E ) , continuous in t o , for each E, satisfying (2.20.2) /I x(t,. 2, , 4) - Z ( t , 20 > 6,II < E

< S ( t o , E) and t 2 to . If 6 in (iii) is independent of t o , the system (2.19.1) [or (2.19.2)] (iv) is said to be uniformly-stable with respect to the system (2.19.2) [or (2.19. l)]. (v) T h e system (2.19.1) [or (2.19.2)] is said to be quasi-equi-ultimatelybounded with respect to the system (2.19.2) [or (2.19.1)] if, for each for all

114 -6 11

68

11. DELAY-DIFFERENTIAL

> 0 and to such that 01

EQUATIONS

2 0, there exist positive numbers B and T ( t o ,a, B ) /I 4 4

to

9

C)

-

x"(t,to , $1

/I < B

(2.20.3)

for all 11 C$ jl 6 01 and t > to + T ( t o ,a, B). If T in (v) is independent of t o , the system (2.19.1) [or (2.19.2)] (vi) is said to be quasi-uniform-ultimately-norm-bounded with respect to the system (2.19.2) [or (2.19.1)]. (vii) When (i) and (v) hold simultaneously, the system (2.19.1) [or (2.19.2)] is said to be equi-ultimately-norm-boundedwith respect to the system (2.19.2) [or (2.19.1)]. (viii) When (ii) and (vi) hold simultaneously, the system (2.19.1) [or (2.19.2)] is said to be uniform-ultimately-norm-bounded with respect to the system (2.19.2) [or (2.19.1)]. (ix) T h e system (2.19.1) [or (2.19.2)] is said to be quasi-equi-asymptotically-stable with respect to the system (2.19.2) [or (2.19.1)] if, for each E > 0, 01 > 0, and to 3 0, there exists a positive number T(to, E, a ) satisfying (2.20.2) for all I/ C$ - 6Ij a and t > to T(to, E , a ) . (x) If T i n (ix) is independent of t o , the system (2.19.1) [or (2.19.2)] is said to be quasi-uniform-asymptotically-stable with respect to the system (2.19.2) [or (2.19.1)]. (xi) If (iii) and (ix) hold simultaneously, the system (2.19.1) [or (2.19.2)] is said to be equi-asymptotically-stable with respect to the system (2.19.2) [or (2.19.1)].

-6

<

+

(xii) If (iv) and (x) hold simultaneously, the system (2.19.1) [or (2.19.2)] is said to be uniform-asymptotically-stable with respect to the system (2.19.2) [or (2.19.1)].

v(t,

s

x"(-)) = 0 and that x" E 5 where is a Now we suppose that nonempty set in En. Let the distance d(x, 5) between a point x and the set be defined by (2.20.4) d(x, S) = infill x - 2 11, 2 E S}.

s

Definitions (i) to (xii) can be easily reformulated. For example, (iii) would run as follows. (iii) The system (2.19.1) is said to be equi-stable with respect to the set if, for each E > 0 and to >, 0, there exists a positive function S ( t o ,E ) , continuous in to , satisfying

s

444 t o ,4), S) < E for all d(4, S) < S(to , E ) and t >, to .

2.21. III. Theorems

69

Corresponding to the above definitions, if we say that the differential equation (2.19.8) has the property (i-s), we mean the following condition is satisfied. (i-s) Given a > 0 and to 3 0, there exists a positive function /?(to, a ) that is continuous in to for each 01 and satisfies the inequality if ro

< a and t 2 t o .

r ( t ; to >

yo)

< B(to 4

(2.20.5)

9

Conditions (ii-s), (iii-s), and (iv-s) may be reformulated similarly. 2.21. Boundedness and Stability: 111. Theorems

I n this section we shall deal with the stability and boundedness theorems due to V. Lakshmikantham. Theorem 2.19 (V. Lakshmikantham [2]). Let the assumptions of Theorem 2.17 hold and assume that there is a function b(r)which is continuous and nondecreasing in r , b(r) > 0 for r > 0, such that

and that

(2.212)

lim b(r) = a. T+CC

Suppose further that the dtfferential equation (2.19.8) satisfies either of Conditions (i-s) or (ii-s). Then the system (2.19.1) is equi-norm-bounded OY uniform-norm-bounded with respect to the system (2.19.2). Proof. Suppose the differential equation (2.18.8) has the property (i-s). Then by definition, corresponding to a > 0 and to 3 0, there exists a positive function /?(to,a ) that is continuous in t for each a and satisfies (2.21.3) r ( t ; to T o ) < B(t0 011, 9

9

if ro < a and t 3 t o . Since b(r) -+ co as L = L ( t o ,a ) such that

W )> B O O 43

I

4

00,

there exists an (2.21.4)

Let x(t, to ,+) and %(t,to ,$) be any two solutions of (2.19.1) and (2.19.2), corresponding to the initial functions and $ respectively, such that

+

< <

IIF"[to] II ro

01.

70

11. DELAY-DIFFERENTIAL

EQUATIONS

Then it follows from the definition of (1 F I , ,(1, by (2.19.4) and (2.21.1), that (2.21.5) I14 - d I/ < b * ( 4 = Y , b* being the inverse function of b. Also, by Theorem 2.17, F ( t , 4 4 t o , 4, qt,

to

,6))< r ( t ; 2, , yo)

(2.21.6)

for t 3 to . Let us now assume that there exist two solutions x ( t , t o ,4) and x^(t,t o ,4) of (2.19.1) and (2.19.2) such that

This implies, as before,

b* being the inverse function of b. If we now assume that there exist two solutions x ( t , t o ,+) and Z(t, t o , of (2.19.1) and (2.19.2) for which

4)

for some t = t , > t o , then using Theorem 2.17 and relations (2.21.1) and (2.21.2) we arrive at a contradiction:

Thus the systems (2.19.1) and (2.19.2) fulfill the conditions of Definition (iii). T h e proof of the second part of the theorem is essentially the same, since, in this case, S(t,, c) is independent of t o . Theorem 2.20 (V. Lakshmikantham [2]). Let the assumptions of Theorem 2.17 hold and assume that there is a function b(r) which is continuous and nondecreasing in r , b(r) > 0 for r > 0, for which (2.21.1) holds. Suppose further that the daflerential equation (2.19.8) satisfies either of the Condition (iii-s) or (iv-s). Then the systems (2.19.1) and (2.19.2) satisfy the corresponding one of the Conditions (iii) and (iv).

2.21. III. Theorems

71

Proof. For each E > 0, if jl x - y 11 = E, we deduce from (2,21.1) that b(E) F(t, x, y ) . If Eq. (2.19.8) has the property (iii-s), given b(E) > 0 and to > 0, there exists a positive function ? ( t o , E) such that

<

r(t; t o , y o )

<

< 44,

(*)

provided yo q ( t o ,E) and t t o . Suppose that there are two solutions x(t, t o ,#) andy(t, t o , $)of (2.19.1) and (2.19.2) with the initial functions and $, continuous at t = t o , such that IIFltolI1

This implies, as before,

II d

-

*

II

<

yo

< b*(dt,

Gdto

7

9

4) =

w, €1, 9

b* being the inverse function of b. If we now assume that there exist two solutions x ( t , to , #) and y ( t , to, $) of (2.19.1) and (2.19.2) for which

II d

-

*

II < w

o

9

4

and

II X(t1

9

to 9 4) - Y(tl Y t o *)

II

=

E

for some t = t, > t o , then using Theorem 2.17 and the relations (2.21.1) and (*) above, we are led to the contradiction

N4 < F(t1 > X(t1

9

to > 41, r(t, to , *)I ?

d r(t,; t o

9

ro)

< 44.

T h e systems (2.19.1) and (2.19.2), therefore, fulfill Condition (iii). T h e proof of (iv) follows immediately, since ? ( t o , E) is independent of to in that case. Theorem 2.21 (V. Lakshmikantham [2]). Let the assumptions of Theorem 2.18 hold and assume that there is a nondecreasing function b(r) > 0 for which (2.21,l) and (2.21.2) are satisfied. Suppose further that the dzgerential equation (2.19.8) satisjies either of the Conditions (i-s) or (ii-s). Then, if lim A(t) = co, (2.21.8) t --tm

the system (2.19.1) is quasi-equi-ultimately-norm-boundedor quasi-uniformultimately-norm-bounded with respect to the system (2.19.2). Proof. We first show that quasi-equi-ultimately-norm-boundedness is implied by (i-s). Let x(t, to ,#) and n(t, t o ,8) be any solutions of (2.19.1) and (2.19.2) such that

<

IIFrtoiA~t0l I1

yo.

(2.21.9)

72

11. DELAY-DIFFERENTIAL EQUATIONS

Then it follows from Theorem 2.18 that A(t)F(t,x(t9 to

?

41, w,to

9

6,) < r ( c t o

9

yo)

for t 2 t o . Since Eq. (2.19.8) satisfies (i-s), given a there exists a positive number p ( t o , a ) such that r ( t ; t o , yo)

if ro

< a. Since b(r) -+

Now, choosing ro

00

as r

-+

< B(t0 ,

.>

(2.21.10)

> 0 and

to 3 0,

co, there exists an L such that

< a, we obtain the inequality I1 d - 6 I/ < b*(a II A& 11) = Y.

Suppose that there exist two solutions x(t, t o ,4) and %(t,t o ,6) of (2.19.1) and (2.19.2) satisfying the condition 11 4 - 11 y and such that

4 <

I1 " ( t ,

9

to 9

4)

-qt,

9

to 7

4)II 2 L

Since A(t,) + 00 as t, 3 co and since b(L) > 0, this implies a contradiction; hence, the systems (2.19.1) and (2.19.2) satisfy the conditions of Definition (v). The proof of the second part of the theorem is clear. Theorem 2.22 (V. Lakshmikantham [2]). Let the assumptions of Theorem 2.18 hold and assume that there is a nondecreasing function b(r) > 0 for which (2.21.1) and (2.21.2) are satisfied. Suppose further that the diferential equation (2.19.8) satisfies either of the Conditions (iii-s) or (iv-s). Then, if (2.21.8) holds, the system (2.19.1) is quasi-equiasymptotically-stable or quasi-uniform-asymptotically-stable with respect to the system (2.19.2).

The proof of this theorem follows, combining the proofs of Theorem 2.20 and Theorem 2.21.

2.22. Perturbed Systems

73

2.22. Perturbed Systems

I n this section we shall deal with the boundedness and stability properties of perturbed delay-differential equations. Consider the systems k(t) =

+ v(2, x( . ))

(2.22.1)

. )) + q t , a( . )),

(2.22.2)

V ( t . .x( . ))

and i ( t ) = P(t, ."(

where V , v , ?, and I7 are n vector functionals defined for t to and x, x" E C ( I , En).Let x(t, to ,+) and g(t, to ,$) be solutions of (2.22.1) and (2.22.2) for t 3 to corresponding to the initial functions +(t) E C(llU, G) and $(t) E C(Itu, e)respectively, G and e being compact sets in En.We shall continue to make use of our notation of previous sections. Definition 1.

If for each t 3 to and x , x" E C, the perturbations v and

6 satisfy the inequazity

I! v(t, x( . 1) II

+ /I w,n( . )) II < rlF(t, 4 t h ."(t)),

(2.22.3)

where 7 > 0, and if the solutions x(t, t o ,4) and x"(t,t o ,$) satisfy the conditions of Defnitions (i) to (iv) of Section 2.20, we say that the systems k(t) =

V ( t , x( . ))

(2.22.4)

qt)

q t , n( . ))

(2.22.5)

and =

satisfy Defnitions (i-s) to (iv-s) weakly. Definition 2. If for each t 3 to and x, 2 E C , the perturbations v(t, and G(t, x"(-)) satisfy the inequality (2.22.3) and the solutions x(t, to ,+) and %(t,to ,$) satisfy the conditions of Dejinitions ( v ) to (vii) of Section 2.20, we say that the systems (2.22.4) and (2.22.5) satisfy the conditions (v-s) to (vii-s) weakly. .(a))

We now prove the following. Theorem 2.23 (V. Lakshmikantham [2]). Let the assumptions of Theorem 2.17 hold except that condition (2.19.9) is replaced by

+

+ hV(t, x( . )), i ( t ) + hP(t, a( 1)) - F(t, x ( t ) , ~ ( t ) ) ] (2.22.6) ."(a< q t , F(t, x(4, W)),

1 iim sup - [ ~ ( t h, x ( t ) h+O+ h aF(t, 4 t h

+

*

74

11. DELAY-DIFFERENTIAL

EQUATIONS

where CY = T L , L being the Lipschitz constant at ( t , x(t), f ( t ) ) . Suppose further that there is a nondecreasing function b(r) > 0 such that (2.22.7)

and that lim b(r) =

(2.22.8)

00.

T"=

Then, if the differential equation +(t)=

w(t,r ) ,

r(t0; t o ,

TO)

= ro

(2.22.9)

,

satisfies one of the Conditions (i-s)-(iv-s), the systems (2.22.4) and (2.22.5) satisfy weakly the corresponding one of Definitions (i)-(iv) of Section 2.20. I n Cases (iii-s) and (iv-s) the condition (2.22.8) can be omitted. Proof. Let x, f E C, . Since the functionF(t, x , y ) is locally Lipschitzian with respect to x and y with Lipschitz constant L , we have, for small h > 0,

F(t

+ h, x ( t ) + h[V(t,(. -F(t,

x ( t ) , ."(t))

. ))

+

~ ( 5 x( , .

< hL[ll ~ ( tx(,

'

))I,

)) /I

Z(t)

+ h[r(t,

+ II G ( t , ?( . 1) Ill

+ F(t -t h, x ( t ) 4-hV(t, x( . )), 5(t) + hV(t, 2( . ))) Making use of (2.22.3) and (2.22.6) and noting that lim sup h+O

+

1 ~ ( t h, x ( t ) h

-{

+ h[V(t,

x(

(II

'

-

)) f G(t, .?(

'

)>I)

F(t, x ( t ) , 3(t)).

= Lq,

we get

+ ~ ( t , . ))I, 2(t) + h [ r ( t ,Z( . )) x(

+ q t , a( . I)]) - F (t, x ( t ) , Z ( t ) ) ) < q t , F ( t , x ( t ) , W)). 6)

If x ( t , to , 4) and x"(t,to , are any two solutions of (2.22.1) and (2.22.2) with the initial functions +(t)and $(t) such that I/ Fclol/I ro , we can easily obtain the desired results by applying directly the proofs of Theorems 2.17, 2.19, and 2.20.

<

Lakshmikantham [2]). Suppose that the assumptions of Theorem 2.18 hold except that the condition (2.19.12) is replaced by Theorem 2.24 (V.

2.24. Final Remarks

75

Suppose further that there is a nondecreasing function b(r) satisjies (2.22.7) and (2.22.8).Let

> 0 which (2.22.11)

lim A(t) = co. t +m

If the dzyerential equation (2.22.9)satisfies one of the Conditions (i-s)-(iv-s),

then the systems (2.22.4) and (2.22.5) satisfy weakly the corresponding one of Definitions (v)-(vii) of Section 2.20. In Cases (iii-s) and (iv-s) the condition (2.22.8)can be omitted.

T h e proof of this theorem is a combination of Theorems 2.21, 2.22 and 2.23. 2.23. Systems with Finite Heredity

Systems with finite past history frequently occur in many investigations. For a system with finite heredity the functional V(t,x(.)) is defined and takes values in En whenever t E I [ ~ y ] ,and x ( t ) E C(I[a,y ] , D), where a is a finite number. ObviousIy the theory developed in this chapter can be immediately transferred to this kind of equation. In this case we only need to give the initial function +(t) on I[., to] where 01 < to < ye 2.24. Final Remarks

(I) I n the proof of Theorem 2.6, we have extended the definition of the initial function +(t) to the interval I(t, , a] by setting &t) = +(to) for t E I(tn, a]. Clearly, the proof remains valid for an extension of the form $(t) = +(to 0 ) for t € I ( t o ,a ] , where +(to 0) may differ from +(to - 0). (11) Theorem 2.15 is still valid if the initial function +(t)is Lebesgueintegrable in It0 ; i.e., the continuity of +(t)is not necessary.

+

+

C H A P T E R I11 ?&

Linear Delay-Dzfferential Equations

I n the previous chapter we have developed the general theory of delay-differential equations. I n this chapter we shall particularly deal with linear delay-differential equations.

3.1. General Form of a Linear Delay-Differential Equation

Consider a delay-differential equation of the form

where V is, as usual, an n-dimensional functional defined for t E I [ u ,y ) and x ( t ) E C(I[a,y), En),f(t) is an n-dimensional real-valued function defined for t E ] [ t o ,y), a and y are given numbers which may be finite or 01 = -a and y = +a and are such that 01 < to < y (if 01 = -a, the interval I[m, y ) should read as ](a, y)). Clearly the system (3.1.1) may be written in the form k,(t)

+ Vi(t,xl(

. ), .-.,x,( . )) =fi(t)

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

(3.1.2)

Definition 1. The functional V ( t ,x(.)) is said to be linear with respect to x $ , f o r any pair of arbitrary real constants h and p,,

whenever t

,

E I[to y )

and x, x” E C(1[01,y), En).

Note that we do not include the condition of boundedness of V as a part of the definition of linearity. 76

77

3.1. General Form T h e condition (3.1.3) is equivalent to Vi(t,Ax,( . )

+ p q . ), ... Ax(,

= AVi(t, xl(

for i

=

7

. ),

...

)

x(,

. ))

.1

+ P%d . )>

+ PVi(t, x"1( . 1,

*..I

%( + ))

(3.1.4)

1, ..., n.

Definition 2. A linear functional V(t,x(-)) is said to be bounded there exists a real constant M > 0 such that for all x E C(I[ci,y ) , En)

if

f o r all t E I[a, y ) , where (3.1.6)

The number M is called the norm of the functional V ( t ,x(.)). Clearly, each component of bounded. As is well known:

a

bounded linear functional V ( t , x(*)) is

A linear functional V is bounded

if and only if it is continuous.

Definition 3. The delay-dzgerential equation (3.1.1) is said to be linear with respect to x, if the functional V(t,x(.)) is linear with respect to x. Definition 4. A linear delay-dzfferential equation of the form (3.1. I ) is called homogeneous i j the vector function f ( t ) = 0 for t E I [ t , ,y).

A linear delay-differential equation of the form (3.1. I ) is said to be inhomogeneous if the vector function f ( t ) + 0 for t E I[t, ,y ) . The function f ( t ) is called the right-hand side vector of the equation. Definition 5.

We now state the following:

( F . Riesz [I]). Every bounded linear real-valued in C(I[a,b ] , E l ) can be represented by a Stieltjes integral

Theorem 3.1

functional

((x)

(3.1.7)

where X(T) is a real-valued function of bounded variation uniquely determined by the functional <(x).

78

111. LINEAR DELAY-DIFFERENTIAL EQUATIONS

where the subscript T in dT denotes that the integration is performed with respect to T at a fixed t, and gij(7, t ) is a real-valued function of bounded variation which is uniquely determined by the functional Vi. Thus, by vector-matrix notation,

v t , 4 . 1)

=

jt

4

(3.1.9)

t ) X(T),

G(T,t ) being the n x n matrix with elements gij(7, t ) ; i.e.,

Therefore, we may state the following: Theorem 3.2. A linear delay-dzflerential equation (3.1 .I) with bounded functional V ( t , x ( - ) )has the form

44 t

fd$(T,

(3.1.1 1 )

t)n.(.) = f ( t ) ,

where G(T,t ) is an n x n matrix, all of whose elements are of bounded variation, and the integration is performed in the sense of Stieltjes with respect to T f o r $xed t. Let g,,(T, t ) be any element of the kernel matrix G(7, t). Denote by t ) the saltus function corresponding to the function gi,(7, t ) and let k , , ( ~t,) be the continuous part of gi1(7,t ) . T h u s si1(7,

!?,AT,

t ) = &AT,

4

+

k,,(T,

4.

Denoting by S(T,t ) and K ( T ,t ) the matrices with elements sij(7, t ) and k c l ( r ,t ) respectively, we find G(T, I )

=

s(T,t ) + K(T, t ) .

(3.1.12 )

T h e matrix S will be called the saltus matrix and the matrix K will be called the continuous kernel corresponding to the kernel matrix G. Thus, making use of this relation in (3.1.11) we get *(t)

+ I'd,S(T, t )

X(T)

+ I'd,K(T, t )

X(T)

=f(t).

(3.1.13)

79

3.1. General Form

If the continuous kernel K ( T ,t ) has a Riemann-integrable partial derivative f ( , ( ~ ,t ) with respect to T , then (3.1.14) Let us assume that the kernel matrix G(T,t ) is a function only of the difference t - T , i.e., G(T,t ) = G(t - T ) (3.1.15) (we shall see the importance of this kind of kernel in the next section). Assume further that a

and that t

< h, < h, < h, < ... < h,

> h, . Furthermore, let

+

S(t - T) = Am(t) A,-,(t)

+ ... + A,(t) + A,(t) for

+ Av-i(t) + ..' + Ai(t) A&) + A,-,(t)

s(t

-T )

E Z [ ~- h, , t

-

a]

for T E Z ( ~- h,, t - h,) (3.1.16) for T ~ l (-t hwLp1, t - hm-2)

S(t - T ) = Am(t)

S(t - 7)=

T

= A,(t)

for

T

E I (~ h , ,t

-

hmp1),

where AY(t)are certain n x n matrices. Clearly the discontinuities of S(t - T ) occur at T = t - h , , v = 0, 1, ..., m and the jump of S(t - T ) at the point T = t - h, is equal to the matrix A,(t), v = 0, 1, ..., m. Then, by the usual rules of integration for Stieltjes integrals, we find

jtd,~(t

2~ , ( t ) m

=

- 7 )x(T>

x(t

-

(3. I . 17)

hv).

"-0

Therefore, in this special case, our delay-differential equation (3.1.1 1) is an integro-difference-differential equation of the form *(t)

+ 2 A,(t) x ( t

-

"=O

h,)

+ 1' K(t

- T) X ( T )

d~

=,f(t),

(3.1.18)

where K(t - T ) is a Riemann-integrable matrix function. If K ( t - T ) = 0, Eq. (3.1.18) reduces to the linear difference-differential equation

+ c A$) m

*(t)

lk0

x(t

-

h,) =f(t).

(3.1.19)

80

111. LINEAR DELAY-DIFFERENTIAL

EQUATIONS

If A Y ( t )5 0, v = 0, 1, 2, ..., m, the equation reduces to the integrodifferential equation (3.1.20)

If the functional V ( t ,x(-)) is of the form V ( t ,x( . ))

=

V * ( t , x( . )) - V *(t - h, x( . )),

where h is a positive constant and V*(t,x(.)) is a linear bounded functional with respect to x, then Eq. (3.1.11) takes the form k(t)

+

Jt t-h

G*(7,

t)4 .) = f(t),

(3.1.21)

G*(T,t ) being the corresponding kernel matrix to the functional V*(t,x(.)). Eq. (3.1.21) is of Krasovskii-Razumikhin type. 3.2. Principle of Closed Cycles

I n Section 1.1 we have seen that a hereditary process occurs in a system when the phenomenon depends not only on the actual state of the system or its immediately preceding states, but on all the previous states through which the system has passed; i.e., a hereditary phenomenon depends on the previous history of the system. I n many cases, by some simple hypotheses, we can determine the nature of the hereditary effects, that is, the structure of the functionals V ( t ,x(-)) which appear in the processes. For this reason we now state the following postulates.

(PI) T h e influence of the heredity corresponding to states a long time before the given moment gradually fades out; i.e., x ( t ) being always bounded in norm, Ij x(t)ll < N , the norm of the variation of the functional V ( t ,x(.)) when x ( t ) varies in any way, in the interval I t l with t, < t , can be made as small as we please by taking the time interval I [ t l , t] sufficiently large. (P2) T h e heredity is invariable; i.e., a process always follows the same course if it passes successively through the same conditions, whatever may be the relative position of these conditions in time.

V. Volterra [6] has shown that, if V ( t ,x(-)) is periodic whenever x(t) is periodic, and with the same period T for both, whatever T may be, the heredity is invariable.

3.3. Existence and Uniqueness of Solutions

81

Definition 1. A heredity which satisjies the postulates (PI) and (P2) is said to be of closed-cycle type.

V. Volterra [6] has also shown that, in a linear and bounded hereditary process of closed-cycle type, the kernel G(T,t ) in the representation (3.1.9) of the functional V ( t , x(-)) is a function of the difference t - T ; i.e., G(T,t )

=

G(t

(3.2.1)

- T).

Therefore, for a linear and bounded hereditary process of closed-cycle type, the functional V ( t , x( .)) has the following representation: V(t,x( . )) = J

d,G(t

(3.2.2)

- 7)x(T).

--a,

Thus, by the results of the previous section, the general form of a linear bounded delay-differential equation of closed-cycle type is k(t)

+ 1'

--m

d,S(t

- T) X(T)

+ St

-w

d,K(t

-T)

X(T) = . f ( t ) ,

(3.2.3)

where S(t - T ) and K ( t - T ) are the saltus matrix and the continuous kernel corresponding to the kernel G(t - T ) . If, in particular, the saltus matrix S(t - T ) has the form (3.1.17) with spans 01 < h, < h, < h, < -.. < h , , and the continuous kernel has a Riemann-integrable partial derivative with respect to T , Eq. (3.2.3) has the form

which is an integro-difference-differential equation. 3.3. Existence and Uniqueness of Solutions

We first prove the following:

is continuousin t a n d x Theorem 3.3 (R. D. Driver [2]). If V(t, and linear with respect to x for t E I[a, y ) and x ( t ) E C(I[a,y ) , En), then V ( t , x(-)) satisjies a uniform Lipschitz condition with respect to x for t E I[to ,y ) and x ( t ) E C(I[cx,y), En). %(a))

Proof.

We have to show that

II V(x( . 1) -

V(t9 x"(

. 1) II

<

4 3

/I x

-

x" II,[n.tl

(3.3.1)

82

111. LINEAR DELAY-DIFFERENTIAL

EQUATIONS

for all t E l [ t o, /3], x,x“ E C(l[a ,y ) , En),where L, is a constant whose value depends only on /3 E I [ t o, y ) . Setting y(t) = x(t)

-

qt)

and making use of the linearity of V ( t ,x(-)) with respect to x, (3.3.1) reduces t o (3.3.2) !/ q t 9 Y( . )>/I < L, /I Y /lI[a.tl

,PI,

for t E ] [ t o y E C(l[a,y ) , En),/3 E l [ t o , y). Suppose, on the contrary, that there is no such constant L, . T h e n , for every v = 1, 2, ..., there E C(l[a,B], En)and a number tyE I [ t o ,p] such that exists a function y(,,,(t)

tl

Vt” Y ( d’ 1

1) I1 2 v /I Yb)Il~[,,t”l .

By a suitable extension of the definition of y(”,(t), we may suppose that

Ii Y W Il~[..t~l= I/ Y W llrr..si

.

Then, by the linearity of the functional V , we get for v

=

1 , 2, ... .

(3.3.3)

But, choosing a subsequence {vk} such that lim t ,

k ”00

=

t,

we find from the continuity of the functional V in t and x that

which contradicts (3.3.3). Hence the relation (3.3.2) is true. Since (3.3.2) is equivalent t o (3.3.1), the stated result is thus proved. Theorem 3.4. Let the conditions of Theorem 3.3 be satisjied. Let G be a compact set in En and x ( t , t o ,+) be any solution which corresponds to the initial function +(t) E C(l[a,to], G ) of the homogeneous equation

I[to, y).

(3.3.4)

ll .(t, to , 4) /I < I/4 III[a.t,le~@(t-t(J,

(3.3.5)

k ( t ) = V ( t ,(.

Then

’

)),

t

L, being the Lipschitx constant f o r V ( t , x(.)).

E

3.4. General Form of Solutions

83

T h e proof is immediate taking y ( t ) = 0 in the formula (2.4.41). Theorem 3.5 (R. D. Driver [2]). Let the conditions of Theorem 3.3 be satisfied. Let G be any compact set in En.Assume that f ( t ) is continuous for t E I [ t , , y ) . Then f o r every initial function $(t) E C(I[a,t,], G ) there is a unique solution x(t, to ,4)of E4. (3.3.1) on the entire interval ][to,y ) .

+

Proof. Let x(t, t o ,4) be a solution of (3.1.1) for t € I [ & ,t, h], where h is defined by (2.6.8) (under the assumptions of the theorem, the existence of such a solution is assured by Theorem 2.6). Suppose, on the contrary, that x(t, t o ,4)cannot be extended to I [ t , , /3), where /3 E I [ t , , 7). If we take

we get a contradiction to Theorem 2.8. Therefore, x(t, t , ,+) can be extended to I [ t , , 8). Since /3 is arbitrary in the interval I [ t , , y ) , the solution exists for t E I [ t , , y ) . T h e uniqueness of the solution is a consequence of Theorem 2.5. Theorem 3.6. Let the conditions of Theorem 3.3 be satisfied. Let G be a compact set in En.Assume that the vector function f ( t ) is measurable for t € I [ t , , y ) . Then for every initial function +(t)E C(l[ol,to],G ) there is a unique solution x(t, t, , $), in the extended sense, of Eq. (3.1.1) on the entire interval I [ t , , y).

T h e proof is immediate by Theorem 2.15. Note that the interval ] [ t o ,/3) in the proof of Theorem 2.15 can be extended to I [ t o ,y ) by the procedure of Theorem 2.8. 3.4. General Form of Solutions

First we shall consider a linear homogeneous delay-differential equation of the form *(t)

+ V ( t ,(.

. )) = 0,

t

E

I [ & , y).

(3.4.1)

Let x(t, t , ,4)and x(t, to ,+) be the solutions of Eq. (3.4.1) corresponding to the initial functions +(t) and &t), defined and continuous in I[m, to].Let X and p be any two scalar constants and consider the function

At) = wt,t o , 4)

+

P ( t , to

,$1.

84

111. LINEAR DELAY-DIFFERENTIAL

EQUATIONS

On account of the linearity of Eq. (3.4.1) the function y ( t ) is a solution of this equation. Further, y ( t ) satisfies the condition

r(t)= @(t) + P&)

t E 4%to].

for

Therefore, by the usual notation, At)

i.e., x(t, to

, A4

= x(t,

+ P&

to,

= A&

A4

+

20

4)

P$);

+

CLX(t,

to >

4,.

(3.4.2)

Assuming that the conditions of Theorem 2.10 are satisfied, we see that x ( t , to , +) is continuous with respect to the initial function + ( t ) . Thus: Theorem 3.7. Let the functional V ( t , be continuous in t and x f o r t E I [ a , y ) and x(t) E C(l[a,y ) , En), and linear with respect to x. Then the solutions of Eq. (3.4.1) are linear and continuous with respect to the initial functions +(t)E C(I[ol,to], G), G being a compact set in En. .(a))

Now we consider an inhomogeneous linear equation

where V ( t ,x(.)) is as above and f ( t ) is piecewise continuous (or Lebesgue measurable) in I [ & , 7).Let +(t)E C(I[a,to],G), G being a compact set. Let us denote by x ( t , t o , +,f)the solution of Eq. (3.4.3) corresponding t o the initial function +(t). Let x * ( t , to , +) be the solution of the homogeneous equation (3.4.1), which corresponds to Eq. (3.4.3), with the initial function + ( t ) . Consider the function z(t) = x ( t , t o >

4,f) - X * ( t ,

to,

4).

Since V(t,x(*)) is linear with respect to x, we have

and since x * ( t , t o , 4) is a solution of (3.4.1), %*(t)

+ V ( t ,x*(

. , t o ,4)) = 0

for

t e l [ t Or). ,

Therefore, the function z ( t ) satisfies Eq. (3.4.3). Furthermore, clearly z(t) = 0

for t E I [ ~to]. ,

85

3.4. General Form of Solutions Thus, with the usual notation, z(t) = x ( t ,

20,

0,f);

i.e., z ( t ) is the solution of (3.4.3) which corresponds to the identically zero initial function. Therefore,

Hence, we have the following: Theorem 3.8. Let thefunctional V be continuous in t and xfor t E I[a,y ) and x ( t ) E C(I[a,y ) , E n ) and linear with respect to x. Let the right-hand side vector f ( t ) be piecewise continuous (or Lebesgue measurable) in I[to, y ) . Then the solution x(t, t o ,+,A of Eq. (3.4.3) corresponding to the initial function $ ( t ) E C(I[m,to],G), where G is a compact set in En,is of the form x(t, t o ,

4,f) = x ( t , t o , 0 , f ) + x * ( t ,

to

9

4).

(3.4.4)

Here x*(t, to , 4) is the solution of the corresponding homogeneous equation (3.4.1) with the initial function +(t), and x(t, t o , 0,f ) is the solution of (3.4.3) with the identically zero initial function. We now show that the solution x(t, t o , 0, f ) of the inhomogeneous equation (3.4.3) is a continuous linear functional of the functions f ( t ) . To this end, consider the equation

4t)

+ V ( t,x( . 1)

=

W),

t

E

q t o , Y),

(3.4.5)

and its solution x(t, t o , 0, Af ), where X is an arbitrary scalar constant. Since V ( t ,x(-)) is linear with respect to x, V ( t , Ax( . ))

= AV(t, x (

. )).

Therefore, Xx(t, to,0, f ) satisfies Eq.(3.4.5). Furthermore, Xx(t, to,0,f ) = O for t E I [to]. ~Thus, , x(t,

t o , 0, Af)

= Ax(t,

t o , 0, f).

(3.4.6)

Let us now consider the equation *(t)

+ V ( t,4 . )) =my t c q t o , Y),

(3.4.7)

where the functional V ( t , x(*)) is the same as above, and the right-hand side functionf(t) is piecewise continuous (or measurable) for t E ][to, y ) .

86

111. LINEAR DELAY-DIFFERENTIAL

EQUATIONS

Let x ( t , t o , 0 , f ) be the solution of Eq. (3.4.7) corresponding t o the identically zero initial function. Clearly the function y(t) = x ( t , to , 0,f

)

+ x(f, to , 0, f )

satisfies the equation *(t)

and y ( t )

=

+ q t , 4 . >) = f ( t > + fct,,

0 for t E I [ a , to].Therefore, x(t, to , 0, f

+f )

= x ( t , to

* 07 f

) + x(t7

to > 0, f ) ,

or, making use of the formula (3.4.6), we have x(t, t o ,

0, Xf

+ pf)

=

h ( t , to I 0 , f )

+ P ( t , to O J ) ; ?

(3.44

i.e., the solutions x(t, to , 0 , f ) are linear with respect to the right-hand side vector f ( t ) . Moreover, denoting by L the Lipschitz constant of the functional V ( t , x(.)) and making use of the results of Section 2.11, we find

II 4 2 , to > 0 , f ) - x(t,

t o , 0,f)

II

< (+)(eL(t-to)

- 1)

(3.4.9)

for t E I [ t o ,y ) and ]If - fll < E; i.e., the solutions x(t, t o , 0 , f ) are continuous with respect to the right-hand side vectorf(t). T h u s we have: Theorem 3.8a. Let the functional V ( t ,x(-)) be continuous in t and x for t E I [ a , y ) and x ( t ) E C(l[a,y ) , E n ) and linear with respect to x. Let the right-hand side vector f ( t ) be piecewise continuous (or measurable) f o r t € I [ t o ,y ) . Then, the particular solution x(t, t o ,0 , f ) of Eq. (3.4.3) is linear and continuous with respect to f ( t ) .

We now state the following: Theorem 3.9 (F. Riesz [11). Every linear real-valued functional t,h( f ) in L,(a, 6 ) can be represented by a Lebesgue integral

#(f) = Jbf(.)W d7,

(3.4.10)

where 8 ( t ) is bounded almost everywhere and uniquely determined by the functional t,h( f ) (see,for example, A. C. Zaanen [ 1I). By the above theorem and Theorem 3.1 we may easily establish an integral representation for the solution x(t, to ,(b, f ) of Eq. (3.4.3).

3.4. General Form of Solutions

87

First we note

by Theorem 3.8. According to Theorem 3.7, x ( t , to ,+, 0), which is denoted there by x*(t, t o ,+), is linear and continuous with respect to the functions +(t)E C(l[a,to],G), G being a compact set in En. Therefore, by Theorem 3.1, we can represent x ( t , to ,+,0 ) by a Stieltjes integral of the form x(t, t o

, 4, 0)=

s1"

d,M(T, 9 $(TI,

(3.4.11)

where M(T,t )is an n x n matrix o f bounded variation and the integration is taken with respect to T . On the other hand, by Theorem3.8a, x(t, t o ,0, f ) is linear and continuous with respect to the functions f ( t ) E Ll(to, t), t E I [ t o ,y ) ; therefore, by Theorem 3.9, the solution x(t, t o ,0, f ) can be represented by a Lebesgue integral of the form

where N ( T ,t ) is an n x n matrix bounded almost everywhere. Hence x ( t , to ,

4,f) =

5""

d&'(~, t)4(7)

+ f N ( T ,t)f(~) dT. t@

T h u s we have: Theorem 3.10. Let V(t, be continuous in t and x and linear with respect to x f o r t E I[a,y ) and x(t) E C(l[a,y ) , En).Let G be a compact set in En. Consider the equation .(a))

44 + q t , x( . )) =f(t),

(3.4.3)

+,

where f ( t ) E Ll[to,y ) . Let +(t)E C(l[a,to],G). Then the solution x(t, to, f ) of Eq. (3.4.3), corresponding to the initial function $ ( t )f o r t E I [ ~ to], , is of the f o r m x(t, to , A f )

=

Itod,WT,

+ I' N(7, tlf(7) d~ ,

t)$(~)

(3.4.13)

t0

where M(T,t ) is an n x n matrix of bounded variation with respect to T and N ( T ,t ) is an n x n matrix bounded almost everywhere. The jirst integral is performed with respect to T in the sense of Stieltjes.

88

111. LINEAR DELAY-DIFFERENTIAL

EQUATIONS

3.5. Adjoint Equations of the First and Second Kind

I n this section we shall investigate linear integro-difference-differential equations of the form g(t)

+ X 3AY(t)x(t - h,) + p f K (T,t ) x ( 7 ) d7 = f ( t ) ,

(3.5.1)

tn

"=O

where A and p are two arbitrary real parameters, and 0

=

ho < h,

< h, < ... <

(3.5.2)

We assume that: (i) the matrices A,(t),v = 0, 1, ..., m, are continuous for t 3 to ; (ii) the kernel K ( T ,t ) is continuous in T for T E I [ t o , t] and in t for t >to; (iii) the vector functionf(t) is continuous for t 3 t o . Let +(t) E C( l[a ,t o ] , G), G being any compact set in En and h, . Let us consider the solution x ( t ) = x ( t , t o ,+ , f ) of Eq. (3.5.1) which satisfies the initial condition

a = to -

for

x ( t ) = + (t)

t

E

(3.5.3)

I[a, to].

Write Ht) = x(t, t o ,

4, 0)

and

42)

=

x(t, t o

>

0,f).

(3.5.4)

By the results of previous sections,

42)= ( ( 4 + r l ( 4

(3.5.5)

Further, [(t)

for

=+(t)

t

E

I[&,to]

(3.5.6)

and

Furthermore, q(t) = 0

and 7j(t)

+

2

"=O

AY(t)q(l-

for

h,)

t ~ Z [ ato] ,

+ p f K(7, t ) q ( ~d7) =f(t). tn

(3.5.8)

(3.5.9)

89

3.5. Adjoint Equations

By the results of the preceding section we may write the following representations: (3.5.10)

and (3.5.11) to

where a = to - h, , as above. We now determine the matrices M ( T ,t ) and N ( T ,t). We assume that these matrices are continuous in T and continuously differentiable with respect to t. Clearly,

Substituting (3.5.10) in (3.5.7) and making use of (3.5.12), we get

+ s, I T to

aM(u, t )

A,(t)M(o, t - h,)

+P

st

to

i K(T,t ) M ( o , T ) dTj+(u) do

=0

(3.5.13)

for t 3 to and by (3.5.6)

sto

M(u, t)+(a) do

for

=d(t)

t

(3.5.14)

E I[a,t o ] .

Since the integrand of (3.5.13) is continuous and since this equality holds for all continuous initial functions +(t),by a well-known lemma of analysis, we have aM(u' t , at

+ X 2 A,(t)M(o, t "4

-

h,)

+p

Jt

K(T,t)M(o, T) d~

= 0.

(3.5.15)

to

Clearly the condition (3.5.14) is satisfied for M(o, t )

=

S(t - u)Z

for

u, t E I[a,t,],

(3.5.16)

where I is the unit matrix in En and 8 ( t ) is the Dirac function (3.5.17)

90

111. LINEAR DELAY-DIFFERENTIAL

EQUATIONS

Therefore, the matrix M ( a , t ) is the solution of the integro-differencedifferential equation (3.5.15) which satisfies the initial condition (3.5.16). By Section 2.24, Eq. (3.5.15) has a solution &'(a, t ) corresponding to the initial condition (3.5.16) and, in view of Theorem 3.5, it is unique. We now deal with the matrix N(a, t). By differentiation in (3.5.11), we find (3.5.18)

Further, by Dirichlet's formula, we have

Thus, substituting (3.5.11) in (3.5.9) and using (3.5.18) and (3.5.19), we find N ( t , t)f(t)

+ j'1-

+p

Jt

K(T,t)N(u,7) d * l f ( u ) do

(0

(3.5.20)

2

t h v=u

to

A,(t)N(u,t

-

h , ) f ( ~do )

=f ( t ) .

We now assume that the matrix N(a, t ) satisfies the following conditions: N(o,t )

=

0

for

t

< U,

(3.5.21)

N(t,t) = I ,

(3.5.22)

where I is, as above, the unit matrix in En. I n this case, N(u, t

-

h)

=

0

for

t -h

< cr

< t,

by (3.5.21), and therefore

j'-"A,(t)N(a,t to

-

h,)f(o) do =

j t A,(t)N(u, t

-

h , ) f ( ~do. )

(3.5.23)

t0

Thus, by (3.5.22) and (3.5.23), the equation reduces to

(3.5.24)

3.5. Adjoint Equations

91

Since the integrand in (3.5.24) is continuous and (3.5.24) is satisfied for all continuous f ( t ) , we have

Wuy t , + A 2 A,(t)N(u, t at m

-

h,)

"ZO

+p

s'

K ( T ,t)N(u,T) d~

=

0.

(3.5.25)

0

Therefore the matrix in (3.5.1 1) is the solution of (3.5.25) corresponding to the initial condition (3.5.21)and (3.5.22),which is uniquely determined by Theorem 3.6 and by Remark (I) of Section 2.24. Note that, according to (3.5.1 l), (3.5.21), and (2.3.22), the condition (3.5.8) is satisfied. T hus we have proved the following: Theorem 3.1 1.

Consider the integro-dzfference-dzfferential equation (3.5.1)

"-0

for t 3 to ,where X and p are arbitrary real constants, and 0 = h, < h, < h, < .-. < h, . Assume that the matrices A Y ( t ) v, = 0, 1, ..., m are continuous for t 3 t o , the kernel matrix K ( T ,t ) is continuous in T and in t for t >, to and to T t , and the function f ( t ) is continuous for t >, t o . Let +(t)E C(I[a,to],G), a = to - h, , G being any compact set in En. Then the solution x(t, to ,4, f ) of (3.5.1) corresponding to the initial function +(t) has the form

< <

where the matrix M(u, t ) is the solution of aM(a' t , at

+ /Im

A,(t)M(u,t

-

h,)

"=O

+ p f K(T,t)M(u,

T)

dT

=0

(3.5.15)

tn

with the initial condition t)

=

qt

- 41,

t

E I[%

to], u E I[a, to],

(3.5.16)

where I is the unit matrix in En and s(t) is Dirac's function. The matrix N(a, t ) is the solution of

92

111. LINEAR DELAY-DIFFERENTIAL

with the initial condition N(o,t )

=

[I"

for for

t t

EQUATIONS


(3.5.27)

The matrices M and N are uniquely determined.

We shall call Eqs. (3.5.15) and (3.5.25) adjoint equations of the first and second kind and the matrices M(a, t ) and N(a, t ) kernel matrices of the first and second kind, respectively. T h e assumptions made in Theorem 3.11 for the functions f ( t ) and +(t) may be relaxed. Indeed, if f ( t ) and + ( t ) are Lebesgue-integrable, the formula (3.5.26) with the kernel matrices M and N introduced above still represents a solution of Eq. (3.5.1) corresponding to the initial function + ( t ) for t E I", t o ] . I n this case, the uniqueness of the solution is guaranteed by Theorem 2.16, since Osgood's integral

diverges, because 5(P)

e L*P ,

where

y

being any finite number such that y Theorem 3.12.

> t o . Thus:

Theorem 3.11 is still valid when f ( t ) and +(t) are

Lebesgue-integrable.

T h e above two theorems extend the results obtained for the case 0 (i.e., for difference-differentia1 equations) by R. Bellman and K. L. Cooke 121 and M. N. Oguztoreli [2].

p =

3.6. Construction of Kernel Matrices of the First and Second Kind

Theorem 3.11 reduces the problem of solving Eq. (3.5.1) to the problem of finding the kernel matrices M(T,t ) and N ( T ,t ) of the first and second kind. These matrices can be determined by Picard's method of successive approximations or by Euler's polygonal method (Sections 2.7 and 2.8). I n this section we shall apply Picard's method to obtain the matrices M(T,t ) and N ( T ,t).

3.6. Kernel Matrices of the First and Second Kind

93

First, we shall slightly change the forms of Eqs. (3.5.15) and (3.5.25) writing

and _ aN(cr' __ t, -h at

m

A,(t)N(cr, t - h,) "=I)

+ p itK(r, t)N(u,

T)

dr.

(3.6.2)

U

Thus, M(u, t ) is the solution (3.6.1)corresponding to the initial condition M(u,t )

= S(t -

011

t

for

E I[a,to],

u E I[a, to)

(3.6.3)

and N(u, t ) is the solution of (3.6.2) with the initial condition for

t

for t


(3.6.4)

I being the unit matrix in Efl and 8(t) Dirac's function. Integrating Eq. (3.6.1) with respect to t between to and t , to < t , and taking into account the condition (3.6.3), we find

and, similarly, integrating Eq. (3.6.2) with respect to t between u and t , u, t 3 t o , and making use of the condition (3.6.4), we obtain N(u, t )

=I

+ h C 1 A,(s)N(cr, s - h,) ds I

"

t

"SO

+p

ds ~ ' K ( T s)N(cr, , T ) d7. D

(3.6.6)

O

We begin with the matrix M(u, t ) . Let us take the matrix (3.6.7)

where u

E

I[&,to],as the first approximation.

94

111. LINEAR DELAY-DIFFERENTIAL

EQUATIONS

T h e second approximation will be

\

u

for t

E I[a,to]. A

(3.6.8)

simple computation shows that

N12(u, t )

= S(t

-

0)l

for for

= XA,(o)

=x

where u

to,

2 rn

+ h,)

~ , ( u

E I [ a , to]. Generally,

\

the ( j

< t < t, to + h, < t < t o + h,

a

for

t

> to + h,

,

(3.6.9)

+ 1)th approximation is given by

for t

> t,,

(3.6.10)

where 0 E I[a,to). By Section 2.8, the sequence {Mj(a,t ) } approaches uniformly t o the required solution M(u, t). We now establish the approximations of the kernel matrix N(u, t ) of the second kind. We take (3.6.1 1 )

where u >, t, , as the first approximation. The n the second approximation will be (0

for

t

< u, for

t> u

(3.6.12)

3.7. Parameters and Initial Data

95

A straightforward computation yields Nz(u, t)

=0 =I

+ p fds u

=I

I*

K(T,S) dr

u

+ p fds J s K( T,s) dr + h 2 J u

t

for

u
+ h 1:Ao(s)ds k

t

A,(s) ds

v=o

u

for

=I

+ p s' ds

Js

u

K(r, s) dr

2 t o . Generally, (0

Nj+l(O,

i

t) = I

for

t

the ( j

hk

(k = 1,2,

..., m - 1)

+ h,,

(3.6.13)

t

u

+ 1)th approximation is given by


+ h 2 f A,(s)Nj(u,s - h,) ds + p m

v=o

+ < t < u + h,+,

u

for

where u

u

+ h fAY(s)ds v=O

n


for

u

st

ds J s K ( r ,s) Nj(u,7)dT

u

for

s

t 2 u,

(3.6.14)

where u >, t o . Clearly the sequence (Nj(a,t ) ) approaches uniformly to the required solution N(u, t). I n Section 3.1 1 we shall establish integral equations satisfied by the matrices M and N . 3.7. Dependence upon Parameters and Initial Data

I n this section we shall deal with the problem of dependence of solutions of a linear delay-differential equation k(t) =

q t , (.

*

1 ; A)

+ f(t; A)

(3.7.1)

upon the parameter vector X = (A, , ..., A,) and the initial data (to , +(t)]. Let V ( t ,x(.)) be a linear functional in x and an ordinary function of t for t E 1[a,183, x ( t ) E C(l[a,,8], En),where a < t, < 18. Let us give an increment Et,h(t)to the function x(t), where #(t) E C(I[a,83, En).Let d V be the corresponding increment of the functional V ( t ,x( -)); i.e., Llv

=

V(t, x( . )

+

€#(

.))

-

V(t,X( .)).

(3.7.2)

96

111. LINEAR DELAY-DIFFERENTIAL

EQUATIONS

Owing to the linearity of the functional V(t,x(.)) in x, we obtain d

v = €V(t,+( . )).

(3.7.3)

Since the functional V ( t ,x(*)) may be written as a Stieltjes integral of the form (3.7.4) where G(T,t ) is an n x n matrix all of whose elements are of bounded variation, we find

by virtue of Eq. (3.7.4). Therefore, according to (2.13.5), we have V’(t,4 . )) = 0 3 7 , t ) ,

(3.7.6)

where the left-hand side denotes the functional derivative of the functional V ( t ,x(.)) with respect t o x at the point T. Clearly, higher derivatives of V ( t ,x(.)) are all identically zero. T h u s the functional V ( t ,x(.)) is differentiable to all orders. We now consider a linear functional of the form V(t,x(-;A)), where h is a K-dimensional parameter vector with components A,, ..., A, . We suppose that V ( t ,x(-; A)) is uniformly bounded on the set J-2 = { ( t ,x, A) I t E I[a,131, x E C(I[%131, E”), A E 4&)}, (3.7.7)

where A,

=

(Axo, ..., A,,)

is a fixed vector in the parameter space

As(X,)

= {A E

A 1 I/ A

- A,

II < S}.

(1 and

(3.7.8)

Thus, V ( t ,x(*;A)) is uniformly continuous on 9 and, according t o the theorem of F. Riesz, it can be represented by a Stieltjes integral of the form

v(t,( . ; 4)= J”

t E I [ u ,/3] and A above. Thus,

E &(Ao),

G ( 7 ,

t ) 4 T ; 4,

(3.7.9)

the kernel matrix G(7, t ) being the same as (3.7.10)

t E I [ u ,/3] and A

E

&(Ao).

3.7. Parameters and Initial Data

97

If we have a functional of the form V ( t ,x(.); A), linear in x, continuous in t, and differentiable in A,, ..., A, , uniformly bounded on Q, we have V(t, X( . ); A)

=

J‘ ~ , G ( Tt,; A) x ( T ) ,

(3.7.11)

where G(T,t; A) is an n x n matrix all of whose elements are of bounded variation uniformly on Q. Thus, (3.7.12) where GAv(i-,t; A) is the partial derivative of G with respect to A,. Finally we consider a linear functional of the form V(t, x(-; A); A). Suppose that V(t, x(-; A ) ; A) is uniformly bounded on Q, and so uniformly continuous there. Hence we may write V(t, X( . ; A); A)

=

J‘ ~,G(T,t ; A)

X(T;

A),

(3.7.13)

and the integral is in the sense of where t E I[&, p] and A E &(Ao), Stieltjes with respect to T and G(T,t ; A) is an n x n matrix all of whose elements are of bounded variation uniformly with respect to A. Write AX

=

and AV

=

V ( t ,X( . ; X

(0,..., 0,AX,, 0,..., 0)

+ AX); A + AA) - V ( t ,

(3.7.14a) X(

. ; A); A).

(3.7.14b)

Thus, supposing that all the required differentiability conditions are fulfilled, we may write (3.7.15) Therefore, if the functional V(t, x(*; A); A) is linear in x and continuous in t and uniformly bounded on Q, and if x(t; A) has continuous partial derivatives with respect to A, , ...,A,, then the functional V ( t ,x(-; A); A) is continuously differentiable with respect to A,, ..., A,. Consider now the functional V*(t, X( . 1; A)

= V ( l , *(

. ); A)

+ f(t; A),

where V ( x ( - )A) ; is as above. Clearly, if the functionf(t; A) is continuous in t and continuously differentiable with respect to A,, ..., A,, then V*(t, x(.); A) is also continuously differentiable in A1, ..., A,.

98

111. LINEAR DELAY-DIFFERENTIAL EQUATIONS

From the above analysis and Theorems 2.11-2.14 we obtain the following: Theorem 3.13. Let V ( t , x(*); A) be linear in x, continuous (analytic) in t , continuous or continuously differentiable (analytic) in A,, ..., & , and uniformly bounded on Q, where Q is dejined by (3.7.7). Let f ( t ; A) be continuous (analytic) in t and continuous or continuously dzzerentiable (analytic)in A, , ...,h, .Let G be any compact set in En. Then the solutions x(t, to ,4,A) ofthe equation *(t) v(t,(. ’ ); A) f f ( t ; A), t 2 to (3.7.1)

< <

corresponding to the initial functions $(t) E C7(1[a,83, G) (0 Y oo), are continuous or continuously dzzerentiable (analytic) with respect to 4, A,, ..., and continuous (analytic) in t o . Consider now the integro-difference-differential equation

where A,(t; A), v = 0, 1, 2, ..., m, are continuous (analytic) in t and continuous or continuously differentiable (analytic) with respect to A, , ..., , and the matrix K(7, t ; A) is continuous (analytic) in t and r and continuous or continuously differentiable (analytic) with respect to A,, ..., A, . Then, by Theorem 3.13, the solutions x(t, t o ,4, A) are continuous or continuously differentiable (analytic) in A, , ..., hk , t , t o , and 4. We now consider the following particularly simple equation

where 01 < to < t and \ 01 1 is sufficiently large. Let x(t) be the solution of (3.7.17) corresponding to the initial condition x(t) =+(t)

Thus, putting d t ; P ) = .f(t)

for

t E I[a,to].

+ P ft”

K(77

tW(4 dT,

(3.7.18) (3.7.19)

we may write

+ A 2 A,(t)x(t - h,) + J m

n(t) = g(t; p)

v-0

for t 3 t o .

t b0

K(T,~)x(T)d~ (3.7.20)

3.7. Parameters and Initial Data

99

By the results obtained above, the solution x ( t ) is an analytic function of the parameters h and p, and therefore we have the expansion

2 A"%&). m

x(t) =

(3.7.21)

2,3=0

Let

2 PCXij(t), m

yi(t;p)

=

2 =

0, 1 , 2, ... .

(3.7.22)

j=O

Therefore, (3.7.23)

Substituting (3.7.23) in (3.7.20) and equating the coefficients of the same power of h in both sides, we obtain Y&

P) = g(t; P) -1- P

Jt

(3.7.24)

K(7, t)ro(7;El) dT

t0

and

2 A,(t)y,-l(t m

Y&; P) =

"=O

-

h,; P)

+ P f K(T, tlrj(7;P)

dT,

(3.7.25)

t0

f o r j = 1,2, 3, ..., and t 3 t o . Integrating (3.7.24) between to and t, and setting

we find for t E I[a,to] (3.7.27)

Applying Dirichlet's formula to the integral in (3.7.27) and putting (3.7.28)

we find the following Volterra integral equation of the second kind: (3.7.29)

100

111. LINEAR DELAY-DIFFERENTIAL

EQUATIONS

to . We now introduce, as usual in the theory of Volterra integral for t equations, the iterated kernels

for j

=

1, 2,

...,r

+ s = j + 1, r

=

1, 2, ...,j , and the resolvent kernel

Thus, by a well-known result of integral equations, we have Yo(C PI

=

+o(t; P I

+P

(3.7.32)

F(@,t ; P)$o(@;P) &T to

for t to. Integrating (3.7.25) between to and t , and setting (3.7.33) for t

2 t o , we obtain

Therefore,

for t >, t o . I n the same manner, we may write for -Yi+l(t;

where

P) =

t

E I[a,to]

3.8. Equation of Closed-Cycle Type j

=

101

0, 1, 2, ... . Hence,

t o . Thus, if we have already obtained the function y j ( t ;p), for t the function y5j+l(t;p) can be determined by (3.7.37) and the function y j + l ( t ;p) can be obtained by (3.7.38). Since yo(t;p ) is given by (3.7.32), this inductive process works for a l l j = 0, 1, 2, ... . Clearly the procedure is rather easy if all AY(t)are constants.

3.8. Equation of Closed-Cycle Type with Constant Coefficients

Consider the equation

where all the matrices A , , v = 0, 1, ..., nz, are constant, while the kernel K depends only on the difference t - T , and f is as usual. We suppose that K and f are of bounded variation and at each point they are equal to their mean values respectively. We also assume that all the integrals below are absolutely convergent in some strip in the s plane. T h e equation (3.8.1) is very suitable for the application of Laplace transform methods. We now sketch briefly this method in purely formal fashion. For a detailed investigation we refer the reader to the book of E. Pinney [l] and, especially for the case K = 0, to the book of R. Bellman and K.L. Cooke [l]. Let x(t) be the solution of (3.8.1) corresponding to the initial condition for

~ ( t= ) +(t)

Multiply Eq. (3.8.1) by

e-Sl

t

< t, .

and integrate from to to

lim x(t)ecst = 0 , t --tm

Res

(3.8.2) 00.

We assume that

> 0.

(3.8.3)

Integration by parts yields

J

i ( t ) c s tdt = to

+(to

+ 0) + s J

x(t)e-st dt, to

(3.8.4)

102

111. LINEAR DELAY-DIFFERENTIAL

+

EQUATIONS

+

where we have taken ~ ( t , 0) = #(to 0). By a change of variable and taking into account the condition (3.8.2), we find j:x(t

-

h,.)e-st dt = e-hvs

t0

Jto-hv

+ ( t ) c s dt t

+ e c h d jm x ( t ) r s t dt.

(3.8.5)

to

Further, according to the convolution theorem, we may write

(3.8.6)

Thus, putting ~ ( s ) = sl

-

2

A,chvs -

"=O

m

K(t)ePstdt,

(3.8.7)

0

where I is the n x n identity matrix and

(3.8.8)

we find (3.8.9)

Thus, for ~ ( s )# 0, we may write

j"

m

x ( t ) c s t dt = x-l(s)Q(s),

(3.8.10)

t0

where x - I ( s ) denotes the inverse matrix of ~ ( s ) . Hence, applying the general complex inversion formula (see, for example, B. van der Pol and H. Bremmer [l]), we obtain 1 ctzm x(t) = estx--'(s)Q(s) ds, t > to, (3.8.11) 27ri

c--ico

3.9. Kernel Matrix of the First Kind

103

where c is an abscissa in the half-plane of the absolute convergence of the Laplace integral of the function x(t). This formula represents the solution x ( t ) for t > to in the integral form. 3.9. Kernel Matrix of the First Kind for Equations of Closed-Cycle Type with Constant Coefficients

Consider Eq. (3.8.1). T h e kernel matrix M(7, t ) of the first kind belonging to this equation satisfies the equation

___ aM(u' t, at

-

2 A,M(u,

t

-

h,)

+f

"=0

K(t

-

T)M(u,T) d7

(3.9.1)

tU

and the initial condition M(u, t ) = S(t

-

u)l

for

CT,

t E I[=, to],

where 1 0 1 I is sufficiently large. Note that M(u, to u ~ I [ a rto]. , By a simple observation we see that x(s) = sl -

m

A,echvs -

(3.9.2)

+ 0) = 0 because of

j K(t)ecstdt CO

(3.9.3)

0

"=0

and (3.9.4) "=0

Since

1"

for u $ I[to - h, , to]

8(t - u ) l c g tdt =

(3.9.5)

tu-h,

where v Q(s)

=

1, 2, ..., m, we have

= e-('+hv)sl

for u e I [ t O- h , , to], v

=

1 , 2, ..., m.

(3.9.6)

Thus, by the inversion formula (3.8.1 1) for x ( s ) # 0, we have M(u, t )

=-

2ri

(s) ds

+

1

1

c+im C-iW

e-hvsX-l(s) ds,

(3.9.7)

for t > to and u E I [ t o - h, , to],v = 1, 2, ..., m. T h e formula (3.9.7) clearly shows that M(u, t ) = M ( t

-

u).

(3.9.8)

104

111. LINEAR DELAY-DIFFERENTIAL

EQUATIONS

3.10. Kernel Matrix of the Second Kind for Equations of Closed-Cycle Type with Constant Coefficients

Consider Eq. (3.8.1). T h e kernel matrix N(a, t ) of the second kind belonging to this equation satisfies the equation -aN(CJ't ,

at

-

2

AJ(o, t

-

h,)

"=O

and the initial condition N(U,

t)

=

[;

+ j t K(t

-

T

T)N(cJ, T ) d7

t < CJ t = u.

for for

(3.10.1)

(3.10.2)

A straightforward computation shows that

2 m

~ ( s )=

sI

-

A , c h V s

-

"SO

juK ( t ) c s tdt

(3.10.3)

0

and that Q(s)

=

(3.10.4)

e-S"I.

Thus, by the inversion formula (3.8.1 1) for x ( s ) # 0, we have N(a, t) = -

2ni

c+im e-im

&-")x-l(s) ds,

t

> to.

(3.10.5)

Clearly, N ( u , 2)

= N(t

-

a).

(3.10.6)

3.11. Integral Equation Satisfied by the Kernel Matrices M and N

I n Section 3.6 we have considered the problem of construction of the kernel matrices M(a, t ) and N(a, t ) of the first and second kind and for this purpose we have used the method of successive approximations. In this section we shall deal with the same problem but with a different approach. First we consider the kernel matrix M(u, t ) of the first kind. Let the equation

3.1 1 . Kernel Matrices M and N (t

> to) be given.

105

Consider its adjoint equation of the first kind

where t > to , a E If., to], I CY. 1 being sufficiently large. T h e kernel matrix M(5, t ) of the first kind is the solution of (3.1 I .2) corresponding to the initial condition M(u, t )

= S(t

-

u)Z

for

U,

t

E

(3.1 1.3)

Z[a, t o ] .

Let u s integrate Eq. (3.11.2) with respect to t from to to t and observe that M(a, to 0) = 0. We find

+

2/ m

M(u, t ) =

v=o

t

A,(s)M(u, s

-

h,) ds

to

+ f K1(7,t)M(u,

T) dT,

(3.1 1.4)

to

where

K1(7, t) =

St K ( T ,

s) ds.

(3.1 1.5)

7

Let us denote by e(t) the unit step function; i.e., for

e(t) =

Since t

> t o , we

A,(s)M(o, s

-

r-”’

may write for t h,) ds

=

<0

> to + h,

A(T

t0--h,

t

+ h,)M(u,

T) dT

(3.11.6)

106

111. LINEAR DELAY-DIFFERENTIAL

EQUATIONS

Thus, introducing (3.11.8)

and (3.1 1.9)

we obtain

It:

M(o, t ) = F ( u ) f

+

(3.1 1.10)

G(T,t ) M ( o ,T ) dT,

where t > to h, and u E I[a,to]. We now consider the adjoint equation of the second kind (3.1 1.11)

where t > u and u 3 t o . T h e kernel matrix N(u, t ) of the second kind is the solution of (3.1 1.1 1) corresponding to the initial condition N(o, t )

for

=

t


Integrating the equation with respect to t from of the initial condition (3.11.12), we obtain N(u, 1 )

=I

+ 2 i A,,(s)N(u,s n,

t

v=o

v

-

h,) ds

+

where K,(T, t ) is defined by (3.11.5). For t

1’A,(s)N(u,

s

-

h,) ds

I

t-h,

=

o-h

U

= o-h,

t

(3.11.12) u to

K,(T,t)N(u,T ) dT, (3.1 1.13)

>u

+ h,

+ h,)N(o,

A”(T A,(T

t and making use

T)

+ h,)N(o, ).

, we may write

dT dT

+

f - h v A p ( ~ h,)N(a, T ) dT 0

=

f-”’ A,(T + h,)N(o,

T) dT

D

=

+ h,)e(t - h, - T)N(o,

/:A”(T

T)

dr.

3.1 1. Kernel Matrices M and N Therefore, N(u, 2)

=I

+

+

st

107

G(T,t)N(a,T ) dT,

(3.11.14)

U

where t > u h, , 0 3 t o , and G(7, t ) is defined by (3.11.9). Now let us construct the iterated kernels

(3.11.15)

and construct the resolvent kernel rn

A(o, t ; A)

=

(3.1 1.16)

'CAjAjtl(U, t). 3=0

By the well-known results on Volterra integral equations, we may write M(o, t )

= F(a)

+ j t A(u,

T ; ~ ) F ( TdT )

(3.1 1.17)

t0

and N(u, t ) = I

+

st

A(a, T ; 1) dT.

(3.11.18)

0

Thus we have proved the following: Theorem 3.14. The kernel matrices M(u, t ) and N(U, t ) of the Jirst and second kind corresponding to the delay-dtflerential equation 172

A,(t)x(t - h,)

Ji(t) = "=O

t

> t o , satisfy the

+ f K(7,

t)x(.)

dT

t0

+

.fct)

(3.11.1)

Volterra integral equations

M(o, 2)

= F(u)

+

st

G(T,t)M(u, T ) dT

(3.11.10)

t0

for t

> to , u E I[.,

to], and N(o, t )

=I

+ 1: G(T,t)N(o,

T)

dT

(3.1I. 14)

108

111. LINEAR DELAY-DIFFERENTIAL

EQUATIONS

for t > u, u >, t o , respectively, where the matrices F(o) and F ( r , t ) are dejined by m

F(u) = x A , ( u

+ hv)[e(u

-

to

+ h,) - e(o - to)]

(3.11.8)

and

and I is the n x n identity matrix. M(u, t ) and N(u, t ) are given by (3.1 1.17) and (3.1 1.18) respectively. 3.12. Stability and Boundedness

I n Sections 2.19-2.22 we have investigated stability and boundedness of delay-differential equations in general. I n this section we shall briefly study, with a different approach, some stability and boundedness properties of linear delay-differential equations. Our method will be based upon the integral representation of solutions in terms of the kernel matrices of the first and second kind. Consider a linear delay-differential equation of the form

for t > t o , and its solution x ( t , t o ,q 3 , f ) with the notation introduced in the previous sections. By Theorem 3.1 1 we have the integral representation of the solution x(t, t, ,4, f ),

u

where M ( o , t ) and N(o, t ) are the kernel matrices of the first and the second kind, respectively. Theorem 3.15. Let the function # ( t ) be bounded for t € ] [ a ,to],and the function f ( t ) and the integrals m(t)

=

fil

I1 M ( u , t ) /I do,

I/ N( o, t ) Il du

n(t) = to

(3.12.3)

109

3.12. Stability and Boundedness

Then the solution x ( t , t o , 4,f ) is also bounded

> to .

be bounded for t for t > to .

Proof. T h e boundedness of the functions 4, f,m, and n imply the existence of positive numbers Si , i = 1, 2, 3, 4, such that

I1 4(t) II

< 8,

t

for

and m(t)

E

4%toll

< 6,

and

l ! f ( t )/I

n(t)

< 6,

< 6, for

for

t

t

> to (3.12.4)

> to.

(3.12.5)

to1

(3.12.6)

From the formula (3.12.2) we find

,< 8, 6,

+

82 64.

This proves the theorem. Theorem 3.16.

Let

lIC(t)

-i

( t ) II

and lIf(t) - A t )

II

< 6,

for

t

E "a,

< 8,

for

t

> to

(3.12.7)

*

If the functions m ( t ) and n ( t ) satisfy the inequalities (3.12.5) then It x(t, to , 4, f) -

to

,8,P)/I < 616,

+

6284

.

(3.12.8)

Proof. By (3.1 1.2) we may write

44 t o , 4, f) - 4 2 , t o , 8,P)=

f"M ( a , t"(4 +

to

-

i ( 4 1 do

N(o, t)[f(O> -J(41 do.

T h e above argument yields the result immediately. Now we consider a second equation of the form rn

Av(t).?(t - &)

k(t) = "=O

+ 1'R(T,t ) . ? ( ~d~) + f(t) f"

(3.12.9)

110

111. LINEAR DELAY-DIFFERENTIAL EQUATIONS

for t > io , and its solution x”(t, io ,$,f). Let M(a, t ) and N(cr, t ) be the kernel matrices of the first and the second kind, respectively, of Eq. (3.12.9). Then we may write the solution 2(t, io in the form i(t,

to,

Suppose that io

&f) = j i@(o,t)$(o)do + i0

< to :

4,f) - 44 to ,4, f ) = j: [&I(o, M a ) + N o , f)ml do

q t , fo >

U

and

we have

and similarly

st

i”

,B,f)

&(u, t)J(o)do.

(3.12.10)

3.12. Stability and Boundedness

111

Thus, Let

Theorem 3.17.

< 81, IIf(4/I < m(t) < 6,, II+(t)II

82,

and

t ~ l [ ato], , IIC(t)!I < 8 1 , IIC(t) -$(t)l/ < a,*, II < 8, IIf(4 - f(t> /I < a,*, t > to, %(t) < S3, n(t) < 6,, 6 ( t ) < 8,, t > to,

(3.12.14) (3.12.15) (3.12.16)

/ GI II M(o, t )

IIm

9

St ll N(o,

- M(a, t ) II da

t ) - N(o, t ) /I do

< 6,*, < S,*,

> i,,

(3.1 2.1 7)

t

>to.

(3.12.18)

t

> to ,

(3.12.19)

t0

Then

I/ 44 to 4, f) - q t , fo > 6,j ) I/ 7

where L

t

= 81 s5*

to < u,

+

Sl* 6 3

<


+ 8, 6,* + S,* 6, + [S, II M(o1 , t ) I1 +

82

II N(Q2> t ) lll(to

- ZO)?

(3.12.20)

t o ,v = 1,2. I t should be noted, according to the inequalities (3.12.16), that M(o, t ) and fi(a, t ) are boundedfor t 3 to and lim N ( a , t ) = 0, t +m

lim N(o, t ) = 0.

(3.12.21)

t +m

W e denote by /3 and B the bounds of

11 M(a, t)l\ and 11 f i ( u , t)il respectively: I/ M ( c , t ) I/ < B for t 2 t o , (3.12.22)

and

jl M(o, t ) 11


for

to.

t

(3.12.23)

Also, it can be easily veriJied that 181

- 61

I

< a,*,

182 -

8,

I

< s,*, I s3-

63

I

< 65*, I a4 -

64

I

< a,*.

(3.12.24)

Theorem 3.17 reduces the boundedness and stability questions for linear delay-differential equations to the boundedness problem of the integrals m(t) and n ( t ) defined by (3.12.3). T h e interpretation of the inequality (3.12.19) with (3.12.20) in terms of the definitions of Section 2.20 is immediate. We only consider the following case. Let E be an arbitrary positive number. If

112

111. LINEAR DELAY-DIFFERENTIAL

EQUATIONS

and

(3.12.26)

and (x) t"

-

i" < 2 ~

6

+ ,PJ'

where the numbersp andflare defined by (3.12.22) and(3.12.23) respectively, then

I/ x(b, t o , 4,f)

- .qt, f"

, $,f)I1

E*

That is, the system (3.12.1) is unijormly stable with respect to the system (3.12.9). Many other interesting stability and boundedness properties can be derived from Theorem 3.17. We leave this t o the reader. Remark. Stability and boundedness problems concerning the systems with constant coefficients can be investigated by Laplace transform method. For difference-differential equations with constant coefficients we refer to the books of R. Bellman and K.L. Cooke [I] and E. Pinney [I].

CHAPTER IV

+

Nonlinear Delay-Dzflerential Equations

Having in the previous two chapters studied first the general theory of delay-differential equations and then the linear theory, we shall in this chapter briefly discuss some nonlinear equations. For the sake of simplicity, we will restrict ourselves to scalar equations. 4.1. Approximation of Continuous Functionals by Functional Polynomials: Theorem of M. Frechkt

I n Section 3.1 we mentioned the theorem of F. Riesz which gives an integral representation for bounded linear functionals defined over the space of continuous functions. T h e following theorem, due to M. FrechCt, is an extension of Weierstrass’ theorem on polynomial approximations of continuous functions to the continuous functionals. As we have already noted, all the quantities mentioned in the sequel are scalars. Theorem 4.1 (M. FrechCt [2]). Let G [ x ( t ) ]be a continuous functional deJined for x ( t ) E C(I[a,83, D ) , where D is a compact set on the real line and I[a,p ] is a finite interval. Then

(4.1.1)

1 . .

113

(4.1.2)

114

IV.

NONLINEAR DELAY-DIFFERENTIAL

EQUATIONS

the functions G 7 L , j ( ~..., l , T ~ ) being continuous and determined for the functional G independently of the variable function x(t). The expansion (4.1.1) is uniformly convergent in every compact set of continuous functions in C(I[a,PI, 0). Now let V ( t ,x( be a continuous functional over I[a,/3]x C(I[u,/3], D). Then, by virtue of the above theorem, we may write 0

)

)

T h e functions V n , i ( ~,l ..., T~ ; t ) , j = 0, 1, 2, ..., k, , are continuous and determined by the functional V(t,x ( - ) )independently of the variable function x(t). T h e limit (4.1.3) is uniform in every compact set of functions x(i) E C(l[or,/3], 0). T he functional Vn(t,x(.)) is analytic in the functional sense and is called a “polynomial of degree k, .”

4.2. Approximations of Solutions

Let V ( t , be a continuous functional with respect to t and x ( t ) on l [ a ,83 x C(l[ a ,PI, D) and locally Lipschitzian with respect to x. Given E > 0, let n be so chosen such that .(a))

for t E I [ a , /I]. L e t L be the Lipschitz constant for the functional V ( t ,x(.)). Consider the equations (4.2.2) and (4.2.3)

4.3. Functional Polynomials of Second Order

115

for t E ] [ t o , /3], 01 < to < /3, and their solutions x ( t , to ,4)and xn(t, t o , #J), whose existence and uniqueness are assured by the results of Chapter 11. According to the inequality (2.1 1.4), we may write

1 x(t)

- x,(t)

1 < (E/L)(eL(t-to)- 1 )

(4.2.4)

lim .%(t, t o +),

(4.2.5)

for t E ] [ t o , /3], i.e., x(t, to

,4)

=

f

n

uniformly in I [ t o ,/3]. Thus, the solution x ( t , t o , 4) of the original equation (4.2.2) may be well approximated by the solution xn(t, t o , #J)of the approximate equation (4.2.3) in the intervall[t, , /3], with a sufficiently large n. 4.3. Equations Whose Right-Hand Sides are Functional Polynomials of Second Order

This section is devoted to the study of an equation of the form

44

2 A,(t)x(t - h") f f KdT; "=a m

=

+(T)

dT

+ f(t)

t0

+ J-: 0

f

Kd71 7 72; t)'471)X(T,)

dTI d72,

(4.3.1)

to

where A is a real parameter, the functions AY(t),v = 0, 1, ..., m, f(t), K,(T;t), and K,(T, , T , ; t ) are given functions continuous for t E ] [ t o , PI, 7,T, , 7, E ] [ t o , 81. We shall denote by x(t, to , A) the solution of Eq. (4.3.1) corresponding to the initial condition

+,

"V(4

a

t o , 4, A) = +(t>

for

t

E

qa,t o ] ,

(4.3.2)

< to < /3. We assume that the interval I[a, to] is sufficiently large.

Since Eq. (4.3.1) depends analytically upon the parameter A, then, by Theorem 2.12, the solution x ( t , t o , +, A) is an analytic function of the parameter A. We now construct the Maclaurin expansion of the function x(t, t o , 4,A) with respect to A. For this purpose, we define the functions

subject to the conditions zo(t) = +(t),

for t E I[.,

to].

z,(t)

= 0,

n

=

I , 2, ...,

(4.3.4)

116

I V . NONLINEAR DELAY-DIFFERENTIAL EQUATIONS

By Theorem 2.10, the function z,(t) satisfies the linear equation

Let M(u, t ) and N(o, t ) be the kernel functions of the first and second kind associated with Eq. (4.3.5). By Theorem 3.11, the solution of Eq. (4.3.5) satisfying the first condition of (4.3.4) may be represented in the form (4.3.6) z,(t) = f0 M(o, t)$(u) do + N ( u , t)f(o)do

Lo

, for t ~ 1 [ t ,PI. Differentiating Eq. (4.3.1) with respect to A, putting X = 0, and making use of the results of Section 2.14, we find the following linear equation:

2 A.(l)%(t 7P

%(t> =

-

”=O

A,) -k

f Kl(7; t)zl(~)dT t0

where the function z,(t) is given by (4.3.6). Since we are seeking the solution of the linear equation (4.3.7) which satisfies the initial condition zl(t) = 0 for t EI[CI, to], Theorem 3.11 gives the following integral representation of the solution zl(t): zl(t)=

j”‘ N(u, t) do j”ro Jyo

K2(71

I

T 2 ; u)z0(71)20(T2)

dT1 d72

(4*3.8)

t0

for t E 1[t, , PI. Differentiating twice Eq. (4.3.1) with respect to h and putting X we obtain the following linear equation:

-

to

=

0,

J

(4.3.10)

4.3. Functional Polynomials of Second Order

h

117

+

Differentiating (n 1) times Eq. (4.3.1) with respect to h and putting 0, we obtain the following linear equation:

=

for t E I[to,PI, where

and Cak are binomial coefficients

- n(n - 1 ) ... ( n - k

Cn -

k!

+ 1)

(4.3.13)

Therefore, taking into account the initial condition (4.3.4), Theorem 3.1 1 yields zn+l(t)

f

.-.,zn)

t ) ~ + l ( o"0; 9

~ ' ( 0 ,

do

(4.3.14)

10

for t € l [ t oPI. , Since we have already found the functions zo(t),zl(t), z,(t), the recurrence formula (4.3.14) holds indefinitely. We now obtain some estimates for the functions z,(t). For this purpose, assume that

I z&t) I < A ,

< B,

I N(o, t ) I

I K(71, 72; t ) I


(4.3.15)

for t e I [ t o PI, , A, B, and C being finite constants. Then, Eq. (4.3.8) yields A2BC (4.3.16) I Zl(t) < 3 ( t - t0)3. Similarly, Eq. (4.3.10) yields

.I a&) I

A3B2C2 < 7( t - to)6.

Likewise, from Eq. (4.3.14) for n

I %(t) I

=

(4.3.17)

2, we find

< 5A4B3C3 504 ( t -

(4.3.18)

tO)9.

Suppose now that the inequalities

I z&) 1

< X(k)Ak+'B"Ck(t

-

tO)3k

(4.3.19)

118

IV. NONLINEAR DELAY-DIFFERENTIAL EQUATIONS

are true for k = 0, 1, 2, 3, ..., n. Then, by the general recurrence formula (4.3.14), we obtain

<

j ~ " + ~ 1 ( t ) x(n

+ 1)An+2Bn+1Cn+1 (t

- f0)3"+3,

(4.3.20)

where

By the inequalities (4.3.15) we have x(0) = 1.

(4.3.22)

A successive application of the recurrence formula (4.3.21) yields with ten figures

x(0)= 1 , X( 1) = *, x(3) = 0.0099206349, ~ ( 5= ) 0.0004481850, x(7) = 0.0000590912, x(9) = 0.0000034308,

x(2) x(4) x(6) x(8)

= 0.0555555556,

0.0019841270, = 0.0000227121, = 0.0000089009, ~ ( 1 0= ) O.OOO0011399. =

(4.3.23)

Contrary to the evidence of the early elements, the sequence x(n) tends to infinity due to the large values of Cnkwhen n is large. T o see this, put (4.3.24) x(k) = ( 3 k 1)

+

Then (4.3.25) Writing aI;

=

K! a"lpk,

(4.3.26)

we find n

(4.3.27) I n (4.3.26) we may choose a arbitrarily, since there is a modified homogeneity in (4.3.25). According to (4.3.22) we have x(0)= mo = 1. Therefore mp0 = 1.

4.3. Functional Polynomials of Second Order First, we choose

a =

119

1/54, Po = 54. We may easily show that pk

>, 54(K

+

(4.3.28)

for all integer k >, 0. This is true when K = 0. Suppose it holds for n. We deduce from (4.3.27) and (4.3.28) that

k

<

=

54(n

l)(n + 3) > 54(n + 2) . 9(3n(n++3)(3n + 4) *

Combining (4.3.24), (4.3.26) and (4.3.28) with

x(4 >

(n

+ I)! (3n + 1) ,

Po = 18. We

pk

for all integer k

=

2),

1/54, we obtain

(n 2 I),

54"

which shows that x(n) -+ 00. Now we choose 01 = 1/18,

01

+

(4.3.29)

may similarly show that

< 18(K + 1 )

(4.3.30)

2 0. Thus we have I[n

> 1).

(4.3.31)

Consider now the following power series m

P(5) = cc?P,

(4.3.32)

n=O

T h e radius of convergence of the series (4.3.32) is R = 18/ABC. Hence P(5) represents an analytic function of 5 in j 5 I < R. Now consider the series (4.3.34)

which is Maclaurin expansion of the function x ( t , to ,4, A) with respect to A. By the inequalities (4.3.19) and (4.3.31), this series is majorated

120

IV. NONLINEAR DELAY-DIFFERENTIAL

EQUATIONS

by the power series P(<).Since P(<) is uniformly and absolutely convergent for all A, t and t, satisfying

I A(t

-

I < R,

(4.3.35)

the series (4.3.34) is also uniformly and absolutely convergent for A, t and to satisfying (4.3.35). Therefore it represents the function x ( t , to , +, A). Thus we have (4.3.36)

for t E I [ t o , 83, where (4.3.37)

provided that the functions f(t), A , ( t) , v = 0, 1, 2, ..., m, K,(T;t) and & ( T ~ , T ~ t); are continuous for t to and A, B, C are given by (4.3.15). It should be noticed that the above results are natural consequences of Theorem 2.12 and 2.14. 4.4. Equations Whose Right-Hand Sides Are Functional Power Series

The method of the preceding section may be used for the study of equations whose right-hand sides are power series in the functional sense of the form

(4.4.1)

where the functions AY(t),v = 0, 1, ..., m , f ( t ) , and K,(T,, ..., 7, ; t), n = 1, 2, ..., are given continuous functions for T~ , t E I [ t o , 81, j = 1,2, ..., n. We denote again by x ( t , to ,(5, A) the solution of Eq. (4.4.1) satisfying the initial condition x(t, t o ,

4, A)

= +(t)

for

We shall consider once more the functions

t

E 1[a,t,].

(4.4.2)

121

4.4. Functional Power Series subject to the initial conditions zo(t)= $(t),

zn(t) = 0,

n = 1, 2,

...,

t

E I[a,to].

(4.4.4)

Since the right-hand side of Eq. (4.4.1) is analytic with respect to in the functional and ordinary sense, respectively, we may apply Theorem 2.12. By successive differentiation with respect to h and putting h = 0, we obtain

x ( t ) and h

and, in general,

where

with G,(t) - f ( t ) .

122

IV.

NONLINEAR DELAY-DIFFERENTIAL EQUATIONS

All the equations above are linear. Let M(u, t ) and N(a, t ) be the common kernel functions of the first and second kind of Eqs. (4.4.5). Then, owing to the initial functions (4.4.4), we may write

(4.4.8)

and, in general, z,(t)

=

f N ( u , t)G,(u; z o ,z , , ..., zk-,)do,

(4.4.10)

'

,I

where G,(t; z o , ..., zkPl)is defined by (4.4.6). By Theorem 2.12 we have (4.4.1 1)

+

for h in some neighborhood of h = 0 and for t e I [ t o , to A], h being sufficiently small. Thus, according to (4.4.3),for these A and t , we have (4.4.12)

where z,(t) are found by the above inductive process. 4.5. Separable Kernels: 1. Equations of Second Degree T h e analysis of Section 4.3 may be further developed in the case in which the kernel function K,(T,, T~ ; t ) has the following form:

CG ; ~ ) ( Tt)Gj2)(~,; ~; t), 7

K*(T~ T ,~ t ;)

=

J=1

(4.5.1)

4.5. I . Equations of Second Degree

123

where G ; ~ ) (tT) ; and G ; ~ ’ ( tT) ;, j = 1, 2, ..., r , are given functions, continuous for T , t 3 t o . Any kernel function K 2 ( ~ 1T~, ; t ) of the form (4.5.1) will be called separable, if it is bounded. Consider the equation in

A,(t)x(t - h,)

*(t) = “=O

+f

K1(7; t)X(T)

dT

+ f(t)

t0

(4.5.2)

where t 3 t o , AY(t),v = 0, 1, ..., m,f ( t ) , K1(7;t ) , and K 2 ( ~ T~ 1 , ; t ) are supposed to be continuous for t 3 t o , T , T ~ T~, 3 t o , and the kernel K 2 ( ~ T~ 1 ,; t ) is separable. In this case Eq. (4.5.2) may be written in the form

I n Section 4.3 we have shown that the solution x(t, t o , 4, A) of Eq. (4.5.2) has the series representation x(t, to >

where zo(t)=

4, A)

= (n

r0

M(u, t)+(u)do

+ 1) 2 f Cnk

k=l

~

(

An

n=o 71.

and zn+l(t>

27 rO

=

t ),doo:J

0

t0

(4.5.4)

zn(t>,

+ f N(u, t)f(o)do

(4.5.5)

to

Jio

~

(

7 > 17 2 ; t)Zk(Tl)Zn-k(Tz) dT1

dTz

7

(4.5.6)

where N(a, t ) and M(a, t ) are the kernel functions of the first and second kind associated with the linear equation

44

c m

=

“-0

A”(t)X(t - h,)

+f

and Cnkare the binomial coefficients.

$0

u 7 ;

t)x(.) d-r

+ f(t)

124

IV. NONLINEAR DELAY-DIFFERENTIAL EQUATIONS

Owing to Eq. (4.5.1), the formula (4.5.6) yields

where Z$"(t) =

It G:F)(T;

d ~ , p

~ ) z ~ ( T )

1 , 2.

(4.5.8)

p = 1 , 2,

(4.5.9)

=

to

Let M;P)(u,t ) =

f G?)(T;t)M(a,

T)

dT,

10

and Nj")(u, t )

=

It

G ; ~ ) ( t)N(u, T; T ) d ~ , p = 1 , 2.

(4.5.10)

,7

Now, let us multiply the function X , + ~ ( T ) by G ~ ' ( T t ) ; and integrate between tn and t . Using the formula (4.5.7) and applying the rule of Dirichlet, we find

T o use the recurrence formula (4.5.11) we need the functions ZgJt), 1, 2. These functions may be computed easily. T o do so, we multiply Eq. (4.5.5) by G~.")(T; t ) and integrate between to and i . Applying Dirichlet's rule we find

p =

Starting from the formulas (4.5.12), by successive application of the recurrence formula (4.5.1 I), we may compute all the functions ZkJt), and, by successive application of the formula (4.5.7), we obtain all the functions z,(t) which we need to form the series (4.5.4). Clearly the special case G ~ ) ( tT) ;= G~'")(T; t),

j

=

1, 2,

is particularly simple for the computation. It should be noticed that Pincherle-Goursat separable kernels.

..., Y ,

(4.5.13)

kernels are special

4.6. II. Equations with Analytic Right-Hand Side

125

4.6. Separable Kernels: II. Equations with Analytic Right-Hand Side

T h e method of Section 4.5 can be extended to the equations whose right-hand sides are functional power series of the form

where

We assume that the functions A,(t), v G%;(T;t), i = 1 , 2 , ..., r, , p = 1, 2, ..., n, n for T, t 3 to and

I K ( 7 1 ..., T

~

t ); I

= 0, 1 , ..., m , f ( t ) , K1(7;t ) , = 2, 3, ..., are continuous


(4.6.3)

for all n = 1 , 2, ..., and for T ~ ..., , T,, t 3 t o , where C is a finite positive number. Any kernel K , ( T , , ..., r, ; t ) o f the form (4.6.2) with the property (4.6.3) will be called separable. I n Section 4.4, we have shown that the solution x(t, t o ,4, A) of Eq. (4.6.1) is of the form (4.6.4)

where (4.6.5)

and

the functions G,(t; zo , zl,..., z ~ - ~being ) defined by (4.4.6). These functions may be computed easily when all the kernels K , , n = 1, 2, ..., are separable.

126

IV. NONLINEAR DELAY-DIFFERENTIAL

EQUATIONS

To do so, we first define the following functions: (4.6.7)

Multiplying the function . z ~ ( T )by G%;(T; t ) , integrating from to to t with respect to T , and changing the order of integration, we find

where (4.6.9)

and (4.6.10) Now, let us multiply the function z~(T)by G$)l,i(~;t ) and integrate from to to t with respect to T . Using the formulas (4.4.6), (4.6.2), (4.6.6), and (4.6.7) and applying Dirichlet’s rule, we find

where (4.6.12)

On the other hand, from the formulas (4.6.2), (4.6.6), and (4.6.7), we obtain

Starting from the formulas (4.6.8), by successive application of the recurrence formula (4.6.1 l), we can compute all the functions ZeA+l,i(t), and successive applications of the formula (4.6.13) yield all the functions zn(t)which we need to form the series (4.6.4).

4.7. Approximation of Continuous Kernels

127

4.7. Approximation of Continuous Kernels by Separable Kernels

Let

where to < fi < GO. By the approximation theorem of Weierstrass, any function K , ( T , , ..., T , ; t ) , continuous in I , , is the limit of a uniformly convergent sequence of polynomials in I,. Therefore, given E > 0, there exists a polynomial P , ( T ~ ..., T , ; t ) such that

K , ( T ~..., , rn;t )

= Pn(rl,

...,

7,;

t)

+ Q , ( T ~ ..., ,

t),

7 ;,

(4.7.2)

where Qn(rl, ..., T , ; t ) is a continuous function defined in I , and

I Q n ( r 1 , ..-,rn; in I ,

. Clearly

t) I

<E

(4.7.3)

each polynomial P,(T~, ..., 1% ; t ) is separable; i.e.,

Pn(r,, ..., r n ; t)

= i+i,+.

2

..+i,=O

aii,,,.i,t irli i ... r2

,

(4.7.4)

aiil.. being constants. Assuming that each K , ( T ~..., , T , ; t ) , n = 2, 3, ..., is written in the form (4.7.2), consider the nonlinear functionals

and

128

IV. NONLINEAR DELAY-DIFFERENTIAL

EQUATIONS

Since the solutions x ( t ) = x ( t , to ,4, A) and Z ( t ) = Z(t, to , 4, A), with the usual notation, are defined and continuous in I [ t o ,j?], there exists a positive number A such that

I x(t) I

< A,

I “ q t )I


(4.7.9)

for t E I [ t o , is]. Let L be the Lipschitz constant of Eq. (4.7.7) for t € I [ t O j,?] and 1 x I A. Then, we have

<

I w t , 4 . )) - W t , C( . )) I

< L 144 - q t ) I

(4.7.10)

and

1 W ( t ,2( . ))

- @ ( t , 2(

.)) I < c A ( t - to)[eAA(t-to)

-

I]

(4.7.11)

for t E ] [ t o , is]. Since

I W t , 4 . )) - @P> a( . 1) I

< I W(t,4 . 1)

-

w, q )) I ’

+ I w, q . )) - w, q . )) I,

the inequalities (4.7.10) and (4.7.1 1) yield

I W ( t ,x( . 1)

-

@ ( t , a( . 1) 1

< L 1 x ( t ) - ~ ( tI )+ c ~ ( t

- to)[eAA‘t-to) -

11

(4.7.12)

for t d [ t o j,?]. We now integrate both sides of Eqs. (4.7.7) and (4.7.8) with respect to t, from to to t . Putting u(t) = I x ( t ) - q

t)I

and using the inequality (4.7.12), we find u(t)

<(c/~)(t

- t,,)eAA(t-to)

+L

(4.7.13)

st

(4.7.14)

u(s) ds

to

for t E I [ t o ,is]. Therefore, by Gronwall’s lemma, the function u ( t ) , i.e., 1 x ( t ) - Z ( t ) l , is an infinitesimal with E for every finite interval I [ t o ,j?]. Thus, the solutions x(t, to , 4, A ) of Eq. (4.7.8) with separable kernels approximate well the solutions x ( t , to ,4, A) of the original equation (4.7.7). 4.8. Extension

In this section we shall deal with an equation of the form n(t) = V(t, X( . ))

+ @ ’ ( ~ ( t ) ,~

(-t hj), ..., x(t

-

hm), t ) ,

(4.8.1)

4.8. Extension

129

where p is a parameter,

and F(u, , u2 , ..., urn+,, t ) is an analytic function of u1 , ..., urn+,, t in a compact set D of the ( m 1)-dimensional u space and for t € I [ t o ,PI, where ,B may be arbitrarily large. We suppose that the parameter h is fixed, and the functions A,(t), v = 0, 1, ..., m , f ( t ) ,K n ( ~,l..., 7, ; t ) , n = 1,2, ..., are continuous for t E I [ t o ,PI. Since Eq. (4.8.1) depends analytically upon the parameter p, by Theorem 2.12, the solution x ( t ) = x ( t , t o , 4, p), with the usual notation, is also an analytic function of p. T h e method of Section 4.3 may be easily modified for the present equation. For simplicity, we shall work on the equation

+

k(t) = f(t)

t A(t)x(t)t B(t)x(t - W)

+ f (" + W t ) ,x(t

$0

&(TI

to

where

w

+ f KI(T; t)x(T)

, 72;

&4T,)x(Tz)

d71d72

to

-

w),

0,

is a positive constant. Let

with the initial conditions

(4.8.3)

130

I V . NONLINEAR DELAY-DIFFERENTIAL

EQUATIONS

Hence, zo(t)can be computed by the method of Section 4.3. Therefore, we may assume that zo(t)is known. Differentiating Eq. (4.8.3) with respect to p and letting p = 0, we find

which is linear with respect to z,(t). Since z,(t) = 0 for t E I[to ,/3], we can calculate zl(t) by the method of Chapter 111. Differentiating twice Eq. (4.8.3) with respect to p and putting p = 0, we find %(t)

=

A(t)z,(t)

+ B(t)z,(t

-

w)

+f

&(7;

t ) 4 4 d7

t0

+ A J";

o

It

KZ(T1 Y 72;

t"O(T1)~2(72)

*n

+ 2[I;,(zo(t), zo(t

-

w),

t)z,(t)

+ 2~1(7,)~1(~2) + %(71)X0(72)1d71d72

+ Fz(zo(t), zo(t

-

w),

t)zl(t

-

w)I,

(4.8.8)

which is linear with respect to zz(t).I n (4.8.8) F, and F, denote the partial derivatives of the function F(u, , u 2 , t ) with respect to u1 and u2 respectively. Since z2(t) = 0 for t E I[t, , /3], we may easily find the function zz(t)for t E ][to, 81. Following this process we obtain all zn(t)as solutions of linear equations satisfying the initial conditions (4.8.5). Having then constructed the series (4.8.9)

we can show, making use of the analyticity of F(u, , ug , t ) and the continuity of A(t),B(t),K1(7; t ) , and K z ( r l ,r 2 ; t ) ,that the series (4.8.9) converges absolutely and uniformly for t E I [ t o ,81. Hence the series (4.8.9) represents the solution x ( t , t o ,4, p). T h e above method may be applied to the general equation (4.8.1) without any difficulty.

4.9. Boundedness and Stability

131

4.9. Boundedness and Stability

I n Sections 2.19-2.22, we have investigated stability and boundedness of delay-differential equations in general. I n this section we shall briefly discuss, with a different approach, some stability and boundedness properties of some nonlinear equations. We shall consider in particular equations of the form

(4.9.1)

where X is a constant parameter and the functions f ( t ) , AY(t),Y = 0, 1, 2, ..., m,K,(T;t ) , and K , ( T , ,T~ ; t ) are supposed to be continuous for T , T , , T ~ t ,3 t o . Equations of the form (4.4.1) can be investigated similarly. Let M(o, t ) and N(o, t ) be the kernel functions of the first and second kind of the linear equation m

(4.9.2)

“=O

Consider the solution x(t, t o , 6,A) of Eq. (4.9.1) corresponding to the initial condition x(t, t o , 4, 4 = + ( t ) for t E I[%t o ] , (4.9.3) where $(t) E C(l[a,to],D), D being a closed bounded set on the real line. I n addition to the above continuity hypotheses, we assume that

s:

I M(o, t ) I do

s’ I

No,

t ) I do

for

t

for

t >, t o ,


for

t >, t o ,


for t >, t o ,

t0

where a,8, y , 6, , and 6, are positive constants.

E I[a,to],

132

IV. NONLINEAR DELAY-DIFFERENTIAL EQUATIONS

I n Section 4.3 we have shown that (4.9.5) where

the function zo(t)is given by

and the functions zJt), n

=

1, 2, ..., satisfy the recurrence formula

for t 2 t o . Using the inequalities (4.9.4) and the formula (4.9.7) we find for 1

2 t o , where

(4.9.9)

I zo(t) I G $0 9.0

= 61B

+ 6,Y.

(4.9.10)

Making use of the general recurrence formula (4.9.8),we obtain by induction (4.9.11) I zn(t) 1 < 8 n for t 3 t o , where $n -__ n2n-l$n+lan

y n,

n = l , 2 ,....

(4.9.12)

Therefore, the series (4.9.5)is majorated by the series (4.9.13) Thus: Theorem 4.2. Let Eq. (4.9.1) be given and assume that the inequalities (4.9.4)are satisfied, all functions involved in (4.9.1)being continuous f o r

133

4.9. Boundedness and Stability

t 2 t o . Then, the solution x(t, t o , 4, A) of Eq. (4.9.1) is bounded by B, y , 4 f o r t >, to ,

4 6 0 , %

I x(t,

to

9

474 I < 4 8 0 , B, Y , 4,

(4.9.14)

where A(9, , a, p, y, A) is the sum of the series (4.9.13);i.e.,

+ 8oayX e~p(28~ayA)l.

a, B, y, A) = $.,[I

(4.9.15)

Remembering that zo(t) = x ( t , tn 3 $7

O),

the above argument also shows that: Under the hypotheses of Theorem 4.2, we have

Theorem 4.3.

I x(t, to 4, A) - x(t, t o , A 0 ) I < A(80, a, B, y , A) 9

- 80

(4.9.16)

for t 2 t o . We now compare the solutions x(t, t o , 4, A) and 2(t, t o , A) of the same equation (4.9.1) corresponding to the initial functions + ( t ) and +(t) in the same initial interval ][a,to]. Let

4,

(4.9.18)

and

lp, = max I zn(t)I ,

= 0,

t>to

I , 2,

... .

(4.9.19)

Since Zo(t) is the solution of the linear equation (4.9.2) with the initial function &t), we may write Zo(t) =

J

t0

M(u, t)&u) du

+

(4.9.20)

Thus, by virtue of Eqs. (4.9.7) and (4.9.20), we have

I for t

- %(t)

!

<

t o . Therefore, f o ( t ) = zo(t)

70 1

$0

(4.9.21)

+

= 81)
3

(4.9.22)

134

IV. NONLINEAR DELAY-DIFFERENTIAL

EQUATIONS

where (4.9.23) Since

and

making use of the equality (4.9.22) and the inequalities (4.9.4) and (4.9.23), we find

I zi(t) - zi(t) I for t 3 t o . Therefore, %(t> = 4

where

4

rt

we obtain

71 >

I 71 !, I q 1 I

Similarly, from

and

+-

"U

"(1

< a2+Y8n + 8,) $1

< a"Y9.0

= 9.1

+

+

$0).

7j1,

(4.9.24)

(4.9.25) (4.9.26)

4.9. Boundedness and Stability

135 (4.9.29)

By induction, we can easily show that (4.9.30)

(4.9.31) (4.9.32) Consider now the function

(4.9.27),and (4.9.30)yield T h e inequalities (4.9.21),(4.9.24), 9.3 ) Since the right-hand side of (4.9.33)is infinitesimal with 6 , we may state the following: Theorem 4.4. Let the inequalities (4.9.4) be satisfied. If all the functions involved in Eq. (4.9.1)are continuous and if

I B ( t ) - 4(t) I < 6,

(4.9.34)

where 6<

41

+ 4 a o + $,)A

c

exp(2v(80

+ 80)A)l’

(4.9.35)

then

I Z ( t , t o , 6,A)

- x ( t , t o , $ , A)

I <E

(4.9.36)

for t 3 t o .

It should be noticed that Theorem 4.4expresses the uniform stability of the system (4.9.1) under small variation of the initial functions. Similar results can be established if to and h also vary. T h e method described above can also be extended to the systems of the form (4.4.1).

CHAPTER V

@& Piecewise Continuous Delay-Dzferential Equations

This chapter is devoted to study of ,delay-differential equations whose right-hand sides are piecewise continuous. For the sake of simplicity we shall consider piecewise linear equations with analytic coefficients. More general equations may be investigated similarly. 5.1. Introduction

Let s be an r-dimensional column vector. We define the row vector sgn s (signum s) by sgn s

=

(sgn s, , sgn s 2 ,

..., sgn sT),

(5.1.1)

where sl, sz , ..., s, are the components of s and sgn sk

=

:i

for s k for sk for sk

undefined

> 0, = 0,

< 0.

(5.1.2)

Consider now the following piecewise linear equation: m

Ay(t)x(t- h,)

x(t) = "=O

+

K(u, t ) x ( u ) do

(5.1.3)

t0

+ B ( t ) sgn{s[x(t - 711,

where T is a positive constant, called switching delay, and the spans h, are as usual, 0 = h, < h, < ... < h,,%. (5.1.4) I n (5.1.3), AY(t),v = 0, 1, functions analytic for u, t

..., m,

and K(a,t ) are given n x n matrix Y matrix function 136

t o , B ( t ) is a given n x

5.1. Introduction

137

analytic for t 3 to ,s(x) is a given r-dimensional column vector function analytic in the n-dimensional Euclidean space En, and x ( t ) is an n-dimensional column vector. We assume that s k ( x ) and grad s k ( x ) , k = 1,2, ..., r , do not vanish simultaneously at any point of the space En.T h e homogeneous equation associated with Eq. (5.1.3) is m

k(t) = z A , ( t ) x ( t - h,) "=O

+ f K(u, t)x(o) do.

(5.1.5)

t0

s k

in En defined by

2, ..., I ,

(5.1.6)

Let us now consider the analytic hypersurfaces the equation

Sk = {X I S k ( x )

=

o},

k

= 1,

which are called switching spaces. T h e space En can be decomposed into domains Dk+ and Dk- in which sk(x) 2 0, respectively, and s k will be the common boundary of the domains Dk+ and Dk-, k = 1, 2, ..., r . Furthermore, let r

9'=

u Sk

and 9 = E n

-

S.

(5.1.7)

k=l

Clearly, the right-hand side of (5.1.3) is discontinuous along the analytic hypersurface S. Let be the set of all n-dimensional analytic vector functions +(t) in the initial interval to - h, - T t to.

< <

Definition 1.

Eq. (5.1.3) if:

A continuous vector x(t) will be called a solution of

(A) x ( t ) satisfies (5.1.3) in 9, ' . (B) x ( t ) has only isolated points in 9 T h e solution of the system (5.1.3) which satisfies the initial condition x(t) = +(t)

for to - h,, - T

< t < to,

(5.1.8)

with + ( t ) E m, will be denoted by x = x ( t , +) or x = x ( t ) . Since, by our hypotheses, the matrix functions B(t),K(a, t), and A Y ( t ) ,v = 0, 1, ..., m, are analytic for u, t >, t o , x(t, +) is analytic in a neighborhood of t = to, if the vector sgnW(t0

is defined.

-

4)

138

v.

PIECEWISE CONTINUOUS DELAY-DIFFERENTIAL EQUATIONS

If, in particular, there is no time delay in (5.1.5), i.e., h, = h, = = 0, K(o, t ) = 0, and if the matrices A, = A and B are constant, we have

... - h,

k(t) = Ax(t)

+ B sgn(s[x(t -

(5.1.9)

T)]}.

This difference-differential equation has been investigated by J. AndrC and P. Seibert [3]. Equation (5.1.9) with T = 0 belongs to the class of differential equations with piecewise continuous right-hand sides. For references we refer to the Bibliography. In the next section we shall extend some basic results on the continuation of solutions which are due to J. AndrC and P. Seibert [3] and to M. I%. O@ztoreli [I]. 5.2. Continuation of a Solution

Suppose that the value +(to - T ) of the initial function +(t)E 'Dl is an interior point of one of the domains 9, say Dk+: +(to - T) E

(5.2.1)

L)k+.

(The case +(to - T ) E Dk- can be investigated similarly.) Let 'Dl, be the set of all initial functions which possess this property. Put (5.2.2)

e+ = sgn{s[+(to - T)]}.

T h e vector e+ has +1 as its kth component:

T h e solution of (5.1.3) which satisfies the initial condition (5.1.8) with +(t)E %nk satisfies the equation

2 AY(t)x(t In

k(t) =

-

"=O

h,) $-

s'

K(u, t)x(u) du

+ B(t)e+

(5.2.4)

t0

for t >, to . Since this solution is analytic from the right in the neighborhood of t = to by Theorem 2.12, it will remain in the domain Dk+ for some further time interval. Let us denote this solution by x + ( t , + ) or simply x+(t). Since the solution of a delay-differential equation with retarded argument can be continued in the forward direction with respect to time, we have to follow x+(t) for increasing t. Suppose the

5.2. Continuation of a Solution

139

smallest t(>to) for which x+(t) reaches the switching space S , is T, the point u that it reaches being given by (5.2.5)

u = lim x+(t, (6). t4Z-0

Of course, T and u depend on the selection of the initial function #(t) from the set %Rk. According to our definitions,

2 A,(t)x+(t - h,) + j t K(u, m

f+(t) =

t)x+(u) du

+ B(t)e+

(5.2.6)

to

V=O

and for

x + ( t ) = +(t)

to - h, - 7

< t < to.

(5.2.7)

For the sake of simplicity we shall denote the right-hand side of Eq. e+):

(5.2.6) by V(t,

%(a),

V ( t ,x( . ), e+)

m

=

A,(t)x(t -- h )

+ jt

K(o, t ) x( o) do

+ B(t)e+.

(5.2.8)

t0

"=O

Let M(o, t) and N(u,t) be the kernel matrices of the first and second kind associated with the linear equation (5.2.6). Then, by Theorem 3.1 1, we have the integral representation x+(t) =

f"

M(u, t)+(o) do

to-h,,,-r

+ f N(o, t)B(u)e+do.

(5.2.9)

t"

Since the function sgn{s[x+(t - T)]} is still continuous and equal to T t < T 7, our solution x+(t, 4) will continue to satisfy the same linear delay-differential equation (5.2.6) in this interval. We put e+ in the interval

u,* =

+

<

lim

t4T+T-O

x+(t, (6)

where X(t) =

f"

M(u, t)+(o) do,

(5.2.1 1)

to-h,-r

which is the solution of the homogeneous equation (5.2.5) corresponding to Eq. (5.2.6). Clearly, the point ur* depends on the choice of the

140

v.

PIECEWISE CONTINUOUS DELAY-DIFFERENTIAL EQUATIONS

initial function +(t) from the set m k . We shall denote by Sie7the set of all points u,*. Note that, according to the analyticity in the interval T t T 7, the solution x+(t, 4) can intersect the switching surface Sk at most at a finite number of points; that is, the intersections of x + (t, +) with Sk cannot have an accumulation point which is reached in the interval T t T 7. In order to follow the further continuation of the solution x+(t, +) after t = T 7, we have to investigate the behavior of the function s k [ X + ( t , + ) ] in the neighborhood of t = T. For this purpose, we first assume that the point u belongs to only one switching space, Sk . Therefore, (5.2.12)

< < +

< < +

+

k = 1,2, ..., r . Let e- be the vector whose components are equal to those of e+ except for the Kth, which is equal to - 1. We now consider the following delay-differential equations: * ( t ) = V ( t ,x( . ), e*),

and put +l(t)

= x+(t,

4)

for

T

-

h,

(5.2.13)


7.

(5.2.14)

This function is analytic in its interval of definition. Let x+(t,+) and x-(t, be the respective solutions of these equations which satisfy the same initial condition ~ ( tz=)

41(t),

T - h,

- T


7.

(5.2.15)

Since all the conditions of the existence and uniqueness theorems are still satisfied, x+(t, dl) and x- ( t , are uniquely determined analytic functions. Obviously, we have x+(T

+

7,41) =

x-(T

+

= u,*,

7941)

(5.2.1 6 )

and from (5.2.3), (5.2.14), and (5.2.15) we find lim

t+T+7-0

sgn(sk[x*(t,

=

+1 .

(5.2.17)

41111.

(5.2.18)

We can easily evaluate the following limit: Yk =

lim

t -1 +r+0

Sgn(Sk[x*(t,

For this purpose, remembering the hypotheses which we laid on the

5.2. Continuation of a Solution

141

functions B(t), K(o, t), AY(t)(v = 0, 1, ..., m), +(t), and s(x), let us at t = T : consider the Taylor expansion of sk[x+(t,

=

~ l ( u ) ( t- T )

+ -21c ~ ( T ’?&)(t ,

-

T)’,

(5.2.19)

where T’ is an intermediate value between t and T , and

It can be easily verified that

and

(5.2.22)

Here by * we denote the operation of transposition, while the operator d/dx is the gradient when applied to a scalar and the Jacobien matrix when applied to a vector. The operator d2/dx2, applied to a scalar, denotes the Jacobian matrix of the gradient. Since u is given by the formula (5.2.10), the quantities cl(u) and cz(u) are well defined and they determine the character of the switching at the point x = u. As we noted above, the switching operation actually happens when the trajectory reaches the point x = u,* E S;2.Tat time t = T T, because of the switching delay T. If cl(u) # 0, on account of (5.2.17), we have cl(u) < 0. Therefore, s,[x+(t, 4)] < 0 for t > T . Thus, we have the delay-differential equation

+

n(t) = V ( t , x( . ), e-)

(5.2.23)

+

3 T T. If cl(u) = 0, c2(u) # 0, on account of (5.2.17), we have cz(u) > 0. Thus, we have the delay-differential equation for t

x(t) =

V ( t , x( . ), e+)

(5.2.24)

142

v.

for t

2T

PIECEWISE CONTINUOUS DELAY-DIFFERENTIAL EQUATIONS

+

T.

If cl(u)

= cz(u) =

lim

T'+T+O

0, the sign to be taken will be

sgn[c,(T', u ) ] .

(5.2.25)

Let x(t, $1) be the continuation of x(t, 4 ) obtained by this manner. x(t, $) will clearly satisfy one of Eqs. (5.2.23) and (5.2.24) for t 3 T + T and the initial condition x(t, $J = x(t, $) for T - h , - T t T T. T, the solution Since x(t, $) is analytic for T - h, - T < t < T x(t,4,) is also analytic for some time interval T 7 t T T p, where p is a positive number. I n this interval the solution x(t, +1) may intersect the switching space S at most only at a finite number of points because of the analyticity of the vectors s(x) and x(t,+,). Therefore, there is no accumulation point in this interval of the intersections of x(t, $,) with S. Let us now assume that the point u belongs to more than one switching space S , . T o investigate this case generally, consider a domain D whose boundary S consists of some parts of the switching spaces S,,, Sk,, ..., S k , (1 i Y). Let us assume that $(to- T) E D and denote by mD the set of all analytic functions $ ( t ) in to - h,, - T t to having this property. I n this case, the function sgn{s[$(t)]} is piecewise defined for t" h,,, - 7 t to . Put

< < + + + < < + +

< <

~

< <

< <

0 =

sgnW(t0

-

< <

.)I>

(5.2.26)

for to - h, - T t t o . Each component u, ( k = 1, 2, ..., Y ) of u has one of the values 3 1, as before, and we can apply the same method as we used at the beginning of this section. Let x(t, 4) be the solution of (5.1.3) which satisfies the initial condition (5.1.8) so that now x(t, +) will take the place of xt(t, $). Let T( > t o ) again denote the first time at which x(t, $) reaches the switching space S (at point u, say) and suppose that u belongs to the switching spaces S,(,, S k z ,..., S, ( p i Y ) . Let us now evaluate the limits y k ( k = k , , k, , ..., k,) by thlformulas (5.2.19) to (5.2.22) and with the help of these limits calculate the components of the vector

< <

0

=

lirn sgn{s[x(i, +)I).

(5.2.27)

$+Ti 0

Then, the continuation x(t, $1) of the solution x ( t , $) for t satisfy the delay-differential equation i(t) =

4)

for

I' - h,,

- T

+

T

will

(5.2.28)

V ( t , x( . ), 6)

and the initial condition x ( t ) = (bl(t) = x ( t ,

3T


T.

(5.2.29)

5.3. Boundedness and Stability

143

Since all the conditions of the existence and uniqueness theorems are is uniquely determined. still satisfied, x(t, From the analyticity of x(t, dl)and s(x), the function x ( t , +1) cannot intersect the switching space S at an infinite number of points in the Consequently: interval of analyticity of x ( t , Theorem 5.1. Each solution x ( t , 4) of the delay-dzfeerential equation (5.1.3) which satisfies the initial condition (5.1.8) with + ( t ) E 1132, can be continued indefinitely in the future; i.e., they have no end points.

5.3. Boundedness and Stability

This section is devoted to the study of some stability and boundedness properties of delay-differential equations of the form (5.1.3) with analytic coefficients. Consider the homogeneous equation (5.1.5) associated with Eq. (5.1.3) and its solution X ( t , 4) which is given by (5.2.1 1) in terms of the initial function + ( t ) and the kernel matrix M(u, t ) of the first kind. Suppose that all continuous solutions X ( t , +) of (5.1.5) are bounded as t -++a:

I1 w , 4) /I < c1

9

(5.3.1)

t 3 to >

where c1 is a finite constant. Let the kernel matrix N(a, t ) of the second kind and the matrix B ( t ) be of the form

and

where c2 is a finite constant. Furthermore, let R(u,t ) and A Y ( t ) be n x n matrix functions analytic for t >, to such that

s+" a,.(,) I/

tU

and

ll du < +a,

v = 0, 1,

..., m,

(5.3.4)

144

v.

PIECEWISE CONTINUOUS DELAY-DIFFERENTIAL EQUATIONS

Consider now the delay-differential equation *(t) =

3

+ A,(t)lx(t

"-0

4-B(t) sgn{s[x(t

--

-

+f

A,)

[K(o,t )

tIJ

T)]}

+ R(o,4140) do

.

(5.3.6)

Then: Theorem 5.2. Under the hypotheses (5.3.1) to (5.3.5) all continuous solutions of Eq. (5.3.6) are uniformly bounded as t -+ 00.

+

Proof.

Let

2 A,(t)x(t m

ai(t) =

-

h,)

+

"=O

st

f i ( a , t ) x ( o ) do

tU

Continuous solutions of (5.3.6) can be obtained by the method of the previous section. For a continuous solution x"(t)of (5.3.6) corresponding to an analytic initial function +(t),the function w ( t ) is continuous in every domain D bounded by the switching spaces S , and is integrable everywhere. Therefore, for t 3 t o , ."(t) = X ( t , 4)

+ 3f N(u, t)A,(o)a(o u=o

+ f N(s, t )

s'

d5

h,) do

1' K(u,

s)S(u) do

t0

t0

+

-

to

N(u, t)B(u)sgn{s[i(u

- T)]}

(5.3.8)

do.

to

Applying Dirichlet's rule in the second integral and changing suitably the variable of integration and using the inequalities (5.3.1) and (5.3.2), the equality (5.3.8) yields

I! a(t)/I

< cl + cz f

E(o) II a(o)I/ do,

(5.3.9)

t,--hm

where E(o) =

3II A,(o + "==O

h,) II

+ e(u

-

to)

1'II K(o,

s)

I1 ds,

(5.3.10)

5.3. Boundedness and Stability in which e(u - to) =

;1

for for

145

< to > to.

(5.3.1 1)

E ( 4 do],

(5.3.12)

u

u

Hence, by Gronwall’s lemma, we have

II W) I1 < c expk.2 J

.t

to-h,

where c = c1

+

c2

I+* I/ B ( 4 I/

(5.3.13)

do.

to

It follows, from (5.3.3), (5.3.4), ( 5 . 3 3 , and (5.3.12), that x”(t)is uniformly bounded as t + +a. Now let us assume that

II N(o, It

< +(t)

for

t 2 to,

(5.3.14)

where +(t) is a monotonic decreasing function such that (5.3.15)

lim + ( t ) = 0.

t ++*

In this case, we have the following result: Theorem 5.3. If (5.3.3), (5.3.14), and (5.3.15) me satisjiied, then the solution x(t, 4) of the system (5.1.3) is asymptotic to the solution X ( t , 4) of the homogeneous system (5.1.5); i.e., lim

t ++*

Proof.

II x(t, 4)

-

x(t,4) II = 0.

(5.3.1 6 )

By Theorem 3.1 1 we have x(t,

4)

=

X ( t , 4)

+ f N(o, t)B(u)sgn{s[x(o

-

.)I>

do

(5.3.17)

t0

and X ( t , 4) is given by (5.2.11). T h e hypotheses (5.3.14) yields

Using (5.3.3) we obtain

II x(t9 4)

-

4) I1 < 6

for sufficiently large t . This proves the theorem.

(5.3.18)

146

v.

PIECEWISE CONTINUOUS DELAY-DIFFERENTIAL EQUATIONS

Suppose that

11 M(u, t ) I1 do

<

~ 3 ,

t

2 to

9

(5.3.19)

where c3 is a finite constant. Consider any two initial functions $(t) and & t ) analytic in the initial interval to - h,, - T t to such that

< <

(5.3.20) Let X ( t , +) and x ( t , $) be the solutions of the homogeneous equation (5.I .5) corresponding to these initial functions respectively. According to (5.3.19) and (5.3.20), the formula (5.2.1 1) yields

/I X ( t , +) Therefore, if, for given

E

-

X(t,

4)I1 < c$.

(5.3.21)

> 0, 6 < e / c 3 , then

for t 3 t o . 6

Theorem 5.4.

If (5.3.3), (5.3.14), (5.3.15), (5.3.19), and(5.3.20)with

< eic3 are satisjied

with an arbitrary

E

> 0, then

I/ 4 t >4)- f(t, $1 I! < E and

/I Z ( t ,

4")

-

X ( 4 4) /I

< 2E

(5.3.23) (5.3.24)

f o r sujiciently large t .

then, under the hypotheses assumed, the inequality (5.3.22) holds. This is equivalent to the inequality (5.3.23). T o prove the inequality (5.3.24), observe that

/I ."(t, 4")

-

x(t,4)II < I1 .c(f,4") - 4 4 4) I/ + I1 x(t,4) -x(t, 4)11.

(5.3-26)

T h e inequalities (5.3.1 8) and (5.3.23) yield the inequality (5.3.24) which proves the theorem. Note that the above results can be easily extended to the case where

5.4. Limits of the Solutions of (5.1.3)

147

the system (5.1.3) is affected by small disturbing terms. In this case the delay-differential equation takes the form

(5.3.27) where

Y

is an analytic function such that

to and x, 7 being a small positive number. for all t It should be noticed that the above theorems extend the results obtained by R. Bellman and K. L. Cooke [2] and M. N. O@ztoreli [l]. 5.4. Limits of the Solutions of (5.1.3) as the Retardations and the Switching Delay Approach Zero

Let D be any domain bounded by the switching spaces S, and let + ( t ) E '%ID , where '%ID is, as before, the set of all analytic functions for to - h, - r t t o . Consider now the distance between the points u and u,* defined by (5.2.5) and (5.2.10). We have

< <

/I u,*

- 14

II < /I X ( T

+

+ jTII

7 1

4)

"u,

- X(?',4) II

+

T

7) -

+ J"'r

I1 N ( o , T

+ 4 I/ I/

M u , T)II II W4Il do.

II do (5.4.1)

t0

Let

E

be an arbitrarily small positive number. From the continuity of

X ( t , +), N(a, t ) , and B(t), we can write and

II X ( T

+

ll N(0, T

+

7 9

4)

-

7) -

for

T

< 6 , 6 = S(r, 4) > 0, and

for

T

<E

-Y(T,4) ll

<43

N ( o , T ) /I < ~/3Tc4

/ ~ c,~where c ~ cz is given by (5.3.2) and c4

= II w ) I l l [ t o , T ] '

148

v.

PIECEWISE CONTINUOUS DELAY-DIFFERENTIAL EQUATIONS

Therefore, if 6

< €/3C2Cq,

(5.4.2)

we have

/ / u,*

- uIj


for

T

< 6.

(5.4.3)

By the same argument we see that

where x,(t, +) denotes the solution of the system (5.1.3) with the switching delay T. Obviously these results are also true for any continuation of x,(t, +) obtained by the method given in Section 5.2. Let K(o, t ) = 0 and put y ( t ) = lim x(t, h,+O

4).

(5.4.5)

It can be shown that y ( t ) is the solution of the equation Nt)

=

40Y(t>

+

sgn{s[y(t

--

41>, (5.4.6)

< <

and the convergence is uniform for some interval to t T o ,provided we stay in a domain D bounded by switching spaces. For the proof of this fact, we refer the reader to the work of R. Bellman and K. L. Cooke [4].Equation (5.4.6) is of the form (5.1.9). Clearly, this limiting process may be accomplished for any continuation of x ( t , +) beyond the domain D obtained by the method of Section 5.2. T h e solution of (5.4.6) gives an approximation to the limiting behavior of the solutions of (5.1.3) for sufficiently small time lags. 5.5. Extension

T h e method described in Section 5.2 for the continuation of solutions of linear delay-differential equations can easily be extended to nonlinear delay-differential equations. We shall briefly outline this problem. Let s(x), as in Section 5.2, be an analytic r vector in En and consider and the set 29 defined by the hypersurfaces S , (k = 1, 2, ..., r ) , 9, (5.1.6) and (5.1.7). Let e = sgn(s[x(t

- T)]}.

(5.5.1)

5.5. Extension

149

The vector e is constant in every domain D bounded by some hypersurfaces S , . Consider the equation k ( t ) = V ( t , x(

. ), e),

x E 9, t

>, t , ,

(5.5.2)

where the functional V is assumed to be analytic with respect to t and x , in the ordinary and functional sense, respectively, in each domain D. Let mD be the set of all initial functions + ( t ) analytic for t E I [ ~to], , where the initial interval is sufficiently large. If we continue to use Definition I of Section 5.1 for the solutions of Eq. (5.5.2), all the arguments of Section 5.2, except the formulas related to integral representation of solutions in terms of kernel matrices of the first and second kind, are still valid and Theorem 5.1 is true for the delay-differential equation (5.2.2). T o extend the results of Sections 5.3 and 5.4 we need further analysis, based on the general theorems of Sections 2.19-2.22 or on the specific theorems of Chapter IV.

CHAPTER VI

+& Delay-Dzflerential Equations Depending on Arbitrary Functions

I n the previous chapters we have investigated delay-differential equations which depend on arbitrary parameters. I n this chapter we shall consider delay-differential equations involving arbitrary functions which belong to an L, space and discuss the first variations of their solutions. We shall restrict ourselves to analytic equations. 6.1. Notation and Definitions

I n this section we summarize some notation, definitions, and results concerning L, spaces, to which we shall refer frequently. For detail, we suggest the books: N. Dunford and J. T. Schwartz [l] and P. R. Halmos PI. We denote by L,(a, b ) or briefly by L,, where p >, 1, the vector space of all scalar-valued real functions u ( t ) measurable in the interval Z[a, b ] , whose absolute values raised to the pth power are integrable in the sense of Lebesgue. T h e quantity

I/ u I/

=

I1 u IlP

=

[I"

I u ( l ) lP

4

llP

(6.1.1)

is called the norm of an element u ( t ) E L,(a, b). Any two functions u ( t ) and u"(t)in L,(a, b ) are identified if the equality u(t) = u"(t)holds almost everywhere; i.e., except possibly for a set of t's of Lebesgue measure zero. T h e norm 11 u /lo satisfies the norm axioms (i)-(iii) stated in Section 2.2. It is well-known that, if 1 1 -+-=1, (6.1.2)

P

P

150

151

6.1. Notation and Definitions

and if v( t ) €&(a, b), then the integral J* v(t)u(t)dt is absolutely convergent and the Holder inequality .h

holds. T h e number q is called the conjugate of p . If we have the Minkowski inequclity

u ( t ) , u“(t)€&(a,

b)

which corresponds to the triangle inequality

II 24

+c

112,

< I1 u

I19

+ II c

119

(6.1.5)

in Lp . Let I [ a , b] be a finite interval and let v ( t ) be bounded on I[a, b]. Then the limit (6.1.6)

exists, is finite and IvW

I GM

(6.1.7)

almost everywhere in I[a, b]. T h e number M is called the essential supremum (ess sup) of u ( t ) and is denoted by 11 v /Im :

/I v [ I r n

= ess sup 1 v ( t ) I. tel[a,bl

(6.1.8)

Note that the vector space &(a, b) with norm (6.1.1) is complete; that is, it is a Banach space. Let w be a linear functional on &(a, b). Then, there is a function w ( t ) in L,(a, b) such that

j b

W(.)

=

a

v(t)u(t) dt

(6.1.9)

for every u ( t ) EL,(u, b). T h e space &(a, b) is called the conjugate space of ,!,,(a, b), and conversely. T h e conjugate space of L,(a, b) is &(a, b). For v ( t ) E L,(a, b) the norm (1 v is defined by (6.1 A). Let u ( t ) = ( u l ( t ) ,..., u7(t)) be an r-dimensional real-valued vector

152

VI. EQUATIONS DEPENDING O N ARBITRARY FUNCTIONS

function. We say that u ( t ) ~ L g ) ( ab), if ui(t)EL,(u, b) for each j , j = 1, ..., r . T h e norm 11 u lip of u ( t ) in Lg)(a,b ) is defined by

(6.1.10)

T h e conjugate space ofLg)(a, b) consists of r vectors v ( t ) = ( q ( t ) ,..., v,(t)), where ui(t)€&(a, b) for each j , j = 1, ..., r, for which the norm is defined by

(6.1.11)

the exponent q being the conjugate of p . Let SZ = ( w l , ..., w n ) be an n-dimensional linear functional on LF)(a, b). Then we have the representation Q(u) =

r

V(t)u(t)dt

(6.1.12)

1I

for each u ( t ) ~ L i r ) ( ab), , where V ( t )is an n x Y matrix function whose elements are in &(a, b). Let R be a compact and convex subset of the r-dimensional Euclidean space E'. We shall denote by C' = L;)(O,6; R )

(6.1.13)

the set of all functions u(t) ~ L g ) ( ab), whose range is R. Where there is no danger of confusion we shall omit the superscript L qv ), I/ lip, and /I llh". in L(7) p We say that a sequence {uk(t)}?, where uk(t)~ L g ) ( ab), , converges to u"(t)strongly if krn I/ uk - zi 11 = 0. (6.1.14) +m I

And we say that (uk((t));"converges to $(t) weak& if, for every n x r matrix function V ( t )with elements in &(a, b), we have (6.1.15)

6.2. Delay-Dtyerential Equations on L, Spaces

153

It is well-known that every strongly convergent sequence converges weakly also. Finally we introduce the notation (6.1.16)

where u(t) E LF)(a,b ) and v ( t ) E L r ) ( a ,b). 6.2. Delay-Differential Equations on L, Spaces

Consider the equation R(t)

=

V(x(.),

44, t )

(6.2.1)

for t e I [ t o , TI, where x ( t ) is, as usual, an n-dimensional real-valued vector function, u ( t ) is an r-dimensional real-valued vector function belonging to the space L g ) ( a , b), and V = V(x(-), u(t), t ) is an n-dimensional real-valued Volterra functional. We assume that: (i) T h e functional V is defined and bounded for t E I [ t , , TI for all u(t) E U and for all x(t) E C ( I [ t o ,TI, G), where U is defined by (6.1.13) and G is a compact subset of the n-dimensional Euclidean space En. (ii) V is Lebesgue integrable in t for each fixed x and u, and continuous in x and u for each fixed t . (iii) There exists a Lebesgue integrable function m ( t ) for t E I [ t o , TI such that (6.2.2) I! V(x(-), 4%1) /I < m(t) uniformly with respect to x ( t ) E C ( I [ t , , TI, G) and u(t) E U. (iv) T h e Osgood integral (2.18.17) associated with the functional V diverges. (v) T h e functional V is differentiable with respect to the components of x ( t ) and u ( t ) in the functional and in the ordinary sense, respectively. Let + ( t ) E C(I[a,to], G) be given, where I a 1 is a sufficiently large number. Then Theorems 2.15 and 2.16 ensure, under the hypotheses (i)-(iv), that for each u ( t ) E U Eq. (6.2.1) has a unique solution in the extended sense, satisfying the initial condition x(t) = d(t)

for

t €{[or,

to].

(6.2.3)

154

VI. EQUATIONS DEPENDING ON ARBITRARY FUNCTIONS

That is, there exists a unique absolutely continuous vector function x(t) which satisfies the initial condition (6.2.3) and Eq. (6.1.1) almost everywhere in I[to , TI. We shall denote this solution by x(t) = x(t, t o , u ) to indicate its dependence upon t , t o ,+, and u. Clearly x(t, t o ,+, u ) is an ordinary function in t and to , but is a functional in + ( t ) and u(t). We shall discuss, in the subsequent sections, the dependence of x ( t , to , 4, u ) upon the functions + ( t ) and u(t). We shall consider first linear equations, then the more general analytic equations.

+,

6.3. Linear Equations: 1. The Functional V I s Linear in x and u

Consider the equation m

2(t) = z 1 4 , ( t ) x ( t

-

"=O

h,)

+

K ( T ,~ ) x ( T dT )

+ B(t)u(t)+ f ( t )

(6.3.1)

to

for t E ] [ t o ,TI, where x ( t ) is an n-dimensional (column) vector function,

AY(t),v = 0, 1, 2, ..., m, are n x n matrix functions continuous for t E l [ t O , TI,

0 = h, < h, < . * * < h , , K ( T ,t ) is an n x n matrix function continuous for T , t € 1 [ t o ,TI, B(t) is a bounded n x r matrix function whose elements are in the space Lq(to, T ) , u(t) E q ? ( t o , T ) , f(t) E L:"'(to , T ) . T h e results obtained in Chapter I11 yield the following representation for the solution x ( t , to , 4, u ) of Eq. (6.3.1):

where M ( T ,t ) and N ( T ,t ) are the kernel matrices of the first kind and second kind, respectively, associated with Eq. (6.3.1). Clearly, the solution x ( t , to , +, u) is continuous with respect to all its arguments.

6.3. I . The Functional V Is Linear in x and

u

155

Note that the matrices M and N are independent of the functions u ( t ) . Consider the matrix function D(7, t )

=

(6.3.3)

N ( 7 , t)B(T),

<

t € I [ t O ,TI, T t . T h e matrix D(T, t ) is an n x r matrix whose elements belong to the space Lp(t,T ) . We shall denote by di(T, t ) the j t h column vector (n-dimensional) of the matrix D(T,t ) . Let e$" be the p-dimensional unit (column) vector defined by T,

(6.3.4) where 8..

=

9'

;1

for i # j , for i = j .

(6.3.5)

A simple computation shows that d9(7,t ) = D(T,t)ej7),

j

=

1, ..., r .

(6.3.6)

Let E be a sufficiently small number. We give an increment crjJ(t), where q 2 ( t )€ & ( t o , T ) ,to t h e j t h component u 3 ( t )of the vector function u ( t ) and write

(6.3.7)

d p = q,(t)ei7)

and 4 x = x(t, to

>

4, 24 + 4 2 4 )

- x(t, t,

,#3 24)-

(6.3.8)

A straightforward substitution yields the first variation au1x of the solution x ( t , t o , +, u ) with respect to the increment ErjJ(t) of the j t h component of the vector u ( t ) : %,x(t)

=

1'

to

d , ( ~h,

( 7 )

dT.

(6.3.9)

Therefore, by the results of Section 2.13, the n-dimensional vector function dJ(T,t ) is the functional partial derivative of the solution x(t, to , 4, u ) with respect to t h e j t h component u J t ) of the vector function u ( t ) at the instant T , T E I [ t o , TI. As we have already mentioned, dj(T, t ) E L r ) ( t o, T ) . Let the Kth column vector of the matrix M(T,I ) be denoted by mk(7, t ) . Clearly mk(7, t ) is an n-dimensional vector function, continuous for

156

VI. EQUATIONS DEPENDING ON ARBITRARY FUNCTIONS

<

T , t E I [ t o , TI, T t. By an argument similar to that used above, we see that the first variation S4kx(t) of the solution x(t, t o , 4,u ) with respect to the kth component &(t) of the initial vector +(t),corresponding to an increment c~9,(t)E C(I[a,to],G), is

84kx(t) = E

10

?o,(T,t)e,(T)

(6.3.10)

dT.

Obviously the vector mk(T,t ) is the functional partial derivative of the solution x(t, t o , +, u ) with respect t o the Kth component +k(t) of the initial vector +(t) at the instant T, T E I[a,t o ] .

6.4. Linear Equations: 11. The Functional V I s Linear in x Only

I n this section we shall consider equations of the form

for t € ] [ t o , TI, where x ( t ) , AY(t)(v = 0, I, ..., m), K(7, t ) are as in Section 6.3 and u(t) E Ll(to, T ) ,$(u, t ) is an n-dimensional column vector continuously differentiable with respect to u, bounded and Lebesgue integrable in t. Since Eq. (6.4.1) is linear in x,we have the representation x ( t , to , 4,u )

=

fa

M ( T , t)4(~) d~

+ f N(T,

t ) # ( u ( ~ )7,) dT

(6.4.2)

10

in which M ( T ,t ) and N ( T ,t ) are, as before, the Kernel matrices. of the first and second kind, respectively, associated with Eq. (6.4.1). T h e first variation of x ( t , to , 4,u ) with respect t o the components of the initial function +(t) can be computed as in Section 6.3. T h e formula (6.3.10) remains valid in the present case. T o compute the first variation of x ( t , t o , 4,u ) with respect to the components of the vector u ( t ) ,let us give an increment dju to the vector u(t), where d p is defined by (6.3.7) with qi(t) € & ( t o , T ) . Consider the increment Ajx expressed by (6.3.8). We have

6.4. II. The Functional V Is Linear in x Only

157

Let us denote by J(u, t) the Jacobian matrix of the vector function +(u, t) with respect to the vector u :

(6.4.4)

all the partial derivatives being evaluated at time t. Clearly J(u, t) is an n x r matrix which is continuous in u and Lebesgue integrable in t . Since +(u, t ) is continuously differentiable with respect to u by hypothesis, we have (6.4.5) R(d,.), $(. 4%t ) - 1 C h l t ) = J(u,

+

+

where

(6.4.6) Therefore, according to (6.3.7), we have

(6.4.7) where

Consequently, the first variation 6,,x(t) of the solution x ( t , to ,4, u ) with respect t o the increment q i ( t ) of the j t h component u j ( t ) of the vector function u ( t ) is given by

sup(t)= E f N(7, t ) / ( U ,

T)ej!"Tj(T)

dT.

(6.4.8)

t0

Let

<

m,t ) = N ( 7 , t)J(%4,

(6.4.9)

t E I [ t o , TI, T t. Clearly P(T,t ) is an n x r matrix whose elements belong to the space L,(to , T). We shall denote by pi(., t ) the j t h column of the matrix P(T,t ) . Since

T,

pj(T,

t)

=

P(T, t)$'),

(6.4.10)

we have

(6.4.11)

158

VI. EQUATIONS DEPENDING O N ARBITRARY FUNCTIONS

Therefore the vector pi(., t ) is the functional partial derivative of the solution x ( t , to , 4, u ) with respect to the j t h component ui(t) of the vector function u ( t ) at the instant T , T E I [ t o , t ] ,t E I[to, TI. It should be noticed that if the function +(u, t ) is linear in u, that is, if

+ f@),

1cr(u(l>,t ) = B(t)u(t)

the matrix P(T,t ) reduces to the matrix D(T, t ) defined by (6.3.3). 6.5. Nonlinear Equations: 1. The Functional V I s Quadratic in x

I n this section we shall deal with scalar equation of the form m

W ) = C A , ( t ) x ( t - A") "=O

+ 11, f0

&(7,

7

+ I" Kl(7, to

t)x(.) dT

7 2 ; t)X(TI)X(T2)

+ $w>, t> (6.5.1)

dT1 dT2 >

t E I [ t o , TI, where x(t), AY(t)(v = 0, 1, ..., m), +(u, K,(T, , T~ ; t ) are scalar-valued functions such that

t),

K,(T, t), and

A,(t),v = 0, 1, ..., m, are continuous in I [ t o ,TI, 0 = h, < h, < . * * < h , , +(u, t ) is continuously differentiable with respect to u and Lebesgue integrable in t, K,(T, t ) and K2(7,, T~ ; t ) are continuous for T , T~ , T ~ t e, I [ t o ,TI, and K2(71 , 7 2 ; t ) = a 7 2 71 ; t ) , u ( t ) E Lj"(to , T ) . 9

We shall denote by G(u, t ) the gradient vector of the function +(u, t ) with respect to the vector u evaluated at time t. T h e j t h component of the r-dimensional row vector G(u, t ) will be denoted by Gj(u,t). Let x ( t , t o ,4,u ) be the solution of (6.5.1) corresponding to the initial condition x ( t ) = 4(t)

for

t ~Z[or,to],

(6.5.2)

where +(t)is a (scalar) function continuous in I[a, to]. We now give an increment A3u = q 9 ( t ) e ! r ) (6.5.3)

6.5. I . The Functional V Is Quadratic in x to the vector u(t), where ?l&) € & ( t o , TI. Let

E

159

is a sufficiently small number and

Since the function $(u, t ) is continuously differentiable with respect to u, we have $(u

j

= 1,

..., r.

+ 4,- $04 5)

t) =

t>%(t),

(6.5.5)

A direct computation yields

where

and lim O(E) = 0. c 4

(6.5.8)

Note that x ( t ) in (6.5.7) is the solution x(t, t o , 9, u ) of (6.5.1). Clearly &t) = 0

for

t €I[&,to],

(6.5.9)

and the equation (6.5.6) has a unique solution f ( t , E) in the extended sense corresponding to the condition (6.5.9). Since f ( t , E ) is absolutely continuous in I[to, TI and K 2 ( ~T1 ~, t ;) is continuous for T ~T , ~ t ,E I[to , TI by hypothesis, we have

where B is a finite positive constant (we assume that T is finite). Now, let us consider the linear equation

160

VI. EQUATIONS DEPENDING O N ARBITRARY FUNCTIONS

Let ~ ( t be ) the solution of this equation which satisfies the initial condition ~ ( t= ) 0 for t ~ l [ c uto]. , (6.5.12) Put (6.5.13) [ ( t ; €) - X(t) = W ( t ; €). From (6.5.9) and (6.5.12) we see that ~ ( te ); = 0

for

t EI[CY, to].

(6.5.14)

A straightforward computation yields

where

By (6.5.10) we have

Let @(T, t ) and N(T,t ) be the kernel functions of the first and second kind, respectively, associated with the linear equation (6.5.15). Then we have W(t;

€)

=

J : o f l ( T , t)H('T)d'T

(6.5.18)

for t e l [ t O ,TI. Therefore

n(~,

Since t ) is continuous, therefore, it is bounded in the interval I [ t o ,TI; thus we have lim ~ ( te ); E-0

=

0

(6.5.19)

uniformly in l [ t o ,TI. Th u s (6.5.20)

6.5. I. The Functional V Is Quadratic in x uniformly for t

E I[to

161

, TI. Combining (6.5.4) with (6.5.13), we find

44t)

= .X(t)

+ y,

(6.5.21)

where y is an n-dimensional vector function whose norm is an infinitesimal quantity of higher order than E . Hence Su,4t)

=

.XW,

(6.5.22)

where suix(t) denotes, as before, the first variation of the solution 4,u ) corresponding to the increment q j ( t ) EL,(^, , T ) of the j t h component of the vector function u(t). The function X(t) is the solution of Eq. (6.5.11) which satisfies the initial condition (6.5.12). Making use of the kernel matrix N(T,t ) of the second kind associated with Eq. (6.5.1 l), we may represent the function X(t) by

x ( t , to ,

for t

EI[to

, TI. Hence, the function (scalar) PA.,

t ) = “7,

t)Gj(47),4

(6.5.24)

is the functional partial derivative of the solution x(t, to ,9, u) with respect to u j ( t ) at time T . Clearly pi(., t ) is the j t h component of the vector (6.5.25) P(T,t ) = N(T,~ ) G ( u ( TT )) ., Now, we give an increment &(t) E C(I[m,to],G ) t o the initial function $(t), E being a sufficiently small number. Put

4.42) < ( t ) = -___

=

x(t, t o 14

+ A 4 - 44 t o

E

E

9

i,4 .

(6.5.26)

Clearly <(f) = 8(t)

for

t E Z[a, to].

(6.5.27)

By an immediate computation we find -0

(6.5.28)

where R(T;t ) is given by (6.5.7). This equation has a unique solution c ( t ; E ) which corresponds to the initial condition (6.5.27). By an argument

162

VI. EQUATIONS DEPENDING O N ARBITRARY FUNCTIONS

similar to that above, we can easily see that the first variation 6,x(t) of the solution x ( t , t o ,4,u ) corresponding to the increment &(t) of the function +(t)is given by 6 + ~ ( 1 )=

E

fo

M(T,t ) e ( ~dT, )

(6.5.29)

where M(T,t ) is the kernel function of the first kind associated with the equation (6.5.30) ”=O

Therefore, the function M(T,t ) is the functional derivative of the solution x(t, to ,$, u ) with respect to the initial function at time T. 6.6. Nonlinear Equations: II. The Functional V I s a Functional Power Series in x

T h e method of the preceding section may be used for the study of equations whose right-hand sides are power series in the functional sense of the form

for t e I [ t O, TI, where x(t), A,(t) (v = 0, 1, ..., m),u(t), #(u, t ) are as in Section 6.5, K,(T, , ..., T, ; t ) ( n = 1, 2, 3, ...) are scalar-valued functions continuous for T ~ ...,, T, , t e 1 [ t o , TI and symmetric with respect to 71

, ..., T n .

Let x(t, t o ,4,u ) be the solution of (6.6.1) corresponding to the initial condition (6.5.2). We now give an increment A p to the vector u ( t ) , where d p is defined by (6.5.3). Let ( ( t ) be defined by (6.5.4). Since the function #(u, t ) is continuously differentiable with respect to u , we have the formula (6.5.5). We can easily verify that

6.6. II. The Functional V Is a Functional Power Series in x

163

where

R(7;t ) = K47; t )

and

A(t)€) uniformly for t

E I [ t o , TI, where

[(t)= 0

(6.6.4)

= O(E)

we assume that T is finite. Clearly (6.6.S)

for t EI[CY, to].

Equation (6.6.2) has a unique solution ( ( t ;E ) corresponding to the initial condition (6.6.5). Let us consider the linear equation

Let ~ ( tbe) the solution of (6.6.6)satisfying the initial condition ~ ( t=) 0 for t E I[a,to].We can easily show that (6.6.7)

lim ( ( t ; E ) = ~ ( t ) , r-10

(6.6.8)

where y is a function such that

. IIYII Iim - = 0. r-0

(6.6.9)

€

Let A?(-r, t ) and N(T,t ) be the kernel functions of the first and second kind, respectively, associated with Eq. (6.6.6). Then (6.6.10)

for t € I [ t o , TI. Therefore E,$x(t) = 6

f N(7, t)Gj(u(7),

7) rlj(7)

to

dT,

(6.6.11)

164

VI. EQUATIONS DEPENDING O N ARBITRARY FUNCTIONS

where SUjx(t) is the first variation of the solution x(t, to ,4, u ) with respect to the j t h component of the vector function u(t)which corresponds to the increment given by (6.5.3). By a similar argument we find (6.6.12)

where the left-hand side denotes the first variation of the solution x(t, t o , 4, u ) with respect to the initial function +(t)which corresponds to the increment A+ = cB(t) E C([la,to],G) with small E. 6.7. Remark

T h e analysis given in Sections 6.5 and 6.6 can be easily extended to the case in which V ( x ( . ) ,u(t),t ) is an n-dimensional functional power series. But, for the sake of simplicity, we shall not go further into this question.

Part II

+

OPTIMAL PROCESSES WITH TIME DELAY

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CHAPTER VII ?&

Formulation of Optimal Problems Involving Time Delay

Having in the previous chapters developed the general theory of ordinary delay-differential equations, we now begin to investigate optimal processes with time delay. I n this chapter we shall formulate our variational problem and in the subsequent chapters we shall discuss this in detail. 7.1. Dynamic Characterization of a Control System Involving Time Delay

Consider a dynamical system ( X )whose state at time t is described by an n-dimensional vector x(t) = (xl(t), ..., xn(t)). We assume that the system ( X )is controlled by certain controllers and let these controllers be characterized at time t by an r-dimensional vector u ( t ) = (ul(t), ..., u,(t)). Let R be a compact and convex set in the r-dimensional space E' of points u = (ul , ..., u,). T h e vector u ( t ) is called the control vector or the control function and the region R is called the control region. t, , be a given compact interval. For measurLet I[t, , t,], where to able functions u(t), defined for t E I[to , q, where t E I[t, , t,], their range in R will be called admissible. T h e set of all admissible controls will be denoted by U. Suppose that the states of (X)corresponding to the time interval 1[a,to],where 1 0 1 1 is sufficiently large, is known and described by an n vector (7.1.1) #(t> E C(4% tol, q

<

where G is a compact region in En, containing the origin as an interior point. 167

168

V I I . FORMULATION OF OPTIMAL PROBLEMS

Let the evolution of the system ( X )be described by a delay-differential equation (7.1.2) * ( t ) == V(X(.)>u(t),4, almost everywhere in ] [ t o , fj, the interval on which the function u ( t ) is defined, where V ( x ( * )u, ( t ) , t ) is a Volterra functional satisfying the conditions (i)-(v), stated in Section 6.2. Theorems 2.15 and 2.16 ensure, under the hypotheses (i)-(iv), that for each admissible control u ( t ) Eq. (7.1.2) has a unique solution in the extended sense, satisfying the initial condition 4 t ) = 4(t), t E I b , to39 (7.1.3) where + ( t ) E C(I[a,to],G); that is, there is a unique absolutely continuous vector function x ( t ) which satisfies the initial condition (7.1.3) with (7.1.1) and Eq. (7.1.2) almost everywhere in ] [ t o , fj. We shall denote by x ( t , t o ,+, u) the solution of Eq. (7.1.2) to indicate its dependence upon t , t o , +, and u. By the physical limitations of the system ( X ) the allowable initial functions +(t)are restricted. We shall denote by @ the set of all admissible initial functions: @ = { d ( t )E C(l[~r, t o ] ,G), 4(t) admissible}. (7.1.4) We assume that the set @ is compact and convex, containing the origin as an interior point. If u(t) and + ( t ) are admissible, we say that x ( t ) = x ( t , t o ,4, u) is the trajectory corresponding to the admissible pair ( + ( t ) ,u ( t ) ) . Thus, any trajectory x ( t ) is an n-dimensional, absolutely continuous vector function satisfying (7.1.3) and Eq. (7.1.2) almost everywhere in ] [ t o , q, the interval on which the control function u ( t ) is defined. An admissible pair { + ( t ) ,u(t)}will also be called an admissible policy. If u ( t ) = 0 during the evolution of the process we say that the system ( X ) is uncontrolled. Finally we shall denote by P the direct product of the sets @ and U : P=@XU.

(7.1.5)

T h e set P is called the policy space of the system ( X ) . 7.2. Constraints

I n the previous section we assumed that the control region R and the region G of values of admissible initial functions were compact and

169

7.2. Constraints

convex, and that they contain, as interior points, the respective origins of the spaces ( E r and En) in which they are defined. We shall relax the convexity condition for the set R in one case only. Control regions are implied by the construction of the systems and they are generally defined by a certain system of inequalities of the form w,(u,

, 242, ..., u,) 3 0,

1, 2,

Y =

..., p .

(7.2.1)

T h e geometrical nature of the set G is also implied by the physical limitations of the system ( X ) . T h e boundaries of the sets R and G will generally be assumed to be piecewise smooth. I n many cases the set U of all admissible controls consists of all functions u ( t ) e L p r ( t o, i), i E Z[tl , t 2 ] , with the property (7.2.2) When p

= 00,

this inequality reduces to

/I u 1 l r n

=

max ess sup 1 ui(t) I l$77$

t0$t$i

< 1,

(7.2.3)

which is equivalent to the inequalities I uj I

< 1,

j

1, ..., Y

=

(7.2.4)

for almost all t in Z[to , fj. T h e inequalities (7.2.4) define the unit cube in Er, whose physical interpretation is obvious. Note that we can always bring bounded controls into the form (7.2.4) by appropriate transformations. During the evolution of a control process it is frequently desired to restrict the trajectories in a certain compact region D of the space En. This region D is determined by the practical purpose of the process, and is given by a system of inequalities of the form tlj(X,,

XC

, ..., x,) 2 0,

j

=

1,2, ..., q,

(7.2.5)

in which no component of the control vector appears. T h e functions are generally assumed to be smooth. We also suppose that

ej

(7.2.6)

G G D.

I n a number of control processes there are also required integral constraints Jr0xj(x, U, t ) dt E

4,

j

=

1,2, ..., S,

(7.2.7)

170

VII. FORMULATION OF OPTIMAL PROBLEMS

where xj(x, u, t ) , j = 1, ..., s, are scalar-valued functions which are supposed to be sufficiently smooth in all of their arguments, and dj are closed subsets of E l . I t should be noticed that the constraints mentioned above are of a nonclassical character. For detailed discussions about constraints we refer to the books: R. Bellman [2], R. Bellman and S. E. Dreyfus [l], and S. M. Roberts [I]. 7.3. Performance Criteria

Consider. an admissible pair ($(t),u ( t ) }E P in which the control function is defined for t E I [ t o , t),f E I [ t , , t,], to t, . Let x ( t ) x(t, to , 4, u) be the trajectory of the system ( X ) , corresponding to this {+(t),u(t)}. T o measure the performance of the policy {$(t),u ( t ) ) we associate with the system ( X ) a functional of the form

<

=;

(7.3.1)

where g and h are scalar-valued functions which are well-defined and sufficiently smooth with respect to all of their arguments. T h e function h depends only on the end values (terminal values) of the controls and trajectories. (In more general problems, the function h depends also upon intermediate values of the trajectories and controls taken at certain moments.) T h e functional J is the performance criterion for the system ( X ) ,which we shall call the cost of the admissible policy {+(t),u(t)}. 7.4. Target Sets

A target set is a family f of closed sets T , C En defined for t E I[tl , t,]. Let (+(t),u ( t ) ) be an admissible pair in which u ( t ) is defined for t € I [ t o ,q, i E I [ t , , t 2 ] .We say that the control u ( t ) transfers the initial function $(t) to the target set F,if the trajectory x ( t ) = x ( t , t o , +, u ) corresponding to the pair {+(t), u ( t ) } satisfies the relation

~ ( Ei TT. )

(7.4.1)

Generally the sets T ivary continuously with t . T h e sets T,may be independent of t or each one may consist of a single point. In the first case we have afixed target; in the second case the target is a single point [ ( t )in En,.

7.5. A n Optimum Problem Involving Time Delay

171

7.5. Formulation of an Optimum Problem Involving Time Delay Let a controlled system ( X ) be given, which is described by a delaydifferential equation *(t)

for t E ] [ t o , fj, f E Z[tl , t2],to and

=

V(x(.),

4 t h t>

(7.5.1)

< t, , where u(t) is defined for t E I[to , tf

x(t) =

m,

t

6 4 % to],

(7.5.2)

with +(t)E @. Let x(t) = x ( t , t o , 4, u ) be a trajectory corresponding to an admissible pair {+(t),u ( t ) ) E P . We suppose that: T h e control region R is compact, convex, and contains the origin of Er as an interior point, and the members of the set U of all admissible controls consist of all measurable functions defined for t E Z[to , tf, f E Z[t, , t,], whose ranges are contained in R. T h e region G is compact, convex, and contains the origin of En as an interior point, and the set @ of all admissible initial functions, which is defined by (7.1.4), is compact, convex, and contains the origin as an interior point. T h e trajectories x ( t ) which correspond to admissible pairs {+(t),u ( t ) } remain in a given compact region D in En, defined by (7.2.5). During the evolution of the process the integral constraints (7.2.7) hold. Let, as before,

be the associated performance criterion with the system ( X ) which measures the cost of an admissible policy {+(t),u (t)} E P . Let the target set 7 be defined as in Section 7.4. Then our optimal problem may be formulated as follows. Optimization Problem. Under the above hypotheses find an admissible policy {+(t),u(t)} so that the control u ( t ) transfers the initial state +(t) to the target set F with minimum cost.

172

V I I . FORMULATION OF OPTIMAL PROBLEMS

Thus, {+O(t>, uo(t)}is an optimalpolicy, if: (i) { + O ( t ) , uo(t)}is admissible; (ii) uo(t)transfers do(t)to F ; i.e., 4 t o , to

7

4O,

(7.5.3)

uO)E Tto >

where to E I [ t l , t,] and uo(t)is defined for t E I [ t o , t o ] ; (iii) the constraints stated above are all satisfied; I(+,u) for every admissible policy {+(t),u(t))which (iv) ](do, uo) transfers the initial function +(t) to the target set 9.

<

Therefore, if { + O ( t ) , uo(t)}is an optimal policy, we have

provided that the foregoing constraints are all satisfied. In the next section we shall deal with the problem of reduction of our optimal problem to an equivalent one for which no integral constraints of the form (7.2.7) are required. 7.6. Reduction of Integral Constraints

T o simplify our investigation, it is very convenient to reduce the integral constraints (7.2.7) by the following classical artifice. Let us add to our system (7.5.1) the equations %+j(Z)

u(4, t ) ,

= Xj(X(4,

i = 1, 2,

*.a,

s,

(7.6.1)

with the initial conditions ~ ~ + ~=(0t ) for

t

j

E I[a,t o ] ,

=

I , 2, ..., s.

(7.6.2)

Defining the vectors and

and combining Eqs. (7.5.1) and (7.6.1), we arrive at the system i ( t ) = V(2(.),u(t), t ) ,

t

E

I[tO, t].

(7.6.5)

7.7. Terminal and Averaging Control Problems

173

T h e initial condition for Z(t) is Z(t)

Let and

= (A(t),

...,&(t), 0, ..., 0)

for

t s I [ a , to).

T,= l', x Jzl x ,A2 x ... x A,,

-

5 = {Tt : t E I [ t l ,t&.

(7.6.6) (7.6.7) (7.6.8)

Note that the sets p l are closed. We can formulate an optimization problem, similar to that stated described by Eq. (7.6.5). I n this in Section 7.5, for the system modified problem the integral constraints will not explicitly appear. Thus, we may assume, at the beginning, that in the original optimal problem there are no integral constraints. It should be noticed that phase space constraints of type (7.2.5) cannot be reduced by the above technique. I n the next sections we shall briefly mention a number of typical examples of optimal problems.

(x)

7.7. Terminal and Averaging Control Problems

Consider the optimization problem stated in Section 7.5. If there is no integral term in the performance criterion I($,u), that is, if we have

my.) W )i),, =

(7.7.1)

then the optimization problem is called a terminal control problem. In a terminal control problem the behavior of the system over the time interval I [ & , fl is of no interest in itself. T h e minimum value of the cost I($,u ) at terminal conditions is desired, regardless of the past history of the process. If, in contrast to the above, there is no h term in the performance criterion I(+,u ) depending on terminal values of the trajectories and controls, that is, if we have (7.7.2)

then the corresponding optimization problem is called an averaging control problem. I n an averaging control problem the behavior of the system (X)over the time interval I [ t o , fl is important. I n this case the optimization is carried out over the entire time interval I [ t o , q; but, in a

174

V I I . FORMULATION OF OPTIMAL PROBLEMS

terminal control problem the optimization is concerned only with satisfying terminal conditions. By the following artifice, we can easily convert an averaging control problem into a terminal one. Suppose that we wish to minimize the functional J(+, u ) defined by (7.7.2). Let us introduce a new variable ~ ~ + ~setting ( t ) , G + 1 ( t ) = g(x(t),4%t )

for

t

Eq t o

, tl

(7.7.3)

and xn+,(t)= 0

for

t E I [ to]. ~,

(7.7.4)

Since (7.7.5)

the minimization problem of the original averaging control problem (7.7.2) has been transformed into the terminal control problem of minimizing x,+~( t). 7.8. The “Bang-Bang” Controls

Consider a control process ( X ) described by (7.5.1) for which the control region R is the r-dimensional unit cube IuJ
k = l , 2 ,...)r.

(7.8.1)

If the components u,(t) of the control vector u ( t ) (the controlparameters) are allowed to assume only the values f l , the control problem is called “bang-bang.” Bang-bang controls occur in a number of important optimization problems.

7.9. The Time-Optimal Problem

I n many optimization problems, it is of interest to minimize the time needed to achieve the goal of the process. I n this case, g(x(t), u (t), t ) EE 1 and h ( x ( t ) ,u(f),t ) = 0 in the performance criterion I(+,u), that is,

j(+,u )

=

t - to = transfer time.

(7.9.1)

An optimization problem, which is simply the minimization of the time f - t, required to achieve the goal of the process subject to the constraints

7.1 1. Lebesgue-Stieltjes Integrals

175

stated in Section 7.5, is called a time-optimal problem. I n the literature, time-optimal problems frequently occur with bang-bang controls. A time-optimal problem for which the target set consists of a single moving point y ( t ) in En is called a pursuit problem. If the moving particle y ( t ) is a trajectory of a controlled system ( Y ) which is described by a delay-differential equation with specific constraints, the optimal problem for the system ( Y ) is to maximize the time for capture of ( Y ) by the system (X). 7.10. Minimum of Maximum Deviation

Let t ( t ) be the desired response of a control process ( X ) and let x ( t , t o , +,u) be the actual trajectory of ( X ) . I n many problems the performance criterion J(+, u ) is defined by

.>

l(4,

=

max

t,
I/ x ( t , to 4, u ) - 5(t) I0 9

(7.10.1)

and the related optimization problem to this performance criterion is min

{4,UI€P

I(#,u ) =

min

max

(4,UjEP t,
/I x ( t , t o , 4, u ) - [ ( t ) / /

(7.10.2)

subject to the constraints stated in Section 7.5. Therefore, the optimal problem, in this case, consists of minimizing the maximum deviation. 7.1 1. Performance Criteria Expressed by Lebesgue-Stieltjes Integrals

Consider a functional (7.11.1) where f is a smooth function in all its arguments, ~ ( t is) a function of bounded variation, and the integral is taken in the Lebesgue-Stieltjes sense. If the function a ( t ) has only one discontinuity at t = t, then the functional J(+, u ) reduces to an expression of the form (7.3.1). If the function a ( t ) has other singularities on I [ t o , q, then the above functional J(+, u ) explicitly depends upon intermediate values of the trajectories and controls at those discontinuity points of a ( t ) . In recent publications performance criteria of the form (7.1 1.1) frequently occur.

176

VII.

FORMULATION OF OPTIMAL PROBLEMS

7.12. Deterministic, Stochastic, and Adaptive Controls

I n the preceding sections we have considered physical systems involving time delay characterized by the following properties: (i) the structure of the system is completely known; (ii) a given admissible pair {+, u } E P completely determines the corresponding trajectory; (iii) the duration of the process is known in ndvance by the given admissible pair; (iv) the criterion of performance is completely known. Processes of this type are called deterministic. I n many cases, it is very difficult or practically impossible to know the exact pair {+, u}, the exact criterion of performance, and the exact structure of the system considered. For example, in a realistic pursuit problem, as mentioned in Section 7.9, if both systems ( X ) and ( Y ) (the pursuer and pursued) are controlled, it is almost impossible to have deterministic knowledge about the behavior of the systems ( X ) and ( Y ) during their evolutions in time. All the knowledge we can get about these systems is of a probabilistic character.

A process for which: (i) an unknown initial function with given probability distribution, (ii) a distribution for admissible controls, (iii) a distribution for the duration of the process depending upon the control vector considered, or (iv) a distribution for the criterion of performance, are allowed is called a stochastic process. An important type of stochastic process is the adaptive or self-learning process. I n an adaptive process, the system, by an updating procedure, observes the behavior of the process and uses the information obtained in order to find a more realistic representation of the process itself. 7.13. Questions Related to the Optimization Problem

T h e optimization problem formulated in Section 7.5. gives rise t o the following questions. (i)

Does there exist an admissible pair @ ( t ) , u(t)} such that the control u ( t ) transfers + ( t ) to the target set F ?

7.14.. Convex Sets

177

Assuming that at least one such pair exists, does there exist an optimal pair {+o(t),uo(t)}? (iii) If there exists an optimal pair, is it unique ? (iv) How can we construct an optimal pair ? (ii)

I n many problems the first question has a negative answer. It may be impossible to transfer the system ( X ) to the target set F . In this case the optimization problem has no meaning. Thus, we shall always assume that there exists an admissible pair {+, u } with the property that u ( t ) transfers + ( t ) to F . T h e second question is the main topic to be investigated in the subsequent chapters. T h e third question has a positive answer in certain cases. T h e last question which is practically very important needs some powerful analytic methods such as the dynamic programming of R. Bellman. In many cases, we shall restrict ourselves to the qualitative aspect of the question. It should be noticed that the above questions are very difficult to answer if the processes under consideration are not deterministic.

7.14. Convex Sets

I n this section we recall, without giving any proof, some results about convex sets t o which we refer in the subsequent part of the book. For detail we suggest to the reader the books: H. G. Eggleston [ l] and S. Karlin [2]. Let x = (xl, ..., xn) be any vector in En. A set of vectors x is 1 imply ax1 ( 1 - a)x2E r. called a convex set if xl, x2 E r and 0 01 Clearly, if rland r2are convex then rl f and ATl are convex, X being a real number. Let a be a nonzero vector in En and let a be a real number. T hen the set

r

< <

H

= {x

I (a, x)

3

r,

a},

+

(7.14.1)

where ( a , x) denotes the scalar product of the vectors a and x, is called a hyperplane. A hyperplane H separates En into two closed half-spaces

both of which are convex. A hyperplane H is called a supporting hyperplane to the convex set I', if is contained in one of the half-spaces of H and the boundary of r

r

178

VII. FORMULATION OF OPTIMAL PROBLEMS

has a point in common with H . Therefore, H is a supporting hyperplane r if inf ( a , x) = a (7.14.3)

to

XEr

the infinite point being admitted as a possible point. A convex set is called strictly convex if every supporting hyperplane to this set has only one point in common with the set. Lemma 1.

closure of

Let I' be a convex set and let y E En,y

r. Then there exists a vector a such that

4 2", where f is the

inf (a, x) > (a,y).

(7.14.4J

X F r

Lemma 2. If y is on the boundary of the convex set r, then there exists a supporting hyperplane through y ; i.e., there exists a %onzero vector a such that (7.14.5) inf (a, x) = ( a , y ) . X E r

Lemma 3. A closed convex set r is the intersection of all its supporting halj-spaces, and every boundary point of r lies on a supporting hyperplane. Theorem 7.f (Separation Theorem). If I' and Q are two convex sets with no interior point in common, then there is a hyperplane H which separates r and 9 ; that is, there exists a nonzero vector a and a real number O! such that ( a , x ) 3 01 f o r all x E r and ( a , y ) a for a l l y E Q.

<

Theorem 7.2. If r and 9 are closed convex sets having no point in common and at least one of them is bounded, then there exists a hyperplane H determined by the parameters {a, m} which strictly separates r and Q; that is, ( a , x ) > a for all x E I' and ( a , y ) < 01 f o r a l l y E Q.

T h e intersection of all convex sets containing a given set A is called convex hull of A and is denoted by co(A). T h e set E ( A ) ,that is, the closure of the convex hull co(A) of the set A , is called closed convex hull o f A. T h e set co(A) is the set of all convex combinations of points of A, that is, c o ( A ) = / k$= L l Y p ~ k l x k E ~k=l k , CX!,.=l,a

k > o ,

k=1,...,r

I,

(7.14.6)

where r is an arbitrary positive integer. I t is known that, for A C En, co(A) can be represented as a convex combination of at most n + 1 points of A.

7.14. Convex Sets

179

If A is a bounded set in En and

Lemma 4.

r=1

where XiEA,

then x

E

ai

> 0,

& = 1, a-

i=l

co(A).

r

r

A point x in the convex set is called an extreme point of if there are no points x1 and x2 in I' such that x = Ax1 (1 - X)x2 for some X (0 < X < I), where x1 # x2.

+

Lemma 5. A closed, bounded, convex set r is spanned by its extreme points. That is, every x in r can be represented in the form

where

XI,

..., xr are extreme points of r.

A set K in En is called a convex cone if, for any x and y in K , we have

A convex cone is clearly convex. T h e smallest convex cone which contains the set r is called the convex cone spanned by and is denoted by K ( r ) . This set is the intersection of all convex cones which contain the set r. Note that

r

K(r)=

12

akxk 1 ak 3 0, x k E , 'I

k = I , ..., r ; Y arbitrary

k=l

T h e set of all directed separating hyperplanes to the set I' which contain the origin is called the polar cone and is denoted by K + ( r ) :

K + ( r )= { a 1 ( a , x)

0

for all x E

r}.

K + ( r )is a closed convex cone. Finally we state the following well-known lemma.

(7.14.9)

180

VII. FORMULATION OF OPTIMAL PROBLEMS

Lemma 6. Let H be a compact and convex set in En and p be a measurable subset of the interval I [ t , , t , ] , mes T (Lebesgue measure of T ) # 0, and let x ( t ) E H , t E rl'. Assume that

1 ___ J mesi'

exists. Then $ I - -

'

mesT

F

x ( t ) dt

x(t)dtE p

H

CHAPTER VIII

& Existence of Optimal Pairs

I n this chapter we shall deal with the existence of optimal pairs for the optimization problem formulated in Chapter VII. T o solve this variational problem, i.e., to establish the existence of an absolute minimum of the cost functional assumed by an admissible pair, we shall extend certain basic results, obtained for systems described by ordinary differential equations, to control systems characterized by delaydifferential equations. Throughout this chapter we shall use the notation introduced in Chapter VII. 8.1. Reduction of the Original Problem into a Terminal Control Problem

Let our controlled system (X)be described by the delay-differential equation

4 1 ) = V(x(*),4 t h 1) for t

E I[to

(8.1.1)

, t), f E I[tl , t,], to < t , , u ( t ) E U and

with $(t) E 0.We suppose that the functional V satisfies the conditions (i)-(v) stated in Section 6.2. Let x ( t ) = x ( t , t o , 4, u ) be the trajectory of the system (X)corresponding to the admissible pair {$, u } E P and let

m,4

= h(x(9,

t)

+ s’ g(.(t), to

181

4 t ) , t ) dt

(8.1.3)

182

V I I I . EXISTENCE OF OPTIMAL PAIRS

be the cost of this pair. We assume that the function h(x, t ) is absolutely continuous in x and t , and the function g ( x , u, t ) is measurable in t for each fixed x and u, and absolutely continuous in x and u for each fixed t . Since the trajectory x ( t ) is absolutely continuous in the interval I [ t o ,t), then there exists a measurable function ho(x(t),t ) such that

Hence, the functional J may be written in the form .,

i

(8.1.5)

where V"(X(4,4% t ) = ho(x(t),4

+ g(.(t),

4% t).

(8.1.6)

Clearly V o ( x ( t ) ,u(t), t ) is measurable in t for each fixed x and u, and continuous in x and u for fixed t. Writing (8.1.7)

*"(t) = Vo(x(t),u(t), t )

with the initial condition x,(t) = 0

for

(8.1.8)

t E [[a, to],

we find

I(+?4 = X O ( 9 .

(8. I .9)

We now define the vectors x

= (XI), x) = (X", x1

,

a , . ,

(8.1 10)

%)

f

and

v = ( v ,, v ) = (vo, v, , ..., vn).

(8.1.11)

Then, Eqs. (8.1.1) and (8.1.7) can be combined into the equation X ( t ) = V ( x ( . ) ,u ( t ) , t ) ,

t E Z[t0 , i]

(8.1.1 2)

and Eqs. (8.1.2) and (8.1 .8) can be combined as x ( t ) = (O,+(t)) = (O,+i(t)p ...,+n(t)),

t ~ I [ ato]. ,

(8.1.13)

Clearly, by the above method, our optimization problem, i.e., the minimization of the functional J(+, u ) , is reduced to the minimization

8.2. Attainable Sets. Notation and Definitions

183

of x o ( f ) ,which can be interpreted as a terminal control problem for the system described by the delay-differential equation (8.1.12). 8.2. Attainable Sets. Notation and Definitions

Suppose that the initial instant to is fixed. Define the vectors f

=

( t , x) = ( t , xo , x) = ( t , xo ,

and f0

=

( t o , xO)= ( t o > 0, +I(tO),

, ..., xn) E En+2,

(8.2.1)

...> bn(t0)).

(8.2.2)

We now introduce the following definitions, differing slightly from those of E. 0. Roxin [ I ] and L. W. Neustadt [ 6 ] . Let D C En be the region mentioned in Section 7.2, in which the trajectories of the system (X)are constrained.

The point

Definition 1. f' = ( t l

, x')

=

(tl , xol, x') = (tl , xol, x,',

..., xn')

E

En+2

(8.2.3)

is called attainable if there is a n admissible pair (+(t),u(t)} E P which

transfers the point io to the point 9, where xo' ==

./(A 4,

+, u), t

and the trajectory x ( t , t o , contained in D.

x' = X ( t , E

3

20

14, a ) ,

(8.2.4)

I [ t o ,fJ, corresponding to this pair is

The set

Definition 2.

.d= { ( t , x) 1 t E Z[tl , t,], ( t ,x) attainable}

(8.2.5)

is called the attainabfe set. Definition 3. Let T be a fixed time such that T E I [ t l , t,]. Then the set dTof all points x E En+' for which ( T , x) E d is called the fixed-time

cross section of&. Definition 4.

Let the initial function +(t)E @ be fixed. Then the set = ( ( t , x) E &

I+

E

@ fixed}

is called the fixed initial function cross section of .d.

(8.2.6)

184

VIII. EXISTENCE OF OPTIMAL PAIRS

Definition 5.

Let the initial function +(t)E @ be fixed. Then the set ==

{ ( t , x) E A?, 14 E

@J

(8.2.7)

fixed}

is called the @xed-time and initial function cross section of d. I n the next sections we shall investigate the topological structures of the attainable set and its cross sections. 8.3. Properties of the Sets A?. Theorem of Roxin

I n this section we shall establish extensions of certain results due to

E. Roxin. We first introduce the notation V(X(.),

Lemma 1.

R, 4

= W(4.),u(t), t )

I4t)E

w.

(8.3.1)

The set V ( x ( - ) R, , t ) is compact for any fixed x and t.

Proof. By hypothesis (i) of Section 7.5 the control region R C Er is compact, and by Section 8.1, V ( x ( - ) u(t), , t ) is continuous in u for fixed x and t. Since the continuity preserves compactness, the set V ( x ( - )R, , t ) is compact for each fixed x and t. We now prove the following extension of a theorem due to E. 0. Roxin [ l ] , which generalizes the results obtained by E. B. Lee and L. Markus [2].

Theorem 8.1. In addition to the conditions (i)-(v) of Section 6.2 and conditions (i)-(iv) of Section 7.5, we assume that the set V ( x ( . ) ,R, t ) is conuex for each t E Z[to , TI, T E Z[tl , t,], and x E D and that there exists a Lebesgue integrable function mu(t)in Z[to, q, t‘ E Z[tl , t,], such that

I

Vo(x(t),

4th 4 I < %(4

(8.3.2)

for t E Z[to, f ] uniformly with respect to all trajectories x(t, to , 4, u) and all u(t) E U. Then, the set a2 is closed. Denote by En+, the space of vectors 4 defined by (8.2.1). a sequence of points of E n i 2 such that t VE d for v = 1, 2, 3, ... and limv+n,6“ = to.We shall show that to~ d . Since 6’’ E d for each v (v = 1, 2, 3, ...), there exists an admissible pair (#”(t),u”(t))E P such that the corresponding trajectory Proof.

Let

tl, t2,t3,... be

x’(t) = X “ ( t , t o , +”, u’),

of the system

(8.3.3)

8.3. Properties of the Sets d . Theorem of Roxin

185

with the initial function +v(t),is contained in D and satisfies

and

xu(T,) = t',

T , = toy, 7'" E Z [ t , , t2]

(8.3.6)

for v = 1, 2, 3, ... . Clearly, the interval I [ t o, t,] contains all the intervals Z[to, TY]for v = 1, 2, 3, ... . We now define the functions for

t E Z[a,to],

for t E Z[to , T J , for t E I[T , , t,],

V(Y( .), uv(T,), 7'")

(8.3.7)

and extend the definition of the trajectory x"(t)to the interval Z[to,T ] by (8.3.8)

which coincides with (8.3.5) in the interval I [ a , T Y ] . First we shall show that the sequence {xY(t))Fis uniformly bounded on I [ t o ,t , ] . Indeed, by the hypothesis (ii) of Section 7.5, the set @ of all admissible initial functions + ( t ) is compact, therefore bounded. Hence

and according to (8.1. I3), we have, in general,

II 4 4 ) !I < II Q, II. Thus

II #"(t) II

< II

Q,

IL

!I

#"(to)

I!

< 11

Q,

II

(8.3.10) (v = 1,

2, 3, --.).

(8.3.1 1)

Further, by the hypotheses (iii) of Section 6.2 and (8.3.2) with (8.3.7), there exists a Lebesgue integrable function @i(t) such that

11 ~

<

( t(1 ) f i ( t )

+

for

t E Z[tn , t 2 ] .

(8.3.12)

[The function mo(t) m(t), with mo(t) of (8.3.2) and m(t) of (iii) from Section 6.2, may be taken as %(t).] Then, on account of Eq. (8.3.8) we can write

I1 x

w

II < II @ I1

+f

tl,

do

(8.3.13)

I86 for t

VIII. EXISTENCE OF O P T I M A L PAIRS E

I [ t , , t,] and v

1 , 2, 3,

=

*

=

... . Putting

I1 a)/I

+ j ?Ti(). t2

do,

(8.3.14)

'I,

we find (8.3.15)

for t EZ[CY, t,] and v = 1, 2, 3, ... . Since w is a finite constant, the inequality (8.3.15) proves the uniform boundedness of the sequence { x ' ( t ) } y on the interval Z[a,t z ] . Since the functions x " ( t ) are uniformly bounded and the functions w v ( t )are measurable inI[t,, , t 2 ] , the inequality

shows, considered as elements of the space L l ( t , , ,tz),that the functions w"(t)form a bounded sequence in Ll(t,,, t,). Since the space L l ( t , , ,t z ) is weakly complete, we can extract a subsequence from the sequence {wv(t)};" which converges weakly to a certain measurable function wo(t). Assume that the indices have been changed in such a way that v = I , 2, 3, ... refers to the weakly convergent subsequence obtained above. Therefore, for each measurable set E, E C I [ t o , t,], we have lim " AT,

j~

( tdt) =

j wn(t)dt. E

(8.3.17)

Let (8.3.18)

Since T , € I [ t l ,t2] for any v = 1, 2, 3, ..., we have To € I [ t l ,t,]. We again change the indices v so that T , + T,, . Consider now the sequence {+"(t)};". Since, by the hypothesis (ii) of Section 7.5, the set 0 is compact, then every infinite sequence of elements in 0 has a subsequence converging to an element of the set 0. Therefore there is a subsequence of {@(t)}?which converges to a function #'(t) E 0. We once more assume that the indices have been changed so that v = I , 2, 3, ... refers to this new convergent subsequence of initial functions; that is, IimQ'(t) Y'ai

= +O(t) E

@,

t E I[n,t o ] .

(8.3.19)

Owing to the norm introduced in Section 2.2, the convergence in (8.3.19) is uniform. Note that, with this new system of indices, the equalities (8.3.17) and (8.3.18) are still true.

8.3. Properties of the Sets d.Theorem of Roxin

187

If the measurable set E in (8.3.17) is contained in an interval I [ t o , TI with T E I[tl , To),To E I [ tl , t 2 ] , then E C I [t o, T,] for almost all v, and according t o (8.3.7), we have lim v-m

J,

V(x’( .), uv(a),u) du

=

J ,wo(u) du.

(8.3.20)

Therefore, the equality (8.3.20) is valid for any interval I [ t o , TI with T E I[tl , To). We now define the function

As an integral of an L-integrable function, the function xo(t)is absolutely continuous in I [ t o ,To]. Combining (8.3.5), (8.3.19), and (8.3.20) we see that lim “’00 x”(t)

= xo(t)

for

t

E Z[a,To).

(8.3.22)

T o show that lim x”(T,) = xo( To), &,--‘a

(8.3.23)

consider the inequality

T h e first integral on the right-hand side of (8.3.24) tends to zero because of the weak convergence (8.3.17). According to the inequality (8.3.12) we can write (8.3.25)

Since &(t) is L-integrable, the right-hand side of (8.3.25) tends to zero for T , + T o . Therefore the right-hand side of the inequality (8.3.24) approaches zero for Y -+ 00; that is, we have the equality (8.3.23). In particular, we have To = 6,”. (8.3.26) Since x‘(t) C D ( V = 1, 2, 3, ...) by hypothesis, the limit (8.3.22) implies that xo(t) lies in the region D or on the boundary of D for

188

VIII. EXISTENCE OF OPTIMAL PAIRS

t E ] [ t o, To]. Clearly, the initial function + O ( t ) constructed above is admissible. We now show that xo(t) is the trajectory of the system (8.1.12), corresponding to a certain admissible pair {+O(t), uo(t)}E P. For this purpose we shall show that there exists an admissible control uo(t)such that 9 ( t ) = V(X".), U O ( l ) , t ) (8.3.27)

almost everywhere in I [ t o , To].This means, according to (8.3.21), that w"t)

=

V(x"(.),u"(t), t ) ,

t E Z(t,

, To).

(8.3.28)

First, let us note that the weak convergence of the sequence {eu'(t)}4 that implies, for every vector 9 of the space lim sup[r) . w v ( t ) ]2 r ) . wo(t) 2 lim inf[r)

. zcv(t)]

(8.3.29)

almost everywhere in ] [ t o , To].I n fact, if we suppose, for example, that on a set E of positive measure lirn sup[r) . ~ " ( t )<] r) . w"(t),

then we obtain

which contradicts the equality (8.3.20). Obviously, for any value of t , in the interval ] [ t o , To), and almost every v, we have w'(t) E V(x"(.), R, t ) . (8.3.30) Hence, for almost every value of t in the interval ] ( t o , To), we have lim sup {lub[r) . V ( x v ( . )R, , t ) ] } 2 r ) . wo(t)2 lim inf{glb[r) . V(xv(.), R, t ) ] } .

O "'D

) a ' "

Since the functional V(x(.), u(t), t ) is continuous in x and u for each fixed t , by hypothesis, we can write lub[r) . V(xo(.),R, t ) ]

2 r ) . wo(t) 3 glb[r) . V(xo(O,R, t ) ] .

(8.3.31)

Being valid for every vector 9, thin equation imp!ies that w o ( t ) belongs to the closed convex set V(xo(-),R, t ) ; therefore, there exists a value uo E R such that w*(t) = V(x*(-),uo, t ) (8.3.32) for each fixed t in ] ( t o , To).In this way we have defined a function uo(t) at almost every point of the interval Z ( t o , To).Obviously we may extend

8.3. Properties of the Sets d.Theorem of Roxin

189

the definition of this function to the remaining points (for example, putting uo = 0 at these points) without affecting the integral of V(xO(.),uO(t),t). Having proved that uo(t) takes its values in the region R , we need only to show that the values of the function uo(t)can be selected so that, as a function of t , uo(t) is measurable. In this connection, we observe that it may happen that, for some values of t , uo(t) is not uniquely defined by the value of V(xo(*),uo, t ) . To construct the function uo(t)we proceed as follows. (i) Take some E > 0. (ii) Since the function wo(t)is L-integrable in the interval ](to , To), we can determine a bounded function w ( t ) which coincides with wo(t) in the interval ] ( t o , To), disregarding a set of measure less than E . Let A be the set of values t where wO(t) = w ( t ) .

(iii)

Divide the set A into measurable subsets Ai , i = 1 ,2 , . . . , p , such that the oscillation of the function wo(t)on each subset A i is less than E ; that is,

lub{lJwO(t,)- w0(t,) 11)

<E

for t , , t , E A ,

(i = 1, ..., p ) .

(8.3.33)

This can be done, since the function wo(t) is measurable and bounded on A. (iv) Take a sequence (ui>ywhich is dense in R. (Recall that R is a compact and convex set in E r . ) I (v) n each subset A i( i = 1, ..., p ) take a value ti E A i. Call ai =

(vi)

W0(ti).

Denote by Aij the subset of A i in which

/I V(xO(0, u3, t ) - a' I1 < 2 E .

(8.3.34)

Since V(xo(-),u , t ) is measurable in t and continuous in x and since xo(t) is also continuous, the sets Ai,are measurable. (vii) We now prove that a,

UA,,=A

( i = l , 2 , 3 ,...).

(8.3.35)

)=1

For, if t

E

Ai , we have

(8.3.36) /I wO(t>- wO(t,)I1 = I1 w0(4 - /I < 6 , by the condition (8.3.33). Further, as shown above, there exists a value w E R such that V(xo(*),w , t ) = wo(t). From the

190

VIII. EXISTENCE OF OPTIMAL PAIRS

continuity of V(x(.), u , t ) in u and from the fact that the sequence {u2}y is dense in R, there exists some ui such that

/I V(x0(.),u', t )

V(X'(.), W , t ) /I

~

=

11 V(X'(.), u', t ) - ~ ' ( tI/ )< E .

U',

t)

(8.3.37)

Hence

/I V(XO(.), (viii)

-

a* ll

< 26

and t E Aij for that value o f j . A certain value of t may, of course, belong to several sets Aij . I n order to define the function u<(t),we use the sets

(8.3.38) where the symbol C indicates the complementary set. Note that t E Aij if t E Aij , but t 4 Ai, with k < j . (ix) We define the function u<(t)by the equation

(8.3.39)

for t E A;j.

u,(t) = u'

By the above results, the function u,(t) is defined on the whole interval Z(t, , To)except a set of measure less than c. I t has the property that

/I WO(t)

~

V(X"0,

u,(t), 1)

/I

e 11

W"t)

II

-

+ 11 V(X"(.),u', t )

-

a'

(1 < E

+ 2~ = 3 ~ (8.3.40) .

(x) If we make this construction for a nulsequence of values of we obtain a sequence of functions f u t ( t ) ) such that lim V(xo(.),u,(t), t ) = wo(t) 6 +O

E,

(8.3.41)

almost everywhere on I ( t o , To). (xi) T o define the corresponding limit function uo(t) we may, for example, put for each component of the vector uo(t), u:(t)

=

ljz uY,,..(t)

(k = I , ..., r ) .

(8.3.42)

Since the values of u c ( t ) belong to R, and since R is compact, uo(t)E R for each t in I ( & , To). Clearly, the formula (8.3.42) defines uo(t) almost everywhere, but the definition may be completed in a convenient way.

8.4. Optimal Pairs. Theorem of Neustadt

191

T h e upper limit of a sequence of measurable functions, uo(t),is measurable. Therefore, uo(t)E U . This completes the construction of the control function uo(t). Now, using the continuity property of V(x(-),u(t), t ) with respect to (8.3.41)and (8.3.42),we find

u and the equalities

V(x"(.),uO(t), t ) = wO(t)

(8.3.43)

almost everywhere. Th u s the proof of the theorem is complete. Theorem 8.2.

compact.

Under the conditions of Theorem 8.1, the set d is

Proof. Since the set d is closed by Theorem 8.1, we only need to show that d is bounded. Let x * = ( t * , x*) be any point of d.Definition 1 of Section 8.2 implies that t E I [ t , , t,] and that there is an admissible pair {#*(t), u*(t)} such that the corresponding trajectory x*(t) = x(t, t o ,+*, u*) is contained in D and x*(t*) = x*. Since the region D is compact and x*(t) E D for t E I[to, t ] , we have x * E D. On the other hand the interval I[tl , t,] is compact by hypothesis. Therefore attainable points are bounded; hence d is bounded.

8.4. Existence of Optimal Pairs. Theorem of Neustadt

In this section we deal with the existence problem of the optimal pairs described in Section 7.5. First we introduce the following definition. Definition 1. The closed sets T , are said to be upper semicontinuous with respect to inclusion, ;f for every E > 0 and t E I [ t l , t,] there is a positive 6(e, t ) such that T,, is contained in an E neighborhood of T , whenever lt'-t\ ( 6 . Note that every upper semicontinuous mapping is closed. We now prove the following existence theorem which is an immediate extension of a theorem due to L. W. Neustadt [6]. Theorem 8.3. I n addition to the hypotheses of Theorem 8.1, suppose that the closed sets T , are upper semicontinuous with respect to inclusion. Then, there is an optimal pair {#O(t),uo(t))such that uo(t) transfers #O(t) to the target set Y with minimal cost. Proof.

Under the hypotheses of Theorem 8.1, the attainable set SP

192

V I I I . EXISTENCE OF O P T I M A L PAIRS

is compact by Theorem 8.2. On the other hand, upper semicontinuity of the closed sets T , implies that the set 1

7 1

=

{ ( t , x) I t

E I[t,

, 4, x E T,}

(8.4.1)

Y', x0 E E')

(8.4.2)

is closed in E71ifland the set T

=

{ ( t , x,, , X) I ( t , X)

E

is closed in E" t2. Since any closed subset of a compact set is compact, the set sd n T is compact and, by the general assumption made in 7.13, is nonempty. 'Therefore, there is a point (to,xg0, xo) E d A T whose coordinate xo is minimal; that is, there exists an optimal pair {+O(t), uo(t)) such that uo(t) transfers + O ( t ) to the target set Y (specifically, to the point xo E F in time to) with minimal cost xoo. Clearly, when the sets T , vary continuously with t , or are independent of t , then the semicontinuity condition is obviously satisfied. It should be noticed that Neustadt's theorem gives a sufficient condition for the existence of an optimal pair. A Remark of Filippov. T h e following remark is due t o A. F. Filippov [ 13. I n actual mechanical systems, it is impossible to realize a control described by a function u ( t ) if the function is merely measurable without being piecewise continuous. I n spite of this, Theorems 8.1-8.3 are not without interest since any measurable function u ( t ) can be approximated by a continuous function v(t) such that

(8.4.3) where 6 is arbitrarily small. If we assume that u ( t ) EQ, where Q is connected and does not depend on t or x, the function v(t) in formula (8.4.3) can be chosen in such a way that v ( t ) ~ QIf, . in (8.1.12), we replace the optimal control uo(t)by a control v(t) which satisfies (8.4.3), then, by virtue of the continuous dependence of a solution on a parameter, the solution x(t, t o , +O, ZI), although it may not attain the state xo at time t = to, will come arbitrarily close to xo if the number 6 in (8.4.3) is made sufficiently small. T h e practical importance of this fact is clear. 8.5. Extended Filippov Lemma

Let x be a compact metric space and p be a nonnegative, finite, regular measure on the Bore1 subsets of x. T h e following theorem due

8.5. Extended Filippov Lemma

193

to H. Hermes [l], which is an immediate extension of a lemma of A. F. Filippov [I], will play an important role in our further investigation. All the definitions will be consistent with those given in the book of N. Dunford and J. T. Schwartz [I]. Theorem 8.4 (Extended Filippov Lemma). Let the vector-valued function f ( x , a1 , ..., a k ) , or, more concisely, f ( x , a ) , be continuous on x x Q ( x ) to En, where for each x E x the set Q ( x ) C Ek is closed, bounded, and upper semicontinuous with respect to set inclusion in x. Let the vector f ( x , a ) describe the set R ( x ) when a describes the set Q ( x ) . Let z(x) be a p-measurable function such that z ( x ) E R ( x ) . Then there exists a p-measurable function a such that a(.) E Q ( x ) , and f ( x , a(.)) = z ( x ) for almost all x E x. Proof. First of all, let us note that, since p is a nonnegative regular measure, for any E > 0, there is a measurable function which is continuous on a closed subset of x , with measure differing from that of x by at most E . Further, by the definition of a Bore1 field, closed sets of x are p-measurable. For a given value of z E R(x),we shall always take that a = (a1, ..., ak) from among all the values of the vector O( E Q ( x )which satisfy the equation f ( x , a ) = z , for which the coordinate a1 has the smallest value. If there are more than one such a, we shall take that one for which the coordinate ( Y ~ has the smallest value, etc. Clearly, the smallest value exists since, on account of the continuity of the function f , the set of values a which satisfy the equation f ( x , a ) = x i s closed. We now show that the functions m1(x), ..., (Yk(x) are p-measurable. Let us suppose that a I ( x ) ,..., 01~-~(x) are p-measurable (if s = 1, nothing need be assumed), and let us prove that ns(x) is p-measurable. By the preliminary remark in the proof, there exists a closed set E C x , of p measure greater than p ( x ) - E , such that the functions z ( x ) , a l ( x ) , ..., as-l(x), are continuous on E. Let us show that, for any number a, the set x t E for which aJx) a is closed. Suppose the contrary. Then there is a sequence {xn E E 1 n = 1, 2, ...}, such that

<

x,

--z

f E

E,

< .,(a)

a,(x,)

- El ,

El

> 0.

(8.5.1)

<

Since I ai(x)I const for all i and x, a subsequence {xnJ can be extracted from the set {xn} such that a+(x,=) + Gi m-tm

,

-

1,2 ,..., k .

(8.5.2)

194

V I I I . EXISTENCE OF OPTIMAL PAIRS

Since ~(x,) E Q(xn), and Q(x) is closed and upper semicontinuous with respect to inclusion, (GI,...) Gk) = 2 € Q ( 2 ) .

(8.5.3)

It follows from (8.5.2) and the continuity on E of the functions oli(x), i = I , ..., s - 1 , that i = l ) ...)s - I ,

G,=a,(5),

Gs

(8.5.4)

< as(%)

- €1.

Passing to the limit on the identity f(x, a l ( x ) , ..., a k ( x ) ) = z(x), and making use of the continuity of the function f,we obtain f'(2, a1(2),..., as&),

By (8.5.4) and (8.5.5), the equation

G,s, ..., Eik) = z(2).

is not the smallest value

(8.5.5) 01,~

which satisfies

f ( 5 , a1(5),... a?-,(&), a,*,..., o l k ) = z ( 3 . )

(8.5.6)

This contradicts the definition of as(.). Thus, our assumption is false, and the set of x E E for which as(x) a is closed. Hence, ~ , ~ ( is x ) measurable on E . Since E Cx, and the p measure of E is greater than p(x) - t, where E is arbitrarily small, a,(x) is measurable on x. Hence, by induction, we conclude that the ai(x), i = 1, 2, ..., k, are measurable.

<

Remark. In the original lemma of Filippov, El, and the measure is Lebesgue measure.

x is a closed

interval in

8.6. The Range of a Vector Measure

Let x be a compact metric space, and y i , i = I , ..., k, be continuous vector-valued functions on x into En. Let p be a nonnegative finite regular measure on the Bore1 subsets of X. For each x E x, we denote by r(x) the set { y l ( x ) , . . . , y kx)), ( and by co r ( x ) the convex hull of r(x). With each p-measurable function z defined on x, associate the vector (8.6.1) where xj ,j = I , ..., n, are the components of z , and denote by R , the range, in En,of v(z) for measurable functions z such that z(x) E r ( x ) ,

8.6. The Range of a Vector Measure

195

while R,,, denotes the range of u(z) for measurable functions z such that ~ ( xE) co r(x). I t is clear that

HI-c

(8.6.2)

Hror.

We now prove the following lemma due to H . Hermes [l]. Theorem 8.5 (Hermes's Lemma). If z is any p-measurable function on x with values in En,such that z(x) E co r ( x ) f o r each x E x, then z admits a representation of the form

(8.6.3)

where the scalar-valued functions ai are p-measurable and 0 < ai(x)

k

< 1,

c a i ( x )= 1

for each

xE

x.

(8.6.4)

i=l

Proof. Since z(x) E co r(x), for each x the representation (8.6.3) is certainly valid, by Section 7.14. I t remains, therefore, to show that this can be accomplished with p-measurable functions ai(x). T o this end, consider the simplex defined by

(8.6.5)

T h u s Q is closed and bounded. Define .f'(x, a ) =

z -?,

a,y*(x)

for

(x, a ) E X

x Q.

(8.6.6)

i=l

Clearlyf(x, a ) is continuous. If z is any measurable function with z(x) belonging to the set f ( x , Q), the extended lemma of Filippov ensures us that z can be represented as z(x) = f(x, a(.)) for almost all x, where 01 is a p-measurable function of x with a(.) E Q. Our main goal is to prove the following theorem. (H. Hermes [I]). R , = R,,,, . T h e proof of this theorem is based on an important extension of a theorem of A. Liapounoff [ l ] given by A. Dvoretzki, A. Wald, and J. Wolfowitz [ I ] (briefly D. W. W.). First we introduce some additional notation. We shall also give details of the proofs of some results to which Theorem 8.6

196

V I I I . EXISTENCE OF OPTIMAL PAIRS

we shall refer in the future. For further investigation we suggest to the reader t h e papers of P. K. Halmos [I] and D. Blackwell [l]. Let {x} = x be an arbitrary space, and let {S}= B denote a Borel field of subsets of the space x. By the definition of a Borel field, B i s a nonempty family of subsets of x which is closed with respect to operations of complementation with respect to x and countably union. S is measurable means that S E B. A real-valued countably additive set function defined for all measurable sets is called a measure. A measure i s j n i t e if it assumes finite values for all measurable sets and it is nonnegative if it assumes nonnegative values for all such sets. A measurable function f ( x ) is a real-valued function defined for all X E X such that the set f ( of all X E X for which.f(x) < c is measurable for every real number c . A step function is a measurable function which assumes only aJinite number of values. If n is a positive integer and qj(x) ( j = 1 , ..., n ) are nonnegative measurable functions satisfying -ql(x) -1- ... 4 ~,,(x) = I

for every

x E X,

(8.6.7)

then q(x) = (ql(x), ..., q T L ( x )is ) called a probability n vector. If all the components q l ( x ) of q ( x ) are step functions, then q ( x ) is called a (probability) step n vector. We shall denote such vectors by q o ( x ) . If, in particular, all components of q(x) assume only the two values zero or unity, that is, if for every x E x one qi(x) is equal to unity and all the others vanish, then q ( x ) is a pure n vector. Such vectors will be denoted by q*(x). If t h e j t h component ( j = 1, ..., n ) of q * ( x ) is considered as the characteristic function of a set S, , then the sets S,, ..., S , are measurable and disjoint and their union is x. Conversely, if S, , ..., S, is a decomposition of the space x into n disjoint measurable sets, and q l * ( x )is the characteristic function of S , , then q * ( x ) = (ql*(x), ..., q n * ( x ) ) is a pure n vector. q * ( x ) is also called a decomposition n vector or, more concisely, a decomposition n vector corresponding to the decomposition

x

=

s, u ... u s, .

Let p,,(S) (k = I , ..., p ) be a finite set of measures, and let q ( x ) be a probability n vector. We denote by V(T> = V ( 7 ;Pl ..')P J 7

the np-dimensional vector (or point in np space),

8.6. The Range of a Vector Measure

197

T h e set of all points V(q)corresponding to all probability n vectors q ( x ) is called the n range of p1, ..., p p and is further denoted by Rn(pl , ..., ,up), or, more concisely, by R, . In the same way we define the step n range of pl,..., pp as the set of all points V(qo(x))= V(qo;pl, ..., pn) corresponding t o all step n vectors qo(x), and denote it by R n o ( p l , ..., p p ) or Rn0.Similarly R,* or Rn*(q*; p1 , ..., p p ) corresponds to all pure n vectors q*(x) and is called the decomposition n range of p l , ..., pp . A measurable set E is called an atom of measure p if p ( E ) # 0 and if for every measurable F C E we have either p ( E ) = 0 or p ( F ) = p(E). If the measure p ( E ) has no atom it is called atomless. We now give a series of theorems due to A. Dvoretzki, A. Wald, and J. Wolfowitz [I]. Theorem 8.7 (D. W. W.). If pl , ..., p7, arejinite measures, then for every n the range R,(pI , ..., pi)) is a compact and convex set in En”. Proof. Let A = V(q) and A’(q‘) be any point of R,. Then every point of the segment joining them is represented vectorially by OLA (1 - a)A’, with 0 < OL < 1. But such a point is clearly (aq (1 - a)q’) and, since aq ( 1 - a)q’ is a probability n vector, the point V(aq (1 - a)q’) also belongs to the range R, . T h u s the set R, is convex. T h e proof of compactness is based on the following lemma.

+ +

+

+

Lemma 1. (D. W. W.) Let {B}= 8 be a Bore1 field of subsets of x generated by countable many sets. Let p‘ ( t = 1, 2, ...) and p be measures over 8 satisfying, for all B E 8 ,

0 < p t ( B ) < p ( B ) < OC)

(t

=

1,2, ...).

(8.6.9)

Then, there exists a measure v over 8 satisfying 0

< v(B) < p ( B )

for all B E 8,

(8.6.10)

and a sequence of integers t, ( q = I , 2, ...) satisfying 0 < t , < t , < ... < t , < f,,,

< ‘..

(8.6.1 1)

such that lim pts(B) = v(B)

9-rn

Proof.

for every B

E

8.

(8.6.12)

I,et B, , B, , ..., B, , ... be acountably basis of 8.Then,according

198

VIII. EXISTENCE OF OPTIMAL PAIRS

to diagonal procedure of Cantor, there exists a sequence (8.6.11) for which (8.6.13) exists for r = 1 , 2, ... . Let B the complement of B with respect to obviously, P W = PYX) - P t ( W

x :B + B

=

x. Then, (8.6.14)

Clearly pCL1g(x) has the limit P o . Thus, if p"(Z3) tends t o a limit as q then so does pts(B). Let B S ( s = I , 2, ...) be disjoint sets of 8 for which

-+00,

lim pb(BS) 'I 4m

exists for s = 1, 2, ... . Since the functions p' are countably additive and (8.6.9) holds for ail B E d, we have ptW)

< P(BS)),

(8.6.15)

and the series of nonnegative terms

2

s=l

P(B?

is convergent. By the standard bounded convergence arguments, we see that (8.6.16) Therefore the limit in (8.6.12) exists and this limit is measurable. Since (8.6.9) and (8.6.12) imply (8.6.10), the proof of Lemma 1 is completed. Let now ~ ' ( x( )t = I , 2, ...) be any sequence of probability n vectors. T h e compactness of R, will be proved if we show that there exist a probability n vector arid a sequence (8.6.1 I ) satisfying (8.6.17)

( j = I , ..., n ; R

=

1,

Bj.v = {x I ~ , ~ ( x<) p )

..., p ) . Consider the sets ( t = I , 2, ...;j = 1 , ..., n ; p rational with 0

-< p < 1)

(8.6.18)

8.6. The Range of a Vector Measure

199

and let { B } = b C (1. be the smallest Bore1 field containing these sets. We write I pk I for the absolute measure associated with pk ; i.e.,

I Pk I(S)

=

SUPU PP(S’) I

+ I PdS”) 1)

(8.6.19)

for all decomposition of S into two disjoint measurable sets S’ and S”. Put (8.6.20) p ( B ) = 1 p1 j(B) + ... + 1 pP 1 (B) for every B E 23. Then, 23, p, and p l , defined by (8.6.21)

satisfy the conditions of Lemma 1. Hence, there exists a nonnegative measure vI over 8 and a sequence (8.6.1 1) for which lim q+m

I*

(8.6.22)

yltq(x) dp(x) = v l ( ~ )

for every B E b. Again applying Lemma 1, we can extract from the sequence (tq} a further subsequence for which (8.6.22) holds with the subscript 1 replaced by 2. Repeating this n - 1 times, and again denoting, for simplicity of writing, the final subsequence by { t J , we see that there exist nonnegative measures v1 , ..., V, over b and a sequence (8.6.1 1) satisfying lim a+m

for every B

JB q>(x) dp(x) = v , ( ~ )

E d. Clearly

vlp)

( j = I,

..., n)

(8.6.23)

we have

+ ... +

vn(B) = P(B)

(8.6.24)

( B E ‘23).

By the Radon-Nikodym theorem there exist &-measurable functions fi(x) ( j = 1, ..., n) such that (8.6.25)

for every B E b. Since the vj are nonnegative measures, we may assume that the f i are nonnegative functions; and, because of (8.6.24), we may further assume that fi(x)

+ ... +.f,(x)

=

1

for every

x.

(8.6.26)

200

VIII. EXISTENCE OF OPTIMAL PAIRS

T h e f , are %-measurable and, therefore, &-measurable; hence,

(fi(x), ...,fa(x j j is a probability n vector. We denote this vector by ~ ( x ) and proceed to show that (8.6.17) holds with this 7 and the above-

considered sequence (8.6.1 1) satisfying (8.6.23). Let g L ( x )( k = 1, ..., p ) denote a %measurable Radon-Nikodym derivative d p k ( x ) / d p ( x ) Then, . replacing f, in (8.6.25) by 7, , we have

j ?,(4 dcrr(4 J’ =

?I(X)RI(X)

=

444

j PA(4 d j

?I@)

X

444

=

j

R k ( 4

X

4(4.

(8.6.27)

Similarly, the left side of (8.6.17) may be rewritten as (8.6.28)

and thus (8.6.17) follows from (8.6.23). This completes the proof of Theorem 8.7. Theorem 8.8 (D. W. W.). If the measures p1 , ..., p p arejinite, and V(7)is an extreme point of R,, , then the set of x f o r which 0 < ~ ~ ( < x )1 for a t least o n e j ( j = I , ..., n ) is a null set for each of the measures p1 , ..., p,, , In particular, all extreme points of R , belong to the decomposition range R,, *. Proof. First, note that a measurable set S is a null set for the measure if p(S’) = 0 for every measurable S‘C S. Let Y denote the set of x defined in the theorem. If Y is not a-null set for pk0 with 1 k, p , then there exist integers j , ,jlwith 1 j o <j , n, a number 6 > 0, a measurable set 2 C Y , such that p

<

<

S

< <

< q,(x) < I

-

6

for X E Z and j = j , , j , ,

(8.6.29) (8.6.30)

Let

5

= c(x) =

(cl(xj, ..., <,(x))

be the vector defined as follows: (8.6.31)

and all other components vanish identically.

8.6. The Range of a Vector Measure -1

Because of (8.6.29), q(x) 8 1. Since

< <

J [73"(4 +

c31,(4

dPk"(4 -

+

20 1

+ 85(x) is

a probability vector whenever

J

5&91

[7?3,,(4

-

= 2&,,(Z) f 0,

dP&)

(8.6.32)

the points V(7 5 ) and V(7 - 5 ) are different. But clearly as B increases from - 1 t o + I the point V(q Ot;) moves from V(7 - 5 ) to V ( y 5 ) along the segment connecting them. Moreover, V(7) is the middle point of this segment and thus it cannot be an extreme point of R, . If V ( 7 ) is an extreme point, then Y is a null set for all q,< . Therefore, if we put 7*(x) = ~(x)for x 4 Y and, say, rll*(x) = 1 for x E Y , the decomposition vector q*(x) thus defined satisfies V(q*) = V(7). This proves the last assertion of Theorem 8.8.

+

+

Theorem 8.9 (D. W. W.). If the measures p1 , ..., p,, arefinite, then the step range R,O coincides with the range R , . More precisely, every point of R, may be represented as V(7O),where 7j0 is a step n vector whose components assume not more than 2,p--p+l dtfferent values. Proof. According to Theorem 8.7, R, is a compact convex set in E n P . However, because of the p equations

2f

J=1

x

yj(x) dP,(x) = p k ( x )

(k = 1,

.*.*

PI,

(8.6.33)

R,n lies in an ( N = np - p)-dimensional linear subspace. Hence, by Lemma 5 of Section 7.14, every point P of R, may be represented

vectorially by

P = alp,

+ ... +

aNf"

+

aN+ipN+i,

(8.6.34)

where P I , ..., P,,, are extreme points of R, and c y l , ..., aN+l are nonnegative constants whose sum is 1. According to Theorem 8.8 we have P, = V(7*,) with 7*, a decomposition n vector ( r = I , ..., N + 1). Hence, putting N+l

(8.6.35)

7" = &7*r, 7=1

we have P = V(7O). Clearly, for every x, every component of ~ O ( X ) equals C.,,a,, where K is a subset of { I , 2, ..., N I}. These being 2N+1 such subsets, Theorem 8.9 is proved.

+

202

VIII. EXISTENCE OF OPTIMAL PAIRS

Theorem 8.10 (D. W. W.). If p1 , ..., pp arefinite atomless measures then, for every n , the range R, and the decomposition range R,* are identical.

According to Theorem 8.7, the common range is convex and compact. Proof. In view of Theorem 8.9 it suffices to prove that R,* = R n O’ For this purpose we shall use the following fact. If p1 , ..., pLpare finite and atomless, then, given 0 ,< a 1, there exists a measurable set S for which CLk(S) = CIIIk(X) (k = 1, ..., p ) . (8.6.36)

<

T h e existence of such a set S follows immediately from a theorem of A. Liapounoff [I] (see also P. R. Halmos [l] and D. Blackwell [l]) according to which, under the above stated conditions, the set of points p,(s), ..., p p ( s ) in E p corresponding to all measurable S is convex. Indeed, the empty set A and x are certainly measurable and (1 - a)pCLk(A) % ( X ) = Q P k ( X ) for all k. To complete the proof of Theorem 8.10, we use the following lemma.

+

a,

Lemma 2 (D. W. W.). If p l , ..., pp are finite and atomless and , ..., a, are nonnegative numbers satisfying a , --. a, = 1, then

there exists a decomposition of having the property that

x into n disjoint

Pk(Sj)= a j p k ( x )

(j

=

1,

+ +

measurable sets S , , .

..., n; k

=

1, ...)p ) .

. S,~

(8.6.37)

Proof. According to (8.6.36), there exists a measurable S, satisfying (8.6.37) for j = 1. Similarly, there exists a measurable S , C x - S , satisfying

(8.6.38)

+ + a,)

where we interpret a,/(a, That is, S, satisfies (8.6.37) f o r j

=

as zero if a2 = --.= a, 2. I n the same manner

sjcx - u si

=

0.

j-1

(8.6.39)

i=l

satisfying (8.6.37) may be obtained f o r j

=

1, ..., n

-

1 . But then

203

8.6. The Range of a Vector Measure thus, n-1

us,

s, = x -

(8.6.40)

1=1

satisfies (8.6.37) for j = n as required. Hence, Lemma 2 holds. Now, let qo(x) be any step n vector. T h e n x can be decomposed into a finite number of disjoint measurable subsets Y , over each of which all the components of qo(x) are constant. According to Lemma 2, Y , may be decomposed into n disjoint measurable sets S l , l ,..., Sn,t such that p k ( S j , [ )=

j

Yl

( j = 1 , ..., n ;k

qju(x) d p k ( x )

=

1, ..., p ) .

(8.6.41)

Putting Si= U, S J . , ( j = I , ..., n) we have, from (8.6.41),

( j = 1, ..., n ; k

jYT,"(x) d p k ( x ) = p A ( S l )

= 1,

...,p ) .

(8.6.42)

T h u s the point V(qo;p1 , ..., p!,) coincides with V ( q * ;pl , ..., p p ) where y i * ( x ) = 1 if x E Siand zero otherwise. I n other words, Rn0C RTL*. Since, obviously, R,* C Rn0, whe have Rn0 = R?!*.This completes the proof of Theorem 8.10. Proof of Theorem 8.6. First of all we have the relation (8.6.2). Now, let z be any measurable function with z(x) E co I'(x). We must construct a measurable functiony, withy(x) E r ( x ) , such that ~ ( x = ) ~(z). By Theorem 8.5, z admits the representation (8.6.3), where aj are measurable and satisfy the conditions (8.6.4). For any measurable subset E C x define

pJE)

=

1 E

(i

y J 2 ( xd) p

=

1,

..., k;j = 1, ..., n),

(8.6.43)

where yii is the j t h component of the vector yi. Consider the k2n-dimensional vector V ( a ) defined by ~ ( a= )

1j

X

al(x)

...) j

x

al(.r)dpln ,

j

X

j

X

al(.r) dpal , ...)

4 4 dpll ..., j

j

al(x)

X

dprn X

I.

dprn

(8.6.44)

By Theorem 8.10, there exists a measurable vector a* with mi*(.)

=

10

and

$ ai(x)

i=l

=

1,

(8.6.45)

204

Thus

V I I I . EXISTENCE OF OPTIMAL PAIRS

j z(x) dp = 1 $ a f ( x ) y z ( x ) dp

=

x z=1

Y

12 x

a t * ( x ) y 2 ( x )dp.

(8.6.46)

1=1

Let I,

= {XEX

I OL,*(x)

=

i

I},

=

1,

..., k.

(8.6.47)

Then each set Ii is measurable and

u 1, 1.

=

x,

1, A 1,

= 14

if i # j ,

(8.6.48)

.

(8.6.49)

z=I

where A denotes the empty set. Define y ( x ) = yz(x)

x EI,

for

Clearly y is measurable, y ( x ) E F(x),and v(y) the proof of Theorem 8.6.

= ~ ( z )which ,

completes

(H. Hermes [I]). Let Q(x) be any set in ETLsuch that r ( x ) = co Q(x) for each x E X . If R,, Rco, denotes the ranges, in En,of v ( z ) for measurable z such that Z(X) E Q(x) and z(x) E co Q(x), respectively, then R, = R, = Rc,,, = R,,,, . Corollary

r(x) C Q(x) and co

Proof. Since r(x)

c Q(x) c co Q(x)

=

co

F(X),

it follows that

Rr C R, C RCo,

According to Theorem 8.6, we have

K, Hence R,

=

R,

=

R,,,,

=

R,,,,.

=

R,,,.

=

Reor.

8.7. Extended “Bang-Bang” Principle of LaSalle 8.7. Extended

“

205

Bang-Bang” Principle of LaSalle

<

I n this section x will be the interval I[t,, 4, i € I [ t l ,t,], N < to < t , t,, t, finite, and p Lebesgue measure. A control, as before, will be a measurable function defined on I [ t , , i] with value u(t) in given set R(t) C R C E‘. Consider now the linear system

2 m

k(t) =

i3,(t)x(t

u=u

--

h,)

+ f K(a, t)x(o) do 4-#(u(t), t )

(8.7.1)

t I)

for t E I [ t o, q, t,b being an n vector function continuous on Z[to,q x E‘. Let x ( t , t, , 4,u ) be the absolutely continuous solution of (8.7.1) satisfying the initial condition x(t)

= +(t)E

@

for

t

E I [ a , to].

(8.7.2)

We assume that:

(P,)

there exist continuous controls ul, u2, ..., uk such that if

then co r(t)= co ,n(t).

(8.7.4)

First introduce the following definition. Definition 1. A control u* with range at t restricted to the values ul(t), ..., u k ( t ) is called a restricted control.

We now prove the following extension of the “bang-bang” principle of J. P. LaSalle [l]. Theorem 8.11 (Extended “Bang-Bang” Principle). I f a state x * can be attained in time t E I [ t , , t,] by an arbitrary control, that is for some controlu the solution x ( t , t, , 4, u ) of(8.7.1) issuch that x(t*, t,, 4,u) = x*, then there is a restricted control u* which at each time t E I [ t o ,t*] assumes one of the values ul(t), ..., u k ( t ) and is such that x(t*, t, , 4, u*) = x*. Proof. T o prove the theorem, we shall follow the general lines of the proof given by H. Hermes [I]. Since Eq. (8.7.1) is linear in x, we have the representation

206

V I I I . EXISTENCE OF O P T I M A L P A I R S

where M and N are the first- and second-kind kernel matrices associated with Eq. (8.7.1). Note that M and N are continuous for t 3 t o . Let u be a control such that x (t *, t o ,+, u ) = x*. I t will suffice to show that there exists a restricted control u* such that

”!

14

N(0, t ) $ ( l L ( U ) ,

U)

14

dU

--

tu

I,,

x(0,t*)I/J(U*(D),

U)

do.

(8.7.6)

Define r*(t)= {!V(a, t)$(u’(t), t ) , ...) N ( u , t)llJ(U”(t), t ) )

(8.7.7)

Q*(4 = { N u , t)lG(P, t ) I p E R(t)l.

(8.7.8)

and Since N ( G ,t ) is a linear operator (n x n matrix), and co r(t)= co Q ( t ) ,

it follows that co T*(t) = co Q*(t)

for each

t E I[t,, t*].

(8.7.9)

‘l’he corollary to Theorem 8.6 yields Rp

=

(8.7.10)

RcSon*.

Th u s , there is a measurable function u* such that u*(t) E r*(t) and

/::

.I.{,,N ( a , tf

ti*(u)

Ry Filippov’s lemma,

da

-=

t*)y5(u(u), u) nu.

(8.7.

u* can be realized in the form

(8.7.

u * ( t ) = N ( t , t*)+(u(t),t )

for almost all t in Z[t,, , t * ] , where the measurable function u* is such that u ( t ) F { u l ( t ) , ..., uk(t)).This completes the proof of the theorem. In the special case in which K(0, 1 )

fl

0,

$(u, 1 )

=

R(t)ZL,

(8.7.1 3)

B ( t ) heing an n x Y matrix, the above theorem reduces to a result due to S?.K. Oguztiireli [2]. When there is no time delay in (8.7.1), Theorem 8. I 1 reduces to a result of I,. W. Neustadt [I], which contains, in the linear case, the 1,aSalle “bang-bang” principle.

8.8. Uniform Approximation Theorem of Warga

207

8.8. Uniform Approximation of Solutions. Theorem of Warga

Consider the delay-differential equation t E I [ t o ,t],

(8.8.1)

t EJ[CL,t o ] .

(8.8.2)

k ( t ) = V ( x ( . ) ,u ( t ) , t ) .

where t E Z[t, , t,] and the initial condition x(t) = +(t) E

for

@J

We assume that:

(P2) each component Viof the n vector functional Y is bounded by M in absolute value, continuous in all arguments, and satisfies a uniform Lipschitz condition of the form

I V , ( x ( . ) u, , t)

-

V , ( % ( . )u, , t) I

< K I1 x - 2 11,

(8.8.3)

where (8.8.4)

By Theorems 2.15 and 2.16, there exists a unique absolutely continuous solution x ( t , t o , 4, u ) of Eq. (8.8.1) satisfying the initial condition (8.8.2). W e further assume that:

(P3) there exist continuous controls u l , ..., uk such that if

\ q t , x) ( W ,).

= =;

{ V ( x ( . ) u'(t), , f ) , ..., W.), u"$ {V(x(.), p, 1) I p E K ( t ) )

t)}

(8.8.5)

then co r(t,x)

co q t , x)

1

(8.8.6)

for all ( t , x ) e I [ t l , t,] x E'. We now prove an extension of a theorem due to J. Warga [I], following the method of H. Hermes [ I ] . Theorem 8.12 (Extended Warga Theorem). Under the above conditions, given any E > 0 and an arbitrary control u, there exists a restricted control u* such that

for all t

E I[tl

, fj.

208

VIII. EXISTENCE OF OPTIMAL PAIRS

Proof. Let x(t, t o ,4, u ) be the trajectory to be approximated and, for the moment, let u be an arbitrary control. T h e n

and

Therefore,

1jz,l

4%t o ] , t E / [ t o, i],

'0,

=

t

d V(u)du,

E

(8.8.10)

where Ax(t)

= x(t)

x*(t)

(8.8.1 1)

and A V ( t ) = V(X(.),2f(t), t ) ==

{ V(x(.),

4 t ) l

-

V(x*(.),u * ( t ) , 2 )

t ) - V(x(.),u*(t),t)>

+ { V(.(.), u*(t>, 4

with x(t) = x(t, to , c$, u ) and x * ( t ) condition (8.8.3) we obtain

--

= x(t,

(8.8.12)

V(x*(.), u*(t), t > ) ,

t , , 6, u*). Using the Lipschitz

Subdivide the interval I [ t , , t) into m equal subintervals each of length 6 = ( t to)/rn. Let I j denote the j t h subinterval. By the corollary to Theorem 8.6, for each j there exists a measurable function y j defined on I j with values ~

?/'(t)E r(t,x ( t , t o

+, u))

such that (8.8.14)

8.9. Relaxed Problem. Theorem of U‘arga

209

By Filippov’s lemma, y j can be realized in the form ~ ’ ( t=) V ( x ( . ,to

for almost all t

E

l j , where

vj

u * ( t ) = v’(t)

9 $9

(8.8.15)

u), ~ ’ ( t t) ),

is restricted control on Zj . Define for t E 1, , j

=

I , ..., m.

For any t E I[t,, i] let v be an integer such that T h e n (8.8.14) and (8.8.15) yield

vS

(8.8.16)

< ( v + 1)S.


M being the bound on the components of the vector V mentioned in (P2). Thus, given any el > 0, we can choose m

2ntM

> ---

,

(8.8.17)

€1

thereby

for t

E

I [ t o , q. Using this bound in (8.8. I3), Gronwall’s inequality yields /I x ( t , tn

for all t

E

9

$,74)

-

x ( t , to

,$, u*) I/ < €

1

~

~

~

’

(8.8.19)

] [ t o ,q. Since el is arbitrary, this completes the proof. 8.9. Relaxed Problem. Theorem of Warga

T h e existence theorem proved in Section 8.3 is based essentially on the convexity and compactness of the set V(x(-),R , t ) . However, there is no reason to expect that the delay-differential equations which describe a physical phenomenon will satisfy the convexity condition. J. Warga [ l ] and R. V. Gamkrelidze [ l ] investigated systems described by ordinary differential equations for which the convexity condition does not hold. Warga considered the “relaxed” problem; Gamkrelidze considered the “sliding regime” associated with the original system. I n the relaxed problem the set of allowable values of x(t) = (a,, 2) is enlarged from V(x(-),R , t ) to the closure of the convex hull of V(x(.), R , t). Warga

210

V I I I . EXISTENCE OF OPTIMAL PAIRS

showed, in the case of ordinary differential equations, that the solutions of the relaxed problem can be uniformly approximated by the solutions of the original problem, Gamkrelidze modifying the original-ordinarydifferential equations to achieve the convexity condition constructed solutions of the original problem which approximate the solutions of the “modified problem” arbitrarily and closely. A sliding regime is the limit of these approximating solutions in which the control switches “infinitely often.” We give here the extensions of the results of Warga to hereditary control systems. Let V ( x ( - ) R, , t ) be the set defined by (8.3.1) and W ( x ( * )t ,) be its closed convex hull. T h e optimization problem described in Section 7.5 and reformulated in Section 8.1 may be expressed, with the same notation in Section 8.1, in the following form. Minimize x,(i) subject to the conditions:

Original Problem.

(i),, (ii) (iii) (iv)

? ( t ) E V ( x ( . ) ,R , t ) almost everywhere in T x(t) = + ( t ) E @, t E I [ u , t o ] ; x(t) E D, t E T ; x ( i ) E T.

=

I[t,, , i ] ;

Enlarging the permissible set of choices of i ( t ) from V ( x ( - )R, , t ) to W(x(.), t ) we obtain the following relaxed problem. Relaxed Problem.

(i)r (ii) (iii) (iv)

Minimize xo(i) subject to the conditions:

a ( t ) E W ( x ( . ) t, ) for almost everywhere in T; x(t) = + ( t ) E @, t E Z[a,t,]; x(t) E D , t E T ; x(i) E F.

Any absolutely continuous vector function x ( t ) satisfying conditions (i),, [resp. (i)r], (ii), (iii), and (iv) will be said to be an original [resp. relaxed] admissible trajectory. An original [resp. relaxed] admissible trajectory which minimizes x,(i) will be referred to as an original [resp. relaxed] minimizing trajectory. Let H( W , 6), W C E 7 & + 6 l , > 0 be the union of all (n I )-dimensional balls in E n + ] with centers in W and radius 6.

+

Definition 1. x, t ) function ~(6,

W(x(.),t ) is quasi-continuous at (x, t ) if there exists a > 0 defined for positive 6 and such that 1

t

~

T

I

+ I1 x

--

y

I1 < ?@, x, Q,

(8.9.1)

8.9. Relaxed Problem. Theorem of Warga implies

c H(W(x(.),t ) , S > .

W Y ( . ) , ).

21 1 (8.9.2)

We now prove the following theorems of J. Warga [l].

Theorem 8.13. (J. Warga) Assume that there exists a measurable subset T‘ of T of Lebesgue measure i - to such that W ( x ( . )t,) is quasicontinuous at (x, t ) if (x, t ) E D x T’, and assume that there exists a constant M such that 1) WJl M ;f W E W ( x ( . ) t, ) , ( x , t ) E D x T. Then, f o r every compact set D* C D , the collection of all relaxed admissible trajectories x ( t ) such that x ( t ) E D*, t E T , is sequentially compact in the topoIogy of the uniform norm; i.e., given any infinite sequence of curves in the collection, there exists a subsequence which converges uniformly to a curve in the collection.

<

Proof. Let x i ( t ) , t E T ,j = 1, 2, ..., be an infinite sequence of relaxed admissible curves contained in D*. Since x’(t) E W ( x i ( - ) 2), almost everywhere in T , it follows that

11 ~ ’ ( 11t )< M

T.

almost everywhere in

(8.9.3)

Since x J ( t ) are, by definition, absolutely continuous and since D* is bounded, it follows that x J ( t )are uniformly bounded and equicontinuous over T. Thus, by Arzela’s theorem, there exists a subsequence [which we shall continue to designate by x’(t)] which converges uniformly to a trajectory x ( t ) in D* which is Lipschitz-continuous with constant M . Hence 2 ( t ) exists almost everywhere in T. We now show that i ( t ) E W ( x ( . ) t, ) almost everywhere in T . Let ~ ~ ( 1=)- x ( t ) - ~ ’ ( t ) , t E T , j = 1, 2, ... . (8.9.4) For every t have

E

T‘, t

< i, for

1

every h > 0, t

+ h E T , and for every j we

1

1

- { ~ (+ t h ) - ~ ’ (+ t h)} - { ~ ( t-) ~ ’ ( t ) } h { ~ (+t h ) - ~ ( t ) } h -

1

=

- {cl(t

h

+ h)

By hypothesis, given any t E T’, t > 0 such that

~ ( 8t,) = ~ ( 8x,( t ) , t )

W(.Y(.),).

-

c,(t)]

+

1

l+h

t

a’(.) dT.

(8.9.5)

< f, and any 6 > 0, there exists

H ( W ( x ( . ) ,4,8)

(8.9.6)

212

VIII. EXISTENCE OF OPTIMAL PAIRS

provided

1

I

t -

+- II

x(t)

-

y(t) I/ f 7(S,t ) .

Since c,(T) converges uniformly to 0 for all T E T , there exists an integer jo(h) = jo(h, t, 6) which is the smallest positive integer such that

li 4 .)I1 < min{4‘%

3 j,,(h).

for all T E T and for all j h + 0. Let now

o
T

min \

1

4@,

(8.9.7)

t)S

Clearly j o ( h ) is nondecreasing as

+

1)

?(is, t ) , t-

tI l .

(8.9.8)

4 4 I/ .

(8.9.9)

-

E I [ ~t + , h],

11 x’(7)

- 44

I/ < I/ 4.) -d

t ) /I -t Ii

As previously observed, we have

11 X ’ ( T ) - X’(t) 11 < M 1

-

f

1;

(8.9.10)

hence, by (8.9.7), (8.9.8), and (8.9.9),

I7

-

1 $- 11 X’(7) - x(t)

11 < 1 T

-

t

1 4 n/l 17 - t f +??(is, f)

s-( M + 1 )h + &Iis, ( t)

<.

t)

for all j >j,,(h).

Thus, by (8.9.6) and 8.9.1 I ) , W(X’(.),7) c fI( W(X(.),t ) , $81,

zit, 1

+ h].

(8.9.1 I )

(8.9.12)

Now A?(.) E W(x’(.),T ) almost everywhere in T , and it can be easily verified that H ( C , c) is compact and convex if C has these properties. Further (8.9.13)

Hence, by (8.9.4) and (8.9.13)

8.9. Relaxed Problem. Theorem of Warga

213

for any t E T' and 6 > 0 provided h satisfies relation (8.9.8). Since x ( t ) is absolutely continuous, its derivative i ( t ) exists almost everywhere in T and I ~ ( t= ) lim - { x ( t h j - x ( t ) } E H( Mf(x(.), t ) , 6 ) h+O h

+

for every 6 > 0 and for almost all t E T'. I t follows that there exists a set T* of measure f - to and contained in T such that k ( t )E W ( x ( . ) ,t ) ,

t

E

7'*.

This completes the proof of the theorem. Theorem 8.14 (J. Warga). Assume that there exists a constant M and a subset T' of T of Lebesgue measure f - tosuch that

and that V ( x ( - )p, , t ) is continuous in (x, t ) uniformly in p f o r (x, t ) E D x T'. Then W ( x (.), t ) satisfies the assumptions of Theorem 8.13. Proof. By assumption, if V E V ( x ( * )R, , t ) and ( x , t ) E D x T , then V 11 M . Thus, if P is in the closure of V ( x ( . ) ,R, t ) then 11 P // M. I t follows that every WEW ( x ( . ) ,t ) is such that 11 WZ.1 < M . Let 6 be any positive number and let (x, t ) E D x T ' . Then there exists q = ~ ( 6x,, t ) > 0 such that, for every p E R,

<

/I

<

(8.9.17)

Let now W be any point o f W(y(.),T), where y and T satisfy (8.9.17). Then, by a theorem of Caratheodory there exists points pi E R , j = I , ..., n I , and nonnegative numbers ai such that

+

fl+l

z a j = 1 j=1

and (8.9.18)

214

V I I I . EXISTENCE OF OPTIMAL PAIRS

From (8.9.16) and (8.9.18), it follows that

11'1

(8.9.19)

(6.

Clearly (8.9.20)

It follows that W E H ( W ( x ( . ) ,t), 6 ) . Since W is an arbitrary point of W ( y ( . ) ,t ) , we conclude that W ( x ( * )t, ) is quasi-continuous at (x,t ) for all (x, t ) t D x T ' . As a corollary of 'Theorems 8.13 and 8.14 we deduce the following. Theorem 8.15 ( J . Warga). If there exists a constant M , a compact set D* C I), a subset T' of T of Lebesgue measure i t,, , and a relaxed admissible trujectory y ( t ) such that ~

( 1 V(x(.),p,

t ) 11

M,

(x, p, t ) € 1)

x K x T,

(8.9.21)

and V(x(.), p, t) i s continuous in (x, t ) uniformly in p for all ( x , t ) E D x T', and also x,(t)

<<

-yo@)

implies x ( t ) E I)*,

t E 'I'

(8.9.22)

f o r every relaxed admissible trajectory x ( t ) , then there exists a relaxed minimizing trajectory .

8.10. Linear Systems. Theorem of Neustadt

I n this section we shall deal with a control process described by a system of linear delay-differential equations of the form

where t 2 t,, , K and A , (v = 0, I , ..., m ) are n x n matrices, x , $I, K O , and 01, (V = 0, I , ..., m ) are n vectors, xO and $Io are scalars, h,(v 0, 1 , ..., m ) are as usual. Denoting by x the ( n + 1) vector

8.10. Linear Systems. Theorem of Neustadt

21 5

+ 1) vector ( + o , +), we can combine Eqs. (8.10.1)

+

(xo , x ) , by the ( n into the one equation

+

+

where K and AV(v= 0, 1, ..., m ) are ( n 1) x ( n 1 ) matrices whose elements can be written immediately in terms of K and K O ,and A , and a,, respectively, v = 0, 1, ..., m . We assume that:

(HI) K,A, (v = 0, 1, ..., m ) , 4 are continuous functions; (H,) the range R of all admissible controls is compact, not necessarily convex.

Further, for the sake of simplicity, we also assume that:

(H3) D

=

E".

Since +(u, t ) is continuous and the range R of all admissible controls u ( t ) is compact, the sets

+(H,t ) = {+(u, 2 ) 1 u

= u ( t ) admissible}

(8.10.3)

are also compact. W e now prove the following theorem which is an extension of a theorem due t o L . W. Neustadt [6]. Theorem 8.16. Under the above hypotheses, the attainable set d associated with the system (8.10.2) is compact, and the fixed-time cross sections .d, of sd are convex and compact. Proof. Let M ( o, t ) and N ( o , t ) be the kernel matrices of the first and second kind, respectively, associated with Eq. (8.10.2) which satisfies the initial condition

where + ( t ) = (0, + ( t ) ) is an admissible initial function. Since Eq. (8.10.2) is linear in x, we have the representation x(t, to ,+, u )

=

jt"M(a, t)+(u) da $- f N(o, f)+(u(a),a) do. 11,

(8.10.5)

216

V I I I . EXISTENCE OF OPTIMAL PAIRS

According to Definition 3 of Section 8.2, we have cd7 =

4

)J

1

M ( u , T ) + ( u ) du

+ J” N ( u , T)+(u(u), u) do I {+, u } E P 1I , (8.10.6) to < t , , and P is the policy space of the 41

where T E I [ t , , t23, 01 controlled system described by (8.10.2). T o show that dTis compact and convex, it is sufficient t o prove that the sets ;<

A,

= ’Jfo

1 ,

(8.10.7)

M(u,T)+(u)do j +(t) admissible / \

and

’

U,= ,:, N(u, T)+(u(u), u) du 1 u ( t ) admissible,

(8.10.8)

are compact and convex. First consider the set A , . Since the set @ of all admissible initial functions + ( t ) is compact by assumption (ii) of Section 7.5, and the kernel matrix M ( u , t ) is continuous for t 3 t o , a u t o , A , is compact. From the linearity of A , with respect t o it follows that A , is convex. Therefore we only need t o prove that the set Q, is compact and convex. To this end, consider the set

+,

R, =

‘1‘ N(u,

‘

T)q(U)

du I q ( t ) measurable

q(2) t

+(H,t )

< <

for

t

E

/ [ t o ,T ] 1,.

‘1,

(8.10.9) Clearly 52, C R , . W e shall show that R , is compact and convex, and that R , = SZ, . By Theorem 8.7, the set R , is convex. As we have already mentioned, the set +(R,t ) is compact and the matrix N is continuous for t > to ; hence the set R, is bounded. W e now show that R, is closed. Let vl, ..., q n , be n 1 linearly independent vectors in En+l, and A1, ..., A , l + l be numbers defined by

+

A,

=

min 7 , . r, rE R

h , = rninT;r, rcs> 1

S,

=-

{r I r t R

and 7, . r

= Al}

(8.10.10) S , = { r l r ~ S , , and

q;r=X,}.

Consider the point f E E r r + l satisfying the relations y i F = h i , == I , ..., n I . It is clear that r E R , , the closure of R, . Let { r 2 )be a sequence of points in H, such that I-, r as j GO, where

i

+

--f

--f

(8.10.1 I )

8.10. Linear Systems. Theorem of Neustadt

217

Then, there exists a subsequence {qj,) of Isi}, and summable scalarvalued functions xi(u, t ) for i = 1, ..., n I , such that

+

(8.10. I 2)

almost everywhere in I[t,, , T], i

=

I , ..., n

+ 1; and

Since the vectors 7i are linearly independent, the vectors v i . N(a, T) for i = 1, ..., n 1 are linearly independent for every u. Thus, (8.10.12) implies that qj,(t) -+ q(t) almost everywhere as k co for a certain measurable vector function q(t). Since qj,(t)E +(R, t ) for every t , and the sets +(R, t ) are closed, we may assume that q ( t ) E +(R,t ) for every t. Finally

+

---f

so that f

=

I,,

N(o, T)q(u) do E K, .

(8.10.15)

By Filippov's lemma, mentioned at the end of Section 8.4, every measurable function q ( t ) satisfying the condition q ( t ) E +(R,t ) for t s I [ t o ,T ] can be realized in the form q(t) = +(u(t),t ) , where u ( t ) is admissible for t E I [ t o ,TI. T h u s R , = Q, . It remains to show that the attainable set a?'is compact. Because of our assumptions of and R, it follows that the sets dTfor T E I [ t , , t,] are uniformly bounded, since the interval Z[tl, t,] is bounded; so that d is bounded. We now show that d is closed. T o this end, consider a sequence of points (ti , xi)E d converging to the point ( t * , x*). Since the interval I [ t , , t2] is compact, t* d [ t l , t,] so that it is sufficient to show that x* E . We now construct a sequence of points y i E dt* such that I( y i - x i 11 + 0 as i + co, and our conclusion will follow from the fact that d,* is closed. Since x iE d l zthere , is an admissible pair {di(t),ut((t))E P such that

+

d 1 4

xZ = ,:M(o, t l ) @ ( o )do

+

st' 1,

N(u, ~,)+(zP(u),0) do,

(8.10.16)

218

i

=

V I I I . EXISTENCE OF OPTIMAL PAIRS

I , 2, ... . I f we define y' by

where Z"(t)

\u'(t) IV

for for

t, < t t, i t

< t, < t*

(8.1 0.1 8)

[unless ti > t*, in which casc the first relation fully defines v2((t)] where is an arbitrary point of R, it is easily seen that the sequence {yi}has the desired properties. T h i s completes the proof. I t is clear that Theorem 8.3 is still true under the hypotheses of Theorem 8.16. v

CHAPTER IX

4P optimization of Control Systems Involving Time Delay Using Dynamic Programming

I n the previous chapter we studied the existence of an optimal pair for the original and relaxed optimal problems. We found certain sufficient conditions- namely Theorems 8.3 and 8.15-for the existence of an optimal pair. In this chapter we wish to investigate the properties of optimal pairs and establish certain necessary conditions for the optimality using dynamic programming techniques. For the sake of simplicity we shall only consider linear systems. T h e reader is assumed t o be familiar with the theory of dynamic programming. For the general theory and its applications to control processes we refer the reader to the following books: R. Bellman [ l , 21, R. Bellman and S. Dreyfus [I], S. M. Roberts [ 13, C. M. Merriams, I11 [l], J. 7’. ‘i’ou [ l ] , and R. Aris [I]. Further references are listed in the Bibliography.

9.1. Reformulation of the Optimization Problem

Let a control system be given which is described by a linear delaydifferential system of the form

where t

, T I, u ( t ) E IJ defined for t E Z [ t , , TI, and (9. I .2)

220

I X . OPTIMIZATION OF CONTROL SYSTEMS

4 being n-dimensional column vectors, u an r-dimensional column vector, K and A , (v = 0, I , ..., m) n x n matrices, OL a finite constant, and x,+, and

0

=

h,

< h, < ". < h,,,.

(9.1.3)

We assume that:

(H,) K and A , (v = 0, I , ..., m) are continuous; (H2) $ is twice continuously differentiable with respect t o u, bounded and 1,ebesgue integrable with respect to t ; (H3) there is no position constraint for the process; that is D = En; (H4) the performance index associated with the system (9. I . I ) is of the form (9.1.4)

where h and g are scalar-valued functions, twice continuously differentiable with respect t o all of their arguments.

By Section 7.5, for an optimal pair problem, we have

{+O,

uo) E

P for t h e above-described (9.1.5)

where the notation {+, u ) E P, I [ t , , TI denotes that the minimization with respect t o {+, u } E P is performed on the interval I [ t , , TI. Accordingly we define f

(77>

min

= { 4 , T L ) 5 P , / [ l , ) , T ] (' h ( x ( T ,t,, ,

4,u ) , T ) -t

1

T '0

R(.Z(~, to ,

1

+,u ) , 4 t ) , t ) dt , (9.1.6)

where x ( t , t,, ,+, u ) is a trajectory of (9.1.1) which corresponds to an admissible pair {+, u } . I,et M ( a , t ) and N ( a , t ) be the kernel matrices of the first and second kind associated with the system (9.1. I ) . Then we have the representation

4 4 4, , A 4

=

9.3. Application of Functional Equation Technique

22 1

(9.1.I 0)

T h e n Eq. (9.1.6) may be put in the form (9.1.11)

We now define the function

.f(T7) =

]w7 +

{m,,)$$i,T1

7)

J7

[

G ( t ,T) dt ,

(9. I . 12)

where T € Z [ t o , TI. I n the next sections we shall examine the properties of the function f(T,T ) by using dynamic programming technique. I t should be noticed that the above problem is an extended version of one first considered by R. Bellman and R. Kalaba [ I ] (see also R. Bellman [2]). 9.2. Principle of Optimality

T h e basic property of optimal policies is expressed by the following principle due to R. Bellman [I]. Principle of Optimality. An optimal policy has the property that, whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision. This principle will play an essential role in the subsequent analysis. 9.3. Application of Functional Equation Technique Consider now Eq. (9.1.12) and let us break the interval Z[T, TI into two parts I [ T ,T d] and 1 [ ~ d, TI, where d is a small discrete time interval. Then we may write

+

+

222

I X . OPTIMIZATION OF CONTROL SYSTEMS

Observing that the kernel matrices M(o, t ) and N(a, t ) are continuous in u and t, and continuously differentiable with respect to t for u E I[a,to] and t >, to , and that the functionsg and h are continuously differentiable, we have, for small A ,

J

G(t,T ) dt

=

~ G ( TT ), + O(d’),

JT

G(t,T ) dt

=

J‘

TIA

G(t,T

T iA

T+A

(9.3.2)

+ A ) dt - A

G,’(t, T ) dt

JT TtA

+ O(A2),

(9.3.3)

where G,’(t, T ) denotes the partial derivative of G(t,T ) with respect to T , and H(?’, T ) = H(7’, 7 + d ) - dHT’(z’,T d) o(d2), (9.3.4)

+ +

where H,‘ denotes the partial derivative of H with respect to T . Therefore

(9.3.5)

T h e principle of optimality now yields f ( T ’ T, -d

=

{Q,u)€??$,T

JT r+A

kA]

G,’(t, T) dt

\ /

min

{$,u)€P,1[7+d,T]

(dG(7, T )

-

dH,’(T,T

+ O(d‘) + H(7’, + d) + 1‘ T

G(t,T

T+A

+ d) + d) d t ) l . (9.3.6)

Therefore, according to the definition (9.1. I2), we may write

IT

+ O(A2)[, (9.3.7) where, again by the principle of optimality, the termf(T, + A ) on the right-hand side is independent of the choice of the pair {#+ u ) for the interval I[T,+ A ] . Consequently we have d

7 i A

G,‘(t,T ) dt

7

T

9.3. Application of Functional Equation Technique

223

Th i s equation is the discrete form of the dynamic programming condition for an optimal pair. Now, taking the termf( T , T + 0) on the left, dividing both sides by 0 and passing to the limit as d -+ 0 in (9.3.8), we obtain the functional equation f,'(T,

T)

=

min

{4,u:€P

1G(T,

T) -

where T E I [ t o ,TI. By formula (9.1.6), we have

Since the function h(x, t ) is continuous and since the set Q, of all admissible initial functions + ( t ) is compact, the minimum of the righthand side of (9.3.10) exists. T h e differentiability of the function h(x, t ) permits us to compute this minimum by the method of calculus. We shall denote this minimum by p:

Let 4° be a point at which the function h(+(to),to) attains its minimum value p ; that is h(+", )2, = P(9.3.12) We now define the sets

I

@" = { + ( t )E @ + ( t o ) =

and PO = {{+, u } E P

14 E @ O ,

4")

(9.3.13)

uE

V).

(9.3.14)

Since the trajectories of (9.1.1) are continuous continuations of their initial functions in the forward direction in time, we may replace the space P by the space Po in Eq. (9.3.9). Hence, f,'( T , T )

=

min

(4,U:EP"

'

G(T,T ) - H:( 7',

T) -

J"' 5

G,'(t, 7 ) dt

'I '

(9.3.15)

Therefore, the solution of our optimization problem is equivalent to the solution of the functional equation (9.3.15)subject to the initial condition .t(to

where p is defined by (9.3.1 I ) .

t

to)

= PI

(9.3.16)

224

IX. OPTIMIZATION OF CONTROL SYSTEMS

Clearly, Eq. (9.3.15) implies two steps. T h e first is to minimize the difference of terms appearing in the braces of (9.3.15). T h e second is to equate the minimum value of this difference of terms to the left-hand side of (9.3.15). In the next sections we shall establish an explicit form for the righthand side of (9.3.15). 9.4. First Variations of the Functionals G(r, z), G,‘(t,

X(T,

t o , 4,

J

.If,

U) =

R/l(a,.)$(a)

da

4

J” N ( o ,

T)

T ) ~ ( U ( U ) u) ,

and H,’(T, 7 )

da.

(9.4.2)

‘11

Further, by (9.3.8), (9.4.3)

where (aglax) is the gradient vector of the function g(x, u, t ) with respect to x, with x = x(t, T , , u). Furthermore, by (9. I .9), we have (9.4.4)

where (dhldx)denotes the gradient vector of the function h with respect to the vector x = (xl , ..., xn) at time t = T ; that is (9.4.5)

all the partial derivatives being evaluated at time t = T with = x ( t , t o , 4,.). Clearly G(T,T ) , G,’(t, T ) , and H7’( T , T ) are functionals of {+, u } and ordinary functions of T . Thi s section is devoted to the computation of the first variations of the functionals G(T,T ) and H,’(T, T ) with respect to and u. Let E be a sufficiently small number and let us give an increment

x

+

A,u

= cT,(t)ej‘)

( j = I , ..., r )

(9.4.6)

9.4. Functionals G(T,T), G,‘(t, T) and H7’( T , T)

225

t o the control vector u , where el‘’ is the r-dimensional unit (column) vector defined by (6.3.4) and ~ ~ ((j t=) 1, ..., r ) is a measurablefunction on I [ t o , TI such that u d,u E U for all sufficiently small E . Put

+

du,x(t) = x ( t , t o

9

4,

As shown in Section 6.4.

.+

d , ~) ~ ( tt o,

4, 24).

(9.4.7)

(9.4.8) where

.

O(c)

c-0

E

Iim -- = 0,

(9.4.9)

and (d$/du) is the Jacobian matrix of the vector function +(u, t ) with respect to the vector u = ( u l , ..., ur), all of the partial derivatives being evaluated at time t = T: (9.4.10) Consequently, the first variation 6,,x(t) of the solution x ( t , to , +, u ) with respect to the increment q j ( t ) of t h e j t h component u j ( t ) of the control vector u ( t ) is given by

We now give an increment Ak#J= EOk(t)ep)

(k = 1 , ..., n)

(9.4. 2)

t o the initial function + ( t ) = (&(t), ..., cjn(t)),where e,(t) (k = I , ..., n) is a continuous function on ] [ a , to] such that Akcj E @ for all sufficiently small E . Put

++

d+,x(t) = x ( t , t o ,

#J t

).

-

. ~ ( tto, +, .)9

(9.4.13)

I t is shown in Sections 6.3 and 6.4 that d,*x(t) = E

j M(a, t ) ep)Ok(o)do + O(E), t,L

(9.4.14)

where O(E) satisfies the relation (9.4.9). Therefore the first variation

226

IX. OPTIMIZATION OF CONTROL SYSTEMS

adkx(t)of the solution x(t, t o , 4, u ) with respect to the increment cB,(t) of the kth component + k ( t ) of the initial function + ( t ) is given by

for k = 1, ..., n. Since the right-hand side of Eq. (9.1.1) is linear with respect to x, it is obviously differentiable in the functional sense with respect to x. By Theorem 2.14 the partial derivative of x(t, t o , 4, u ) with respect to the initial instant to is the solution of the homogeneous linear delaydifferential equation 7,,

a(t) = Z A Y ( t ) z ( t- h,) "20

+ 1'4, K(o, t)z(o)do

(9.4.16)

which satisfies the initial condition z ( t ) = w",

t E I[w, t , ] ,

(9.4.17)

where

I t is obvious that the kernel matrices of the first kind associated with Eqs. (9.1.1) and (9.4.16) are identical. Therefore, the solution of Eq. (9.4.16) which satisfies the initial condition (9.4.17) is given by z(t) =

1

'4,

M ( u , t)wo do.

(9.4.19)

Hence (9.4.20)

After this preparation we may proceed to establish the first variations of the functionals G(T, T) and HT'(T,T ) . Let 8 + k G ( ~ T), be the first variation of the functional G(T, T) with respect to the increment c d k ( t ) of the kth component +,(t) of the initial function + ( t ) . Making use of the continuous differentiability of the function g ( x , u, t ) with respect to the vector x, we obtain S,*G(i,

T)

=

(2-1 x

. hdkx(7,to ,4,10,

(k = I , ..., n),

(9.4.21)

9.4. Functionah G(T,T ) , G,'(t, 7 ) and ElT'( T , T )

227

where Sdkx(7,to , 4, u ) is given by (9.4.15) and (aglax) is the gradient vector of the function g(x, u , t ) with respect to the vector x, (9.4.22)

all the partial derivatives being evaluated at time t = 7, with x = 4 7 , to u). Similarly, corresponding to the increment q j ( t ) of the j t h component u j ( t ) of the control vector u ( t ) , we obtain 141

where (ag/au,) is the partial derivative of the function g with respect to ui and LSU,x(~, t o , +, u ) is given by (9.4.1 1). T h e meaning of the notation s U j G (T~) ,is clear. We now establish the first variations of the functional H7'(T , T ) . First, let us note, according to the defining relation (9.1 .lo), that

4[,](7) = 47,

t" * 4, 4,

(9.4.24)

since T E I [ t o , TI. Hence, using the formulas (9.4.20) and (9.4.24), we may write (9.4.25)

where

which is an n vector, and M I T l ( ~t ), is the kernel matrix of the first kind associated with Eq. (9.1.1) corresponding to the new initial interval ] [ a , T ] . Therefore, by Theorem 3.1 1, the matrix M F ~ ] (t U ) satisfies , the delay-differential equation

and the initial condition

where I is the unit matrix in Enand 8 ( t ) is Dirac's function. I t should be , upon T continuously and has continuous noticed that M ~ ~ l (t u) depends derivatives in T .

228

IX. OPTIMIZATION OF CONTROL SYSTEMS

Note that ~ ( tT ), is an n x n matrix which is independent of +#,u ). Further, since 5 2 ~ ~does 1 not depend on u, the integration variable in (9.4.25), we may write

It follows from (9.4.26) that (9.4.31)

and

Furthermore, a direct computation yields

and

where (d2h/dx2) is the Jacobian matrix of the gradient vector (dhldx) of the function h ( x , t ) with respect t o x at time t = T with x = x ( t , t o , 4, u ) . Clearly (d2h/dx2)is an n x n matrix. ‘The first variations of the functional H,’(T, T ) may be written now. First, from the formula (9.4.4), we deduce dh

dx and

9.4. Functionals G(T,T ) , G,'(t, T ) and H,'( T , T )

229

(9.4.38)

f o r j = 1, ..., r . Finally the formulas (9.4.2) and (9.4.30) yield (9.4.39)

(9.4.40)

(9.4.41)

and

(9.4.42)

230

IX. OPTIMIZATION OF CONTROL SYSTEMS

T h e formulas (9.4.21), (9.4.23), (9.4.37), (9.4.38), (9.4.43), and (9.4.44) are the first variations of the functionals G(T,T ) , HT’(T , T ) , and G,’(t, T ) which we set out to find. 9.5. Necessary Conditions for Optimality

As we mentioned in Section 2.13, the first variation of a functional is zero for those functions which make the functional itself a maximum or minimum. Accordingly, the first variations of the functional appearing in Eq. (9.3.15),

F(+, u )

= G(T,T )

-

H,’(?’,

T) -

J: G,’(t,

T)

dt,

(9.5.1)

must all vanish for an optimal pair, since, by (9.3.15), the functional F(+, u) attains its minimum for an optimal pair. T h u s we have 8,$(+,

U ) = c?,~G(T, T ) - 8+kH,’(T, T ) --

lr6+kC,’(t, dt T)

fl 7

S,,f(+,

u ) = 8 L l l G 7( )~ ,

8,,,H,’(T, T ) -

5

for k

=

1, ..., n , j

=

1 , ..., r , and

T

, TI.

sll,G7’(t, T ) dt

(9.5.2)

9.5. Necessary Conditions f o r Optimality

23 1

(9.5.4)

QE,

= 1, ..., r , v = 1, ..., m, where is the transpose of Q71 and (@/ax), (aglau,), (dhldx), (azg/ax2),(d2h/dx2), and (a$/au,) are, as before, evaluated

j

at time t = T . Then, using the formulas (9.4.21),(9.4.23),(9.4.37), (9.4.38), (9.4.43), (9.4.44),and (9.5.3)-(9.5.6),Eqs. (9.5.2) can be written in the form

and

k = 1, ..., n , j = 1, ..., r , and T E I [ t o , TI; and making use of the formulas (9.3.11) and (9.4.15)in (9.5.7)and (9.5.8),we find

i

2BY(T)M(u,

T -

hJ

+ /'C(t, T

T)M(u,t ) dt 1 ep)O,(u) do

= 0,

(9.5.9)

232

IX. OPTIMIZATION OF CONTROL SYSTEMS

Consider first Eqs. (9.5.9). Since Eqs. (9.5.9) hold for all functions O,(t), continuous on I [ a , to], by a well-known lemma of the calculus of variations, we may write

k = 1, ..., n, T we have

E

l [ t o , TI, and u

E ] [ a , to]. Since el?) are

the unit K vectors

1, ..., n, T E I[t,, TI, and u E I [ a , to]. Now consider Eqs. (9.5.10). Introducing the function e ( t ) defined by (3.1 1.6), and applying Dirichlet's rule for changing the order of integration in double integrals, we may write

k

=

(9.5.13)

j

=

1, ..., K and

T

€ I [ t o , TI, where T

I

B(T,

= c e ( T - h, "=O

-

u)B,(T)N(O,T - h,)

+ j e(t ~ ) c (7)A7(u, t , t ) dt. -

(9.5.14)

We now show that D,(T) = 0,

j

=

I , ..., r ,

T

E I [ t o , TI,

(9.5.15)

and E(T, U ) = 0

for

T

E I [ t o , TI

and for almost all u

Then, the equation becomes

E

I [ t , , TI. To do so, we put

(9.5.16)

9.7. Terminal Controls: I

233

I [ t , , TI, which must hold for all T j ( t ) measurable in Z[to, TI. Equation (9.5.18) is a homogeneous Fredholm integral equation of the third kind with a piecewise continuous kernel. Clearly, if Ej(7, a) 9 0 and Dj(7) 0 Eq. (9.5.18) cannot be satisfied for arbitrary T j ( t ) . Therefore, necessarily we have Ej(7, 0) = 0 for I [ t o , T ] and for almost all u E I [ t o , TI, and Dj(7) = 0 for T E Z [ t o , TI. From the above results we have the following theorem. T E

+

I+,

Theorem 9.1. In order that an admissible pair u> be optimal for the problem described in Section 9.1 it is necessary that {+, u } satisfy Eqs. (9.5.12), (9.5.15), and (9.5.16).

9.6. The Matrix x ( t ,

T)

T h e matrix X(t, T) defined by (9.4.29) has an interesting physical interpretation. T o see this, let us integrate both sides of Eq. (9.4.27) with respect to 0 between a and T . We obtain

for t

E I [ T , TI, T E

Z[to , TI. Further, by Eqs. (9.4.28) and (9.4.29) we have X(t, T)

= 1,

t

E [[a, TI.

(9.6.2)

Therefore, the matrix X(t, T ) is the solution of the delay-differential equation (9.6.1) corresponding to the initial condition (9.6.2). In other words, X(t, T ) is the unit response of the adjoint equation of the first kind associated with the original control system (9.1.1) for the initial interval Z[a,T ] , T € Z [ t o , TI. Thus, we do not need to find the intermediate matrix IM171(u, t ) to construct the matrix X(t, T ) .

9.7. Terminal Controls: 1

T h e conditions for optimality, namely (9.5.12), (9.5.15), and (9.5.16), are relatively simple for terminal and averaging control processes. In this section we consider terminal controls. In a terminal control process the function g(x, u, t ) which appears in the general formula (9.1.4) for performance index is identically zero.

234

IX. OPTIMIZATION OF CONTROL SYSTEMS

Accordingly the formulas (9.5.3)-(9.5.6) and (9.5.14) reduce to C(r, T)

(9.7.1)

= 0,

and

where p E Z [ t o , TI, 7 E Z [ t o , TI, and t E Z[T,TI. Consequently, the conditions for optimality (9.5.1 2), (9.5.1 5), and (9.5.16) reduce to the following equations:

2 B,(T)M(u, 7n

T -

h,)

=

u=O

0,

u E I[m, t o ] ,

T

(9.7.6)

E I[to , 1’3,

Ti1

z e ( ~ h,

-

~ ) B , ( T ) NT( ~ ,h,)

= 0,

T

6 Z [ t O , TI,

“=O

p € Z [ t O , TI

a.e.

(9.7.8)

Clearly the matrices M , N , and x do not depend on x, u , and +. Solving Eqs. (9.7.6) and (9.7.7) with respect to x = (xl, ..., xn) and u = ( u l ,..., ur) and substituting these expressions into the formula

we obtain n Fredholm integral equations of the first kind with respect to the initial function 4 = (41,..., &) by which c $ ~..., , 4n may be determined up to an arbitrary linear combination of the nontrivial solutions of the homogeneous equation M ( 0 , t ) + ( ~do )

= 0.

(9.7.10)

23 5

9.8. Averaging Controls 9.8. Averaging Controls

I n an averaging control problem the function h(x, t ) in (9.1.4) is identically zero. Consequently we have (9.8.1)

(9.8.2)

(9.8.3)

and ??I

E(T, P ) = x e ( T - h,

-

p)R,(T)N(p,

7 -

A),

v=u

+ 1' e(t - T)c(t,T ) N ( p , t ) dt, (9.8.5)

where p E I [ t o , TI,T E I [ t o , TI, and t E I[T,TI. Hence the conditions for optimality (9.5.12), (9.5.19, and (9.5.16) become ZBv(T)N(U, 7 - h,) "-0

+ j Tc(t,

T)n/r(U,

t ) dt = 0,

(9.8.6)

(9.8.7)

and

CP(T - h, - p)BV(~)N(p,- h,) 4m

T

e(t

-

~)c(t, .)Ar(p,

t ) dt = 0

(9.8.8)

v-0

T E ] [ t o , TI and for almost all p E ] [ t o , TI. Solving Eqs. (9.8.6) and (9.8.7) with respect to x = (xl, ..., x,) and u = ( u l , ..., u,) and substituting them into (9.7.9) we again obtain n Fredholm equations of

for

the first kind by which we can determine 4 = (& , ..., 4,) up to a linear combination of the nontrivial solutions of the corresponding homogeneous equation (9.7.10). T h e linear combinations of the nontrivial solutions which correspond to optimal initial functions are determined

236

IX. OPTIMIZATION OF CONTROL SYSTEMS

by (9.3.12) and (9.8.8). T h e above method involves considerable difficulties. I t is thus preferable to reduce the averaging control problems into terminal ones by the artifice explained in Section 7.7.

9.9. Terminal Controls: 2

We now investigate a particularly simple terminal control problem in which $(u, t )

I :

(9.9.1)

W(t)u

and (9.9.2)

where W(t)is a given continuous n x r matrix and

is a given continuous n vector function. Denoting by W j ( t )the j t h column of the matrix W(t),we obtain

Further we have

where

* denotes, as usual, the transpose operation

and (9.9.6)

where I is the n-dimensional unit matrix. Substituting the above equalities in (9.7.2) and (9.7.3) we obtain

9.9. Terminal Controls: 2

237

Hence, the conditions for optimality (9.7.6)-(9.7.8) reduce to

where

T

EI[to

, TI and p E Z [ t o , TI, a. e. Note that, by (9.4.26), we have

Further, we have

Consider now Eqs. (9.9.10) and put

r,(Y',

T)

=

x(T,T)W,(T),

,i = 1, ..., Y .

(9.9.14)

Then Eqs. (9.9.10) express that the vector X ( T ) -- [(T)* is orthogonal to the n vectors T j ( T ,~ ) ,= j I , ..., 7 , for each T € 1 [ t o , TI. Thus the vector ( x ( T ) - ( ( T ) ) * is orthogonal to the linear variety generated by the vectors F j ( T ,T ) , j = 1, ..., Y , for each T € ] [ t o , TI. Let r be the n x r matrix whose j t h column is T j ( T ,T ) , j = 1, ..., r . We assume that r has the rank r , i.e., there exists at least one nonvanishing 7 x r determinant extracted from the matrix F. Thus, the system of linear equations (9.9.10) can be solved with respect t o r components of the vector x = (xl, ..., xJ, say xA.,, ..., xk,. Let us substitute these solutions into (9.9.9) and (9.9.1 l), which constitute 2n equations linear in x and u according t o (9.9.12). If the Y x n matrix (1 1

rv*(T)X(T , T ) h f ( U , 7)

(9.9.15 )

has the rank 7 , we can solve the system of linear equations with respect to u l , ..., u, . Again, from the system of Eqs. (9.9.9) we find n - r linear equations relating to X(T - hi), v = 0, I , ..., m. Since the components

238

IX. OPTIMIZATION OF CONTROL SYSTEMS

xk, , ..., xk, are found above, the remaining n - r components can be found by these n - r linear difference equations. Thus, starting with x ( t o ) = +(to)= +O, where $o is defined by (9.3.12), we can find all the components of the vectors x and u. Substituting these expressions into

we obtain n Fredholm integral equations of the first kind by which = , ..., +), can be determined up to an arbitrary linear combination of the nontrivial solutions of the corresponding homogeneous equation

:j M(o, t)+(o)do

(9.9.17)

= 0.

T h e linear combinations of the nontrivial solutions which correspond to optimal initial functions are determined by (9.3.12) and (9.9. I 1). 9.10. Processes with Quadratic Performance Indices

T h e problem considered in the preceding section is a special case of 0.1)

0.2)

h ( x ( f ) ,4

=

4

n

C[Xk(t) h=l

!X4l2,

(9.10.3)

where W ( t )is a given continuous n x r matrix and ( ( t ) = (El(t), ..., f n ( t ) ) is a given n-dimensional column vector. In this case the conditions for optimality (9.5.12), (9.5.15), and (9.5.16) reduce to ,">

0.4) a,(.)

T )0, -t( ~ ( 7 )- &))*x(T, T ) W ~ (=

i

=

1, ..., r.

0.5)

and

(9.10.6)

9.1 1 . Optimal Cost

239

u E I[a,to], T E I [ t o , TI, and p E I [ t o , TI a.e., where W j ( t )is the j t h column of the matrix W(t).By (9.4.26)

for

c TI1

Q,,, =

-

u=o

A.(7)47 - h, to ,+, 4 - V T ) 7 4 7 ) . 7

(9.10.7)

Further x(t)

x(t, 2 0 9 4, u) =

J

tlb

M ( g , t)+(u)do

+ f N(p, t ) w ( p ) ~ ( p )dp. 1,

(9.10.8)

If we substitute (9.10.7) into (9.10.4) we obtain n linear difference equations in x(t). Starting with the initial condition 4 t O ) = +(to) =

P,

(9.10.9)

where do is defined by (9.3.12), we can solve this system with respect to x ( t ) . Then Eqs. (9.10.5) immediately yield ul(t),..., ur(t),the components of the control vector u(t). Using these expressions and Eq. (9.10.8) we obtain n Fredholm integral equations of the first kind for d l ( t ) , ..., &(t), the components of the initial function d ( t ) . This system of integral equations can be solved up to a linear combination of the nontrivial solutions of the homogeneous equation ~ ' M ( u+#o) , do

= 0.

(9.10.10 )

T h e linear combination which corresponds to the optimal initial function is determined by (9.10.6). 9.11. Optimal Cost

Let {do(t),uo(t)} be an optimal pair and x o ( t ) = x(t, t, , do, uo} the corresponding optimal trajectory. Then, combining Eqs. (9.3.9), (9.4.1)-(9.4.3), (9.4.26), (9.4.29), and (9.4.30), we obtain fT'(7*,T ) = ' d X " ( T ) >

Clearly

74Y7),.)

f ( T , T ) = h(xO(T), T ) .

(9. I I .2)

Integrating both sides of Eq. (9.1 1.1) with respect to T between to and T and using the condition (9.11.2) we find f( T , to) which gives, according to Section 9. I , the optimal cost of the process.

CHAPTER X

+

Time Optirna1 Problem

I n many optimization problems, it is of interest to minimize the time needed t o achieve the goal of the process. This chapter is devoted to establishing the solution of a time optimal problem for a control system whose state is described by a linear delay-differential equation. T h e time optimal problem for systems characterizedby lineardifferencedifferential equations has been investigated by G. L. Haratigvili [ 11 (see also L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko [l], E. Shimeniura [I], and M. N. ORuztiireli 123).

10.1. Formulation of the Problem

Consider a control system whose state at time t is given by a linear delay-differential equation of the form

for t that

2 t o , where, as usual, the constants h, (v 0

=-

h,

< h,

i ...

< h,, ,

=

0, 1, ..., m) are such (10.1.2)

A,(t) (v : 0, I , ..., m ) are given continuous n x n matrices, B ( t ) is a given continuous n x r matrix, u ( t ) is an r-dimensional column vector, f(t) is a given continuous n-dimensional column vector, and x ( t ) is the state vector which is an n-dimensional absolutely continuous column vector. 240

10. I . Formulation of the Problem

24 1

We suppose that the control region R of the system is the r-dimensional unit cube I uI,1 ,< I , k = I , ..., r I (10.1.3)

I

\'

and measurable vectors u ( t ) , t >, t , , whose range are in R, are admissible. We denote by U the set of all r-dimensional vector functions u ( t ) measurable on each finite interval { [ t o , t ] and

u = (\ u ( t ) 1 1 u r ( t ) 1 < I ,

k = I , ..., r ,

2 t , ,I ! .

(10.

I , ..., r , t >, t o1\ .

(10.

t

Let U o be the set of all functions uo(t) in U with

uo= /u,(t) 1 1 uk0(t) I

=

I, k

=

T h e set U is the set of all admissible control functions for our control problem and U o is the set of "bang-bang" control functions. We denote by @ a compact convex subset of the set of all real-valued n-dimensional continuous vector functions in the initial interval ] [ a , t,]. T h e elements of the set 0 are called admissible. A pair {+, u } in which + ( t ) and u ( t ) are admissible will be called an admissible pair. T h e solution of the system (10.1.1) satisfying the initial condition .+)

t

= act),

E

I [ % t"], 4(t) t @,

(10.1.6)

will be denoted, as usual, by x ( t , t , , +, u). Now let us consider a moving target x ( t ) . We assume that x ( t ) is an n-dimensional continuous column vector. We assume that there exists an admissible pair {a,#) such that the trajectories x ( t , t , ,+, u ) and z ( t ) satisfy the equation

.(7, t , for some T

> t,

,c#l,

u) =

z(T)

(10.1.7)

and x(t, t o ,

4,u ) f

.+),

t

< T.

( I 0.1.8)

T h e quantity T depends on the chosen pair {+, u ); therefore we may write T

=

T ( 4 ,u).

( 1 0.1.9)

Our control system (10.1.1) should be controlled in such a manner that the corresponding capture time T = T(+,u ) will assume its minimal value. This minimum will be denoted by

7'" = &@,neu min T(+,u).

( 1 0. I .I 0)

24.2

X. TIME OPTIMAL PROBLEM

T h e n our time optimal problem may be formulated as follows: Time Optimal Problem. Find an admissible control u ( t )E U and an admissible initial function + ( t ) E @ for which 7.(4,

u) =

7'".

(10.1.11)

10.2. Reduction of the Problem to an Integral Equation

Let M ( o , t ) and N ( T ,t ) be the kernel matrices of the first and second kind associated with the system (10. I . I ) . Then, we have the representation

f

J" N ( T ,t ) f ( ~d7. )

(10.2.1)

10

As described in Section 10.1 for the time optimal problem, the particle z ( t ) must be reached by x ( t , t o , 4, u ) in minimum time. We wish, therefore, at some t to have r ( t , t"

). = z(t),

14,

0.2.2)

that is, to have 0.2.3) where

J"

~ ( t= ) ~ ( t-~ ) N ( T ,t ) f ( ~dT. )

(10.2.4)

tIJ

It should be noticed that the vector w ( t ) is completely determined and continuous for t >, t o . Our problem is, by the above consideration, reduced to the study of Eq. (10.2.3).

+, u ) and the Sets r(t) and ro(t)

10.3. The Functional n(t,

Consider the functional

10.4. Theorems of LaSalle

243

Then Eq. (10.2.3) is equivalent to Q(T,+,.)

=

(10.3.2)

w(7-1,

where T = T($, u ) is the time at which the trajectory x ( t , t o , $, u ) first hits the particle z ( t ) ; that is, T is the first root of Eq. (10.2.2) which is greater than t o . We now define the sets

r(t)= W(t,4, ). and

P ( t ) = {sz(t,+, 240) I

I $ E @,

+

E

@,

11

E

uo E

U>

(10.3.3)

UO}.

(10.3.4)

It should be noticed that Q ( t , 4, u ) is the solution of Eq. (10.1.1) with f ( t ) = 0 satisfying the initial condition (10.1.6). 10.4. Properties of

Q,

r, and ?. Theorems

of LaSalle

I n this section we establish some properties of the functional Q and the sets r and To. Lemma 1.

The set r ( t > is convex and compact.

This lemma is a natural consequence of the extended Neustadt theorem (Theorem 8.16). Further, by the extended bang-bang principle (Theorem 8.1 l) , we have: Lemma 2.

r(t) = rO(t).

By Lemma 2, anything that can be accomplished in time t by a u ( t ) E U can also be accomplished in time t by a uo(t)E Uo. Consequently we can state the following theorems due to J. P. LaSalle [ 11 for our time optimal problem. Theorem 10.1. I f , of all uo(t) E U o there is an optimal one relative to Uo, then it is optimal relative to U . Theorem 10.2. If there is an optimal control, then there is always a bang-bang control that is optimal.

T h e importance of these theorems is obvious. We now prove the following theorem which is an extension of a theorem due to M. N. Oguzttkeli [2].

244

X . TIME OPTIMAL PROBLEM

Theorem 10.3. If + ( t ) E @ and u ( t ) E U are in some suficiently small neighborhood N($) o f $ ( t ) E @ and N ( 5 ) of U ( t ) E U , then corresponding to each E > 0 there is a 6 > 0 such that

/I Q(t> 6,4 - Q(t, 4,U ) It < E

(10.4. I )

for each Q(i,$, zi) and all

and all

+

t-sct
Proof.

fi(i, $, 12)

@, u

E

U.

Suppose t -

(10.4.2)

< i and

Q ( t ,4,U)

'to =

consider

(M(u, t)$(u)

-

M(o, t)4(u)>do

{ N(T, t)B(T)c(T)- N(T,t)B(T)U(T)\ d7

+ 1'N ( T ,~ ) B ( T ) ~dT.( T )

(10.4.3)

t

Since the matrix N ( T ,~ ) B ( Tis) continuous and since, according to (10.I .4),1) u(t)lJ I , we have

<

( I 0.4.4)

where (10.4.5)

t being sufficiently large. Using the mean value theorem, we obtain M(a, 1 )

= M(a, t)

and N(T, t )

=

N(7, t )

+ (t

+ (t

~

-

S)M,(O, t'),

t
(10.4.6)

< t,

(10.4.7)

t < t"

t)N,(T, t"),

where t' and t" may be different for different elements of the respective matrices M,(u, t ) = aM(u, t)/at and N ( ( T ,t ) = ~ N ( T t ) ,/ a t . Hence M(fJ,

--

M ( u , M u ) = M ( u , W(4 - +(.)I

+ (t

-

t)Mf(%t')4(.),

and N ( 7 , t ) B ( T ) C ( T ) - N(T, t ) B ( T ) U ( T ) = N(T,i ) B ( T ) [ G ( T ) - U ( T ) ] f (t

-

t)Nt(T,t " ) B ( T ) U ( T ) .

10.5. Existence of an Optimal Pair

245

Therefore

Consequently

/I Q(t; B, C) - w,4%u) I1 < ( t o - 47% I/ 4 - 4J II + (t - t)[m, + (to -- a)m,m,

+ (t -

ll C - 24 II

+ (t - Q n m , ] .

(10.4.1 1)

Let us consider the neighborhood N($) and N(G) defined as follows:

Then

II Q(t,4,4 - Q(4 41.) II < E for

i .- 6 < t < i, and for all 4 E N(&,

uE

N(G), where (10.4.1 3 )

This completes the proof of the theorem.

10.5. Existence of an Optimal Pair We now prove the following existence theorem, which is an extension

of that due to J. P. LaSalle

111.

246

X. TIME OPTIMAL PROBLEM

If there is a pair of functions

Theorem 10.4.

+

E @,

uE

U , such that

x ( t , t o , $1 U ) = z ( t ) ,

then there exists a pair of functions + O ( t ) Proof.

E

@, u"t)

E

U , which is optimal.

By hypothesis the set 9JZ =

~ ~ ' l . x ( Z 7 , t o , +=, zu() T ) ,

I

+E@,

UEUI

(10.5.1)

is not empty. Let To be the greatest lower bound of all T (> to) which belong to the set (%t. By Section 10.3, for T E '%t, we have w( T ) = sz( T , +, u ) E r(7.). Let T' be such that T i -+ To for i

+,

Ez = Q ( P , u')

=

:1

-+ GO. Consider

M(o, T')+(u)do 4

(10.5.2) the sequences

14 N ( T ,T")B(T)u'(T)d~ T'

(10.5.3)

and

+,

Eoz= Q(T0,

Ui) =

t0

M(o, To)f(u) du

+

N ( T ,T")B(T)u~(T) dT. (10.5.4)

Obviously Ei E E( T i ) , Eoi E E( TO), and

Ei - Boi = J ~ { M ( uTi) , - M(u, T0)}+(o)do

+ J7"'{N(~, Ti)- N(T,TO)}B(T)U*(T) dT tU

f J T i N ( T , T*)B(T)u~(T) dT.

(10.5.5)

To

Now, using the notation introduced in the previous section, we find

(10.5.7) (10.5.8)

247

10.6. Continuity of the Functional T(+,u ) for

(10.5.9)

and

(10.5.1 1)

From (10.55)-( 10.5.1 l), we obtain

11 Q(P,p,ui) - .n(TO, d", d ) 11 < €

(10.5.12)

for

Since the set F(T0) is compact, we can select subsequences {@(t)} and {uik(t)} of the sequences {p(t)}and {ui(t)} so that they converge respectively to the functions $ O ( t ) E @ and uo(t)E C'. Therefore w( TO) = .n(TO, $0,

240)

E

r(TO),

( 10.5.14)

that is x( TO, to , +O, uo) = z( TO),where TO is the optimal time. 10.6. Continuity of the Functional T(+, u )

Let P be the topological space of points U with the metric

p

=

(4, u),

4 E @,

where

uE

(10.6.1)

Consider now the function g(t, p ) = II

W,$, u) z(t)!I, -

t 2 to.

(10.6.2)

By Theorem 10.3, (i) t -+ g(t, p) is continuous for t >, to and for fixed p (ii) p -+ g ( t , p ) is continuous for p E P and for fixed t

E P;

to.

248

X. T I M E OPTIMAL PROBLEM

Let p,, E P and consider the equation g(t,p,) = 0. Let T, be the greatest lower bound of all solutions of this equation. Therefore g( To , po) = 0 and g(t, p,) f 0 for t < To , T,, > t, . Suppose that

(iii) g'(t, p ) >, c (> 0) for t -~To j < a,, , and N,, is some neighborhood of p , ;

p E N,, , where 6, > 0

~

(iv)

lim sup I g(t, p ) f ~ + f ~ ( )I

'I(,

-

g(t, p0)I = 0.

Suppose now that p is close top, and consider the solution ofg(t, p ) = 0. For small h we have ,?(To i h , p )

-

g ( T , , p ) t hg'(T,

+ 6h,p),

050


(10.6.3)

By (ii) we can choose a neighborhood N , C No of p , such that 0

If p

E

N , and 0

< g( 7'" ,p ) < k8,c

< h < A,,
p

E

A7,

.

(10.6.4)

then from (iii) we can get

g(T,

Similarly, if -6,

for

+ h, P) 2 ch

< 0, then

+ h, P) <

g(?',,

0.

(10.6.5)

+ ch,

(10.6.6)

. It follows from (i) that there exists which is negative if h = a solution t of g ( t , p ) = 0 satisfying I t - To j z S l . I n fact, the argument shows that a solution t exists satisfying

<

I

It

< '

const g( 7; ,p).

(10.6.7)

which tends t o zero as p + p , , by (ii). If T,, is the lower bound of such solutions we have therefore T,, -<, 'r;

+d p ) ,

(10.6.8)

where q ( p ) + 0 as p -+p , . On the other hand, by (i), given any there exists an A ( ( ) > 0 such that g ( t , PO)

A([)

for

1,

< t < '7'0

~

5-

6 > 0, (10.6.9)

So, by (iv), we have for some neighborhood N , C N , of p , ,

A t , p )'3 : A ( [ ) '0 It follows that TI, > T ,

-

( if

for t , S t

< To

~

6, p E N , .

p E N , . So, finally, 1 T , - T,, 1

(10.6.10) +0

as

10.7. Further Properties of r ( t ) and Optimal Pairs

249

p-p,.Thecaseg'(t,p)
Under the hypotheses (iii) and (iv) the functional

T(+,u ) i s continuous in all its arguments. 10.7. Further Properties of

r(t) and

Optimal Pairs

First we prove the following theorem.

If s;! is an interior point of r ( t ) then there exists an that N,(SZ) C r(7)f o r all 7 in I ( t - E , t), where N,(SZ) is a neighborhood uf SZ of radius E . Theorem 10.6.

E

> 0 such

Proof. Let SZ be in the interior of r ( t ) and let NeI(SZ) be a neighborhood of SZ of radius > 0 contained in the interior of r(t). Let c 2 = E , and let N J Q ) be the neighborhood of SZ of radius c 2 . Let 8 = S(EJ > 0 chosen to satisfy Theorem 10.3, i.e., (10.4.13) with E = c 2 . Consider Q* E NJSZ). Suppose for some t , , satisfying t - 8 < t , < t, that s2* is not in the interior of r(tl). T h e n since r(t,)is convex (Lemma 1 of Section 10.4), there exists a support hyperplane P t l such that there are no points of r(tl)on one side of P , , . Because the neighborhood NCp(Q*) C iVel(Q) C r(t)we see that there is a point p E r ( t ) such that

9

I1 P

-

%)

I/ 2 € 2 .

But this contradicts Theorem 10.3. Therefore NJSZ) C r(7)for all Z [ t - 6, t]. Now set E = min(E, , 8). T h e n E > 0 and NE(SZ) C N,,,(SZ) and l ( t - t , t ] C i ( t - 8, t]. T h u s 7E

N , ( Q C T(T)

for

7E

I(t

- E,

t].

(10.7. I )

This proves the theorem. Theorem 10.7.

point of

Let w(t) begiven by (10.2.4). Then w( TO)is a boundary

r(To), where To is the optimal time.

Proof. Suppose, on the contrary, that w(T0) = SZ(To,#O, uo) is an interior point of r ( T 0 ) .Then, by Theorem 10.6, there exists an E > 0 such that

j~~(w(7'0)) C r(II'0) for all

T o - 6 < t < 7''.

(10.7.2)

250

X. TIME OPTIMAL PROBLEM

T h e continuity of w ( t ) at t that w ( t )E

Let 2y

=

=

T o implies that there exists a 6

N,(w(T"))

To - 6

for all

> 0 such

< t < 1'".

(10.7.3)

min(6, 6 ) . Then w(7'"

-

c F(T"

y) E N<(W(P))

-

(10.7.4)

Y).

But this is a contradiction since T o - y < TO and To is, by hypothesis, the minimum of t such that w ( t ) E r(t).

There exists a vector i,bo = (i,bIo, ..., # n o ) such that

Theorem 10.8.

$0

. Q( T",$, u )

<

$0

(10.7.5)

. Q( TO, $0, u")

f o r all {+, u ) E P . Proof. Since T(TO) is a convex set and since SZ(To,$O, uo) is, by Theorem 10.7, a boundary point of the set T ( T o ) ,there is a support hyperplane through sZo = sZ( To,+O, uo). Let $O be a nonzero vector orthogonal to this support hyperplane directed to the side which contains TO). Therefore, for each a(TO, 4, u ) E TO) and no points of the set for all E @, u E U , we have

r(

+

r(

($", sz( T",4, U ) - sz( T",$0, u")

< 0.

This scalar product is equivalent t o ( 0.7.5).

10.8. Optimal Controls. Theorems of LaSalle

Let {+O, uo} be an optimal pair and TO the optimal time. By Theorem 10.8 there exists a vector i,bo such that (10.7.5) holds for all (4, u } E P. Since

40,uo) =

z u ( ~= ) Q(TO,

ffl

~ ( u z'o)+o(~) ,

da

+-

N(T, TO)B(T)~P(T)

and Q( To,$", U ) =

J'I" M(a, ToMo(a) + 1: da

N ( T ,T")B(T)u(T) dT,

dT

10.8. Optimal Controls. Theorems of LaSalle

the inequality (10.7.5), with

4 = +O,

25 1

implies

for all u E U. Let h(t)

=

$"N(t, P ) B ( t ) .

(10.8.2)

Let A,(t) be the j t h component of the vector A(t). Then jZ($U,

N(7, T0)B(7)U0(7)) dT

< 2 :;Ij 3=1

4(T) I dr

==

$wsgn h3(4 d7.

3=1

(10.8.3)

Hence we see that on any interval of positive length where Ai(t) # 0 it must be (10.8.4)

ujo(t) = sgn A j ( t ) .

Thus, we may state the following theorem which is an extension of a result due to J. P. LaSalle [I]. Theorem 10.9.

A l l optimal control functions uo(t) are of the form uo(t) = sgn +ON(t, T0)B(t),

(10.8.5)

where C0 is a vector for which Theorem 10.8 holds.

We now introduce: Definition 1. The control system (10.1.1) is said to be normal if no component of # N ( t , T o ) B ( t ) ,# # 0, is identically zero on an interval of positive length.

Let (10.8.6)

Y ( t ) = N ( t , T")B(t).

Let y,(t) be the j t h column of the matrix Y ( t ) . Therefore, according to Definition 1, the system (10.1.1) is normal if the functions -Ytj(t)j~ z j ( t ) , ---, Ynj(t)+

j =1

--.r

r,

(10.8.7)

are linearly and functionally independent on each interval of positive length. I n this case the vector uo(t) is uniquely determined by (10.8.5). As a natural consequence of Theorem 10.9, we obtain another theorem of J. P. LaSalle [I].

252

X. TIME OPTIMAL PROBLEM

Theorem 10.10. For normal control systems there is a t most one optimal control function, and i f there is one, it is bang-bang.

Thus, for normal systems, the only way to reach the target in minimum time is by bang-bang controls of the form (10.8.5). For other control systems the most that can be said is that if it is possible to reach the target in finite time then there is a bang-bang control that is optimal. T h e optimal control will be of the form (10.8.5), but there may be an infinity of optimal control functions, some of which will not be bangbang. 10.9. Optimal Initial Functions

Let us consider Eq. (10.2.2) with the optimal functions (. TO, in ,$0,

UO) =

z( TO),

(10.9- I )

where we know the existence of TO, +O, and uo and also the general form of the optimal control uo(t) by the formula (10.8.5) for a normal system. We write this equation in the following form:

J'I" nqu,

TO)+O(a)do = g( T O ) ,

(10.9.2)

where

We wish to find the optimal initial functions +O(t). Assuming that the vector uo(t)is found, all the quantities in g( T o )are known. Then (10.9.2) is a Fredholm integral equation of the first kind. Writing t instead of To, + ( t ) instead of +O(t), and putting M$

we obtain Let

&'*(a,

=

It"M(u, Md

==

t ) + ( ~d) ~ ,

(10.9.4) (10.9.5)

&).

t ) be the transposed matrix of M(a, t ) . Put

n/r,

j M(0, t)X(t) dt, 10

=

a


(10.9.6)

and (10.9.7)

10.9. Optimal Initial Functions

253

I t is well-known that (MA x) =

M*X).

($9

(10.9.8)

Let { A k ) be the set of eigenvalues of the following homogeneous Fredholm equations of the second kind: M$

M*x

= A$,

= AX.

(10.9.9)

Let (V(t)) and {Xk(t)}be the complete sets of the principal eigenfunctions of the kernels M ( a , t ) and M*(o, t ) , which are biorthonormalized: (10.9.10)

where M$h

= A,+,'

h!I*xk

= hkxk.

(10.9. I I )

Let {+"(t)} and {xoi(t)}be the nontrivial solutions of the integral equations M+

= 0,

M*x

= 0.

(10.9.12)

Assume that the solvability condition =0

(X"",)

(10.9.13)

is satisfied for each (boi(t).Consider now the expression of the function

g(t) with respect to the biorthonormal system {#J~,xk): d

At) =ZcdYt),

( 10.9.14)

k=l

(10.9.1 5)

Let us now determine the constants uk so that the series $(t> =

3

k=l

ak@W

+ &),

( 10.9.I 6 )

where &t) is a solution of the integral equation M+ = 0, should be a solution of the integral equation (10.9.5). For this purpose multiply both sides of (10.9.5) by x k ( t ) and integrate with respect to t from to to ; hence, making use of the preceding formulas, we obtain (Y

a k = -c,k

A,

k = l , 2 , 3 ,....

(10.9.17)

254

X. TIME OPTIMAL PROBLEM

Therefore, an optimal initial function is given by the formula ( 10.9.18)

I t should be noticed that the above deductions are considerably simplified using basic properties of the kernel matrices M and N . We summarize this result as follows. Theorem 10.11. Let {Ak} be the set of eigenvalues of the integral equations ( 10.9.9) and {+k(t)), {xk(t ) } be the complete sets of eigenfunctions of the kernels M(u, t ) and its transposed M*(o, t). Let { 4 0 i ( t ) } and (xoi(t)} be the sets of nontrivial solutions of the integral equations (10.9.12). If the solvability condition ( 10.9.13) is satisfied for each xoi(t),then optimal initial functions +O(t) are given by the formula (10.9.18), where the coefficients are defined by (10.9.15) and $ ( t ) is a linear combination of the functions + O i ( t ) .

10.10. A Necessary Condition for Optimality. Theorem of Neustadt

I n Section 10.7 we have established the existence of a vector t,ho such that $O . w(TO) maximizes the function t,ho * Q for Q E T(TO). Consider now the vector function u A t ) defined by u,.(t)

=z

sgn[$N(t, TO)B(t)],

t

E

I [ t o , To],

(10.10.1)

where $ is an n-dimensional unit vector:

ll*ll

=

(10.10.2)

1.

Obviously, more than one $ may determine the same ul,(t). Let an optimal initial function. It is clear that u,..(t)

+O(t)

be

(10.10.3)

= u"(t).

By the definitions of uu,(t)and + O ( t ) and the unique maximum condition, we have

*

'

Q(T",$0, u ) < II, . sz("",

$0,

uJ

(10.10.4)

r(

for all Q( TO, + O , u ) E TO), Q( To,+O, u ) # Q( TO, +O, uv). Consider the vectors $ for which (lO.lO.5)

255

10.10. Theorem of Neustadt and define

qt,+) = where

+ * L ( t$),,

L(t, $4= Q(t,+O, u v ( t ) being

Uyr) -

(10.10.6)

4th

(10.10.7)

the extremal functions defined above. Let KObe the convex set of vectors defined as follows:

+

+ +

KO= {#I I + . 52

< II, . w(T0), Q E I'(To)}.

(10.10.8)

Clearly for E K O ,V(To,+) = 0. Suppose now that V ( t ,+) is strictly increasing at t = To for every E KO. If 4 KOwe have V(TO, 4) > 0 according to the definition of the set K O . On the other hand, we have V ( t o ,+) < 0 by (10.10.5). Therefore, there exists some unique T(+) such that

+

V?'(+), +) = 0

(10.10.9)

+

+

for t in a neighborhood of T o and in a neighborhood of K O .If E KO then T(+)= TO, and if + $ K Othen T(+) < TO. Thus, we obtain the following extension of a local maximum principle, due to L. W. Neustadt [I], which gives a necessary condition for optimality. Theorem 10.12. Let Tobe the minimum time at which any x(t, t o ,4, u), given by a normal system ( 10.10.1) and satisfying the initial condition (10.1.6), can reach the target z ( t ) ; if for every E KOthe function V ( t ,+) dejined by (10.10.6) is strictly increasing with t at t = TO, then for $ in a neighborhood of K O ,and t in a neighborhood of TO, the vectors E KO maximize the time for which

+

+

(G . Q(t, 4 0 ,

Uw) = (CI

. zo(t).

(10.10.10)

This theorem, which is very close to Pontryagin's maximum principle (see L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko [l] and G . L. HaratiSvili [l]), reduces the optimal time problem t o finding the maximum of the function T($). T o find this maximum, we shall, following the general lines of the method due to Neustadt, show first that the function t = T ( $ ) possesses continuous partial derivatives. For this purpose, since t = T(+) is defined implicitly by the equation V ( t ,+) = 0, we shall show that aV/at and aV/la~,L~, i = 1, ..., n, exist and are continuous, and if aV/at # 0, we have

-a+,- - TavI Tav aT -

(2

. = 1, ..., n).

(10.10.11)

256

X. TIME OPTIMAL PROBLEM

Put

where Z(a, t') = N(a, t)B(u).

(10.10.13)

Denote the components of Z(u, t ) by zij(o, t ) and the components of by t,hjG (i = I , ..., n ; j = 1, ..., r ; K = I , ..., n). Hence

i,h

I t can be easily seen that

_ az - e,* aJJ,

-

. f , Z ( u , t)sgn[# . Z(U,t ) ] do,

(10.10.15)

where ei* is a row vector which is defined by

aij being Kronecker's

delta.

Further

a

-/J:t,h 84,

. M(u, t)Co(u)d o / = e,* . f o M ( o ,t)C0(o)do.

(10.10.17)

n

Therefore aV ~

= e,* . L(t,

4).

(10.10.18)

a+i

Similarly, we have (10.10.19) Let us now calculate &/at. Put

Clearly (10.10.21)

257

10.10. Theorem of Neustadt

has measure zero of every 6 such that the set

has a measure less than

E.

4.

Hence for any

E

>0

there is a positive

Let (10.10.24)

Since ~

t , 4) is continuous and continuously differentiable,

~ ( 0 ,

a = max l0
Choose d t such that I A t A S j = At I qj(t

+ 0 At, t +

258

X. TIME OPTIMAL PROBLEM

Since the last integral is less, in absolute value, than 1 A t

\BE,

we have

therefore

and, since 2 ( u , t ) = N(u, t)B(o) and N ( t , t ) = I , the identity matrix, we obtain s~(t,tfr) ___ =

at

as, &

3=1

=

4

’

B(4 sgn[%

’

a q U ,t )

441 + J 7 sgn[+ . Z(%41 do. t

20

(10.10.28)

Consequently

=

P(t, 4).

If P(T($),#)

=

(10.10.29)

Q(#) # 0, from (10.10.18) and (10.10.29), we obtain (10.10.30)

and (10.10.31)

Clearly the vector function L(t, #) is continuous in both t and be a parameter and consider the differential equation

!dr

= Q(+) . VZ’ =

-L(T(+), +).

4. Let r (10.10.32)

Since L( T(#),$) is continuous, this equation has a solution satisfying a given initial condition. If Q(#) # 0, then dT(#)ldr exists, and if Q($) > 0, we have .~

dr

-

= Q(+)

. ( V T . V T ) > 0.

(10.10.33)

259

10.10. Theorem of Neustadt

If + $ K O , V T # 0 and therefore, with the assumption Q(+) > 0, dT(+)/dr > 0. If Q(+) = 0, dT(+)/dr does not exist. But, by the formula (lO.lO.lS), V V is, for fixed t , defined, and

Hence, V ( t ,+ ( r ) ) is monotonically decreasing with r, or T(+(Y))is increasing with I , so that, for all $ KO, T(+(Y)) is increasing with r . Let us denote by 9 the domain of the function T(+).This function is defined for all for which the inequality (10.10.5) is satisfied. Consider now the solution + ( r ) of the differential equation (10.10.32) which satisfies the initial condition

+

+

#(To)

= 40

(10.10.35)

3

where E 9. Note that $(r) E 9 for all values of r then for some r* > ro , leaves 9,

4(r*) . [W" , 6 , 4- w(t0)l

> ro . For,

= 0.

if +(r)

(10.10.36)

Hence V(T(ICl(r*), Il,(r*)))

=

4(r*) . [ Q ( W ( Y * ) ) ,

4 3 4 --

47'(?&*)))1

= 0.

Therefore, by (10.10.9), we have T(+(r*)) = t o . But this is impossible, since T(+(r*)) > T(+(r,)) 3 to if Y > y o . It can be easily seen that the norm 11 4 /I of the solution of the differential equation (10.10.32) is constant. Let +(r) be the solution of (10.10.32) such that $(ro) = €9. If +(Y) approaches a limit as r -+ 03, then this limit is in 9. T h u s T(4) attains its maximum. Note that, on the boundary of 9, T(+) = t o .

CHAPTER XI

4b Optimal Pursuit Strategy

In the previous chapter we have investigated the time optimal problem with a given target z(t). In this chapter we shall consider the case in which the target is also a controlled system with time delay. Pursuit problem for systems characterized by ordinary differential equations has been studied by D. L. Kelendzheridze [I] and for systems described by difference-differential equations by M. N. O@z+oi-eli [3]. 11.l.Formulation of the Pursuit Problem

Consider two control systems X and 2, given, in the n-dimensional phase space, by linear delay-differential equations of the form

and

where ci and dj (i = 0, 1, ..., m and j such that 0

= co < c,

< ... < c,

and 0

=

0, 1, ..., m*)are given constants

= do < d,

< ... < d,*,

(11.1.3)

K(T,t ) , H(7, t), Ai(t), and B,(t) (i = 0, 1, ..., m ;j = 0, 1, ..., m*) are given n x n continuous matrix functions, A ( t ) is a given continuous n x r matrix, B(t) is a given continuous n x s matrix, x ( t ) and z ( t ) are

n-dimensional vectors which describe the states of the control systems 260

11.1. Formulation of the Pursuit Problem

26 1

X and 2, respectively, at time t, u(t) is an r-dimensional column vector controlling the motion of the system X , and v ( t ) is an s-dimensional column vector controlling the motion of the system Z. T h e components of u and v will be denoted by u1 , ..., u, and v u l ,..., v,, respectively. Let R, and R, be the control regions of the system X and Z, respectively. We assume that R, is a compact and convex subset of E'containing the origin, and R, is a compact and convex subset of EScontaining the origin. Let U be a set of r-dimensional vector functions u(t), piecewise continuous for t >, to with range in R, , and V be a set of s-dimensional vector functions v(t),piecewise continuous for t 3 to* with range in R, . We assume that U and V are compact, convex, and contain the null vectors. u(t) E U and v ( t ) E V are admissible controls for the systems X and 2, respectively. Let (9 be a compact and convex subset of the set of all n-dimensional vector functions +(t), continuous in the initial interval I[a, to] having the property (1 I . 1.4) +(to> = xo ?

where xo is given. T h e elements of the set (9 are admissible initial functions for the system X . Similarly, we shall denote by Y a compact, convex subset of all n-dimensional vector functions $(t), continuous in the initial interval I[a*, to*] and having the property *(to*)

(1 1.1 .5)

= zo7

where xo is given. Function $(t) which belong to the set Y are admissible initial functions for the system 2. A pair (4, u} is called an admissible pair for the system X if 4 E @, u E U . Similarly, {$, v } is an admissiblepair for the system Z if $EY,v E V . A system @, u ; $, v> with 4 E @, u E U , $ E Y,and v E V is called a strategy. We shall denote by x ( t ) = x(t, t o , 4, u) and z ( t ) = z ( t , to*, $, v ) the solutions of the systems ( 1 I . I . 1 ) and (1 1.1.2), respectively, whose meanings are obvious. T h e system X will be called the pursuing system and the system 2 the pursued system. For an arbitrary admissible pair {$, v } let us assume that there exists an admissible pair {#I, u} such that the trajectories x(t, to , 4, u ) and z(t, to*,$, v ) of ( 1 1.1.1) and (1 1.1.2) corresponding to the controls u, v , and initial functions 4, $, respectively, satisfy the equation .v( T . t o

,4, u )

= (.

T , to*,

9, v)

( 11.1.6)

262

X I . OPTIMAL PURSUIT STRATEGY

for some

1' > max{t, , to*}

and

, 3 , U) f

~ ( tt o,

z ( t , to*,

#, V)

( 11.1.7) for

t

< 1'.

( 1 1.1.8)

T h e quantity T depends on the chosen controls u ( t ) and v ( t ) and the chosen initial conditions +(t) and # ( t ) ; therefore we write T = T(4,u ; $, v). This time T will be called pursuit time. If an admissible pair {$, v> for the pursued system 2 is chosen, the pursuing system X should be controlled in such a manner that the corresponding pursuit time T(+,u ; #, v) will assume its minimal value. Denote it by T 4 , v= min T(+,u ;,)I v). (11. I .9) &O,UtU T h e system 2 should choose an admissible pair {+, v} which maximizes the quantity T4,t..This maximum will be denoted by T o = * Emax min T ( 4 ,u ; + , v ) . P , V i V &@,UEU

( 1 1.1.10)

We now formulate the optimal pursuit problem. Optimal Pursuit ProbZem. Find admissible controls u(t), ~ ( t )and , admissible initial functions +(t),+(t) for which the corresponding pursuit time T(+,u; $, u) satisfies T ( +,u ; #,v)

=

7'0.

(1 1.1.1 1)

A strategy {+, u ; $, u} for which the equality (1 1.1.1 1) holds is called optimal strategy. 11.2. The Functionals 8 ( t , a,u ) , O ( t , Y, v ) , and the Sets C(t), D(t)

Let M,(o, t ) and M,(o, t ) be the kernel matrices of the first kind and N1(7,t ) and N2(7,t ) be the kernel matrices of the second kind of the systems ( I 1.1.1) and ( 1 1.1.2), respectively. Consider the functionals Q ( t , 4 ,u )

and

=

fu

&i',(o,t)+(a) do

+ j t N1(7,~ ) A ( T ) u (d7T ) tU

(1 1.2.1)

263

11.3. Existence of Optimal Strategies Clearly x(t.t0,4,u)

=Q(t,4,~),

~ ( t , t , * , $ , v= ) @(t,#,~).

(11.2.3)

Thus, SZ and 0 are absolutely continuous with respect to t . Equation (1 1. I .6) is equivalent to Q(T,d, 4

=

@(T,*, v),

(1 1.2.4)

where T = T(+,u ; $, v) is the pursuit time, defined in Section 11.1. By its definition T is single-valued. We now define the sets C(t) = {JW, I 4 E @,

u E U},

( 1 1.2.5)

D ( t ) = { @ ( t ,4, v) I

v E V}.

(1 1.2.6)

49.)

and

4E Y,

In the remainder of the discussion, we shall need the following properties of the sets C ( t ) and D ( t ) , proved in Chapter X. (i)

C ( t ) and D ( t ) are compact and convex.

(ii) If 0 and P are the sets of all bang-bang control functions, and if U and V are defined by

u=

v= then

I I u(t>

I u , ( t ) I < I , i = I , ...,r ,

i l1 v(t)

v3(t) 1

< 1,

t >, t o : ,

( 1 1.2.7) j

=

1 , ..., s, t 2 to*

C(t) = {Q(t,4, .”) \ 4 E @,

D(t) = { @ ( t ,$, 0) I

E

1E

O},

Y , B E V}.

!’ ( 1 1.2.8)

(iii) If SZ is an interior point of C ( t ) , then there exists an E > 0 such that Nt(Q) C C(T) for all 7 in I ( t - E , t ] , where N,(SZ) is a neighborhood of radius E.

It should be noticed that Theorem 10.5 is still valid for T(+,u ; $, v). We assume that the conditions of Theorem 10.5 are satisfied throughout this chapter. Hence T(+,u ; $, v) is continuous in all its arguments. 11.3. Existence of Optimal Strategies

We now prove the following existence theorem which is an extension of that due to D. L. Kelendzheridze [ l ] (see also the book by L. S.

264

X I . OPTIMAL PURSIJIT STRATEGY

Pontryagin, V. C . Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko [I]) and that due t o X. iu. Oguztoreli [3].

.

Theorem 11.l If .for an arbitrary admissible pair {+, v} there exists an admissible pair {$, u>,such that x ( t , t o ,+, u) = z(t, to*, $, z)), then there exists two admissible pairs {$O, uo} and { + O , vo} which are optimal; that is, TO = 7'(&",24"; $0, no>, (1 1.3.1)

where T o is dejined by (1 1.1.10). Proof.

By hypothesis the set

O ) l = { / l l X ( T , t O , ~ , U ) = ~ ( T , t ~ * , # ,&nE) @ , ,

# E Y ,

U E U ,

U E V }

(1 1.3.2) is not empty. Let u s choose arbitrarily an admissible pair {$*, z)*} and consider the following subset of % l : !Ill* = {T* = T(&,u ; #*, .*}

I&

E

a),

u E U}.

By hypothesis 9Jl* is not empty. he the greatest lower bound of all T* Let TbLtl,* T,; Ir*

---

inf

&@,UEU

E

( I 1.3.3)

'91t*:

T(&,u ; #*, v * ) .

( 1 1.3.4)

By definition we have Q ( T * , & , u )= O ( T * , # * , W * ) E C ( T * ) ,

Let the sequence Ti*E W * , i

=

T*EYX*.

(1 1.3.5)

I , 2, 3, ..., be selected so that

lim l',*= Tp * o * .

(1 1.3.6)

Z'oc.

Consider now the sequences ( Q ( T ,, &i,

UZ)),

{Q( T,.*,* ,@, UZ)},

2 =

1 , 2, 3, ...,

where +i = +i(t) and ui = ui(t) are admissible. We can easily show that for Ti* - TW*"* <6

(1 1.3.7)

(IQ ( T f, c", ui) - Q(?l",*"* , f , ui) j / < E,

(1 1.3.8)

we have

1 1.3. Existence of Optimal Strategies where

6

265

is an arbitrary small positive number and

with

where t is sufficiently large. Since the set C(T,*?,*)is compact, we can extract subsequences (Q”(f)],

{u”(t)},

R

2, 3, ...

= 1,

from the sequences { @ ( t ) ) and { u i ( t ) ) so that they converge to the functions do(t) E @ and uo E U , respectively. Therefore O(T,*,,* , 4*, v*)

= Q(TJ**,* ,+a,

u0)

E

c(T,*v*),

( 1 1.3.11)

where T,*,,,* is defined by (1 1.3.4), $* E Y, v* E V being taken arbitrarily so that T* = T(4,u; $*, v * ) E %t*. Obviously T,. * 2)

z=

T ( p ,u”; **, v*).

Consider now the set ‘1110

=

{rr’,, 1

*

E Y,

2)E

V}.

E

mO:

Let T o be the least upper bound of all T,, 7’0

=

sup 7’(9)”, ,€Y,V E V

( 1 1.3.12)

uo; *, v) = sup

l€Y,*€V

( 11.3.13)

l;,,

(1 1.3.14)

which is equal to T o defined by (1 1. l . 10). This optimal time T o may be finite or infinite according as the set Wo is bounded or unbounded. Consider first the case in which T ois finite. Let the sequence T,O E %lo, j = 1, 2, 3 , ..., be selected so that lim Tj0= TO. 3 *cc

Consider the sequences {@(T,”,p,v’)},

(@(I’”,p,v’)},

j

=

1 , 2, 3, ...,

PURSUIT STRATEGY

are admissible. Let % * = sup I1 4 11, Y

t ,

(11.3.15) where t is sufficiently large. Then, for

\ 7'" - T,U j < 6",

( 1 1.3.16)

where S* = min I

€ I 3m,*m,*

'

we have

__

€

+,

11 @(TO,

-

1

€

'

3(TO - t,*)m,*m,*

3(t0* - c(*)m2*m3* ' (1 1.3.17)

+J,4 11 < €.

qr;,

(11.3.18)

Since the set D ( t ) is compact, we can extract subsequences

(+.(tl),

k

{VJk(t)},

=

I , 2, 3, ...

(1 1.3.19)

from the sequences {$'(t)) and (v'(t)} so that they converge to the functions (Cln(t)E Y and vo(t)E V , respectively. Therefore TO

=

T ( p ,uo; $0,

and Q( T",40, UO) = @(TO,

(1 1.3.20)

VO)

p , z q E D(TO).

Consider now the case T o = co. Let the sequence Tko,k be so chosen that

TI;"+ co monotonically.

(11.3.21) =

1,2, 3, ..., (1 1.3.22)

Since Tk0E mo,we have

@(I',",$, v) = Q(Th",#",

k

UO),

=

1 , 2, 3, ...,

(1 1.3.23)

for some ,,!I E Y and u E V , for each k. Let us select an admissible pair {@, vk} which satisfies ( I I .3.23). T h u s

TI;''

=

T(+O,u O ; I,P,a'),

k

=

I , 2, 3, ... .

(1 1.3.24)

1 1.4. Properties of Optimal Strategies

267

Consider the sequences {$k(t)}and {vk(t)}.Since t / ~ ~E( tY, ) vk(t)E V and since the sets Y and V are compact, we can extract subsequences r

{v"(t)>,

{fqt)},

=

1, 2, 3,

... ,

(1 1.3.25)

from the sets { $ k ( t ) } and { v k ( t ) }so that they converge to the functions t,hO(t)E Y and vo(t)E V , respectively. Hence T(+O,uo; $O, vo) = 00. 11.4. Properties of Optimal Strategies

I n all the theorems which we shall prove in this and in the next section we shall assume, without specifying it each time, that the optimal pursuit time T o is finite and that the convex and compact set C( TO) has interior points. Let A be a capture point at which the system X (with an admissible pair {+, u } ) encounters the system 2 (with an admissible pair {$, v}) at time T = T(+,u ; 16, v). Let Ao be the optimal capture point which corresponds to the optimal strategy {+O, uo; $0, vo} and the time t = TO. Hence A

= Q(T,4, u ) =

@(T,4,v) and

A0 =

Q ( P+O,,

u") =

@(To,(cIo, vo). (1 1.4.1)

If $ ( t ) and v ( t ) , selected arbitrarily from the sets Y and V , respectively, are kept fixed, the corresponding optimal policy of the system X will be described by = +O(t) and u = uO(t) and the capture will occur at t = T$?,. Therefore

+

uO) = @(T,,

Q( ?'*,

7

4,v)

(1 1.4.2)

and Q(t,+O,

uo) # @(t,4,v)

t

for

< T,,

,

(1 1.4.3)

where T,,

=

min T(4, u ; 4,w) = T(+O,uo; 4,v).

&,P.UEU

( 1 1.4.4)

By Theorems 10.7 and 10.8, the point A,?, = Q(T,,,

+

7

+O,

(11.4.5)

u"),

which is the point A for u = uo(t), = +O(t), v = v ( t ) , and $ = $(t), is a boundary point of the set C(T,,,) and there exists a unit vector 7 = ( v l ,..., qn) in En such that 7 ' Q(T,,

9

< 77 . qT,,

7

4O3

uO)

(11.4.6)

268

X I . OPTIMAL PURSUIT STRATEGY

for all Q( T,, , +, u ) E C( T,v). Clearly, the vector depends upon the choice of the vectors $ ( t ) E Y and v ( t )E V . By the remark at the end of Section 10.2, Theorem 10.5 is still valid. That is, the functional T,@,varies continuously when the functions $ ( t ) and v(t) vary continuously in Y and V , respectively. Since Q(t, +O, u0) is absolutely continuous in t , if the admissible pair {$, ZI> vary continuously, the point A,,, will vary continuously in such a manner that it will always be a boundary point of the corresponding set C( T4@,). Let S ( t ) denote the boundary of the set C(t). Hence A,v E S(T,J

(11.4.7)

for each admissible pair {+, v}. Let {+j, vj>,j = 1, 2, 3, ..., be a sequence of admissible pairs such that v'(t) + V O ( t )

1/5'(t) + +O(t),

uniformly, where {$O, wo} is the optimal pair for the system 2. Consider the sequence of times {I, I

7

=

T * I . ~=, T(4', u"; 41, v')},

which corresponds to the sequence

{$j,

vj},

j

=

(1 1.4.8)

1, 2, 3, ... . Since

1 O = max T,, = T(+O,?to; $0, no),

I .

*EY,VEY

( 1 1.4.9)

taking a subsequence, if necessary, we may assume that

I; <

and

lim I -m T j = To.

(1 1.4.10)

Consider now the point Ao defined by (1 1.4.1) and the sequence of points

( / I , = n ( ? ' , , ~ o , ~ o ) = - - ( T , , ~ , ~j l=) t1, , 2 , 3,....

(11.4.11)

Since the functionals Q(t,4, u ) , @(t,$, ZI),and T(+,u ; $, v) are continuous in all their arguments, we have limAj l'io

=

/lo.

( 1 1.4.12)

We now prove the following. Theorem 11.2.

The point /lo is a boundary point of the set C(To).

11.4. Properties of Optimal Strategies

269

Proof. Suppose, on the contrary, that Ao is an interior point of the set C(T0). Then, by Property (iii) of Section 11.2, there exists an > 0 such that Nt(Ao)C C(T) for all T in the interval T o - E < T < TO, where N,(AO) is an E neighborhood of Ao. T h e continuity of @ ( t , #O, vo) at t = TO implies that there exists a 6 > 0 such that @ ( t , )Go, vo) E N,(Ao) for all t in TO - E < t < TO. Let 2y = min{c, 6). Then

@(TO- y , lp,v") E N,(AO) c C( TO - y).

But this is impossible, since TO - y < To,and Tois the minimum value of t such that @ ( t , )Go, vo) E C ( t ) . Making use of Theorem 11.2 and Property (i) of Section 11.2, the first part of the following theorem, which is an extension of Kelendzheridze's main theorem can be easily proved. There exists a unit vector qo = (qIo, ..., qno) in En,

Theorem 11.3. such that (9

7)"

.52( TO,f$, u ) < 70 . Q( T",+",240)

for all admissible pairs @, u>, sZ( TO, 4, u ) # sZ( TO, #JO, uo);

< 7" . @(TO,p,vo)

. q z - 0 , +, ).

(ii)

for all admissible pairs {)G,,v} which are suficiently near to the

optimalpair {#O, vo} and @(To,#, v) E C( TO); 7)o

(iii)

. W(A0,v"(I'O),7.0) < 7)" . V(A0,U " ( T O ) , TO),

where nl

V(x(t),~ ( t )t ), = z A i ( t ) x ( t - c,) 1=0

+ f K ( T ,t ) . % ( ~d~) + A(t)u(t)

(11.4.13)

fo

and 111

*

W(z(t),~ ( t )t ), = C B , ( t ) z ( t- dj) j=O

+

tt

H ( T , ~ ) z ( Td~ )

+ B(t)v(t). (1 1.4.14)

Proof. Consider the union 2 of all the sets C ( t ) for t >, t o . 2 is an open set and (1 1.4.1 5 ) A0 E C(TO) c z.

Therefore, we can find a number t* @ ( t ,9",v") E z,

< T o such that i*


(11.4.16)

270

XI. OPTIMAL PURSUIT STRATEGY

For every t in I[t*, TO] let the relation

7

=

~ ( tbe) the number which is defined by

q t , $0,

v0) E S(7),

where S(T) is the boundary of the set C(7). Since the elements of the closure of C(t) are continuous in all the arguments, S ( t ) varies continuously with t. From the continuity of the function @(t,t,ho, vo), we see that T = 7 ( t ) is continuous in the interval I[t*, TO]. We can easily verify that ~ ( t> ) t for t < TO, .(To) = To. (11.4.17) Since @(t,Go, vo) is on the boundary S ( T ) of the convex body C(T), there exists a support hyperplane 17, to C(T)at @(t,t,ho, vo). Let 7,be the unit vector orthogonal to this support hyperplane which is directed from @(t,Go, no) into the half-space which does not contain the convex body C(7). For every point Q(t, 4,u ) E C(7), the vector Q ( t ,4,u ) - @(t,$0, wo) is directed into the half-space which contains C(7). Hence (Q(t,+,

4

- @(t,p,@),v7)

<0

(1 1.4.18)

for Q(t, 4,u ) E C(7). Consider now the sequence of times {T,};defined by ( 1 1.4.8) which has the property ( 1 1.4.10). If j is sufficiently large, t* < Ti< TO. It follows, from (1 I .4.10), (1 1.4.17), and the continuity of 7 ( t ) , that lim 7 ( T j )= TO.

(11.4.19)

7'W

Since in the n-dimensional Euclidean space the unit sphere is compact, there exists a convergent subsequence of the sequence of unit vectors qTcT,,which converges to the unit vector r l T O which is orthogonal to the support hyperplane 17," to the set C( To)at the boundary point Ao, directed into the half-space which does not contain the convex body C( TO). If Q(t, 4,u ) is an interior point of the set C(To),Q(t, 4, u ) E C(T,) for sufficiently large j , by Property (iii), Section 11.2. From ( I 1.4.18) we see that

(W,4, u ) - @(Tj, p,w"), 17,,T,J < 0.

Taking the relations (1 I .4.1) and ( 1 1.4.19) and the continuity of 7 ( t ) into account, for j + 00 we obtain (Q(t,4, u )

@(TO,+",WO), q p )

-~

< 0.

(1 1.4.20)

1 I .4. Properties of Optimal Strategies

27 I

(9,v }

be an admissible pair for the pursued system 2. Then be captured by the pursuing system X with optimal policy at t = T,, < TO. Suppose that the pair {$, v} is sufficiently near to the optimal pair {$O, vo) and is such that @(T+?., $, v) is an interior point of the set C( TO). Then, by Property (iii), Section 11.2, and by (1 1.4. lo), @( T& , 4, v ) is an interior point of the sets C( T,) for all largej. Therefore @(Ti $9 v ) E C(Tj). Consider now the points A j , j = 1 , 2 , 3, ..., defined by (11.4.11). Since A j E S( T j ) ,it follows that @( T, , $ j , v i ) E S( Ti). Since @( T j , $, v ) is an interior point of C( T j ) ,and since @(T j , @,vf) is on the boundary of C ( T j ) ,the vector @ ( T i ,$, v ) - @ ( T i ,+!d, d), which passes through A j , is directed into the half-space which contains the convex body C( T j ) .Consequently Let

@(t, 4, v ) can

9

(@(Tj

7 $9

v) - @(Tjv

Pv v J ) +? T , )

<0

( 1 1.4.21)

for sufficiently large j . Passing to the limit a s j -+ M in (1 1.4.22), we obtain (@(T", #, v) - @(T",#', vo), ? r u )

< 0.

(1 1.4.22)

Using the linearity of the functional @(t,4, v), we may write A@

=

O(T", *> v)

-

@(TO,9 0 , VU)

@( T", # - lp,v - c".

=

Then the inequality (1 1.4.23) can be written in the form ( A @ , 7)p)< 0.

( 1 1.4.23)

T h e formula (1 1.4.23) is true for every admissible pair {$, v } which is sufficiently near to the optimal pair {+O, vo] and is such that @( TO, $, v ) E C( TO). Consider now the function Ij(t)

(Q(t,

d", uo) - @ ( t , #", v"), Ij,(rj)).

( I 1.4.24)

Since Q(t, 4 O , uo) E C ( t ) for every t , it follows from ( 1 1.4.18) that &( T j ) 0. Since Q( TO, 4 O , uo) = @(To,$O, vo), we have &(To)= 0. Since uO(t) and ao(t) are piecewise continuous and since the kernel matrices M , , M , , N l , and N , are continuously differentiable in the interval T j < t To if j is sufficiently large, the function Cj(t) has a continuous derivative in the interval Ti t TO. Hence (,'(A,) >, 0 for some hj such that Ti< A, < TO. Therefore

<

<

< <

1.im &'(Aj) 3 -a,

> 0.

(1 1.4.25)

272

XI. OPTIMAL PURSUIT STRATEGY

(1 I .4.26)

where PU =

W(A0,2,0(1'0), TO) - V(A0,U O ( P ) , TO),

( 1 1.4.27)

I' and W being defined by (1 1.4.13) and ( 11.4.14). Let C T ( T o )be the convex hull of C(To) and w , and let K* be the convex cone, with vertex /lo, of the vectors A 0 which emanate from Ao. Since, by the above argument, the convex set C ( T o ) and the vectors A 0 and w all lie on one side of the support hyperplane I 7 p t o the convex body C*(To)at /lo, the set C*(To) and the vectors - A @ lie in two opposite closed half-spaces defined by 17,". Hence, the vector - A @ , which ernanates from AO, does not pass through interior points of the convex body C*( TO). T h e vectors - A @ form a convex cone K which is symmetric t o K*, with respect to /lo. Therefore, K does not intersect the interior of the convex body C*(To).Since C*(To)has interior points (because C ( T o ) has), C*(To)and K are separated by a hyperplane no.Therefore the convex hull C*( TO) and the convex cone K* lie in one closed half-space defined by no,and the cone K is contained in the other. Let qo be the unit vector which emanates from /lo, is orthogonal t o no,and is directed into the half-space which contains K . Thus, for this vector qo the relations ( 1 1.4.20), ( 1 1.4.23), and (1 1.4.26) are satisfied-namely: (i) (Q(t, 4, u ) - Q ( T o ,P,uo),qo) (ii) ( A @ , qo) < 0 for A 0 E K*; (iii) (w,T O ) .( 0.

< 0 for Q(t,4, u ) E C ( T o ) ;

This completes the proof of Theorem 1 1.3.

11.5. Optimal Controls in a Particular Case

Assume that the systems (11.1.1) and (1 1.1.2) are normal and the matrices N , , N , , A, and B are analytic. If the control regions R,, and R, are unit cubes in E' and E", respectively, that is 1,

i-= 1 ,..., Y !i , (1 1.5.1)

---

I

11.5. Optimal Controls in a Particular Case

273

we can easily show, as in Section 10.8, that optimal control functions uo(t) and vo(t)are of the form uu(t) = sgn[qU. N l ( t , To)A(t)l,

to

and vo(t) = sgn[To . N2(t,?'O)B(t)],

to*

< t < TO

(1 1.5.2)

< t < TO.

(11.5.3)

Thus, if the vector qo is known, the optimal controls are completely determined. T h e analyticity of the matrices N l , N , , A , and B assure the piecewise continuity of uo(t)and vo(t). Let + O ( t ) and $O(t) be optimal initial functions for the systems X and 2. Let u,,(t) and v,,(t) be defined by ( I 1.5.2) and (1 I .5.3), respectively, with 7 replacing T O ; that is, u,,(t) = sgn[T

. N,(t, TO)A(t)],

v,,(t) = sgn[T . N2(t, TO)B(t)].

( I 1.5.4)

Clearly, more than one 7 may determine the same strategy and u,p(t) = u"(t),

v&)

= u"t).

( I 1.5.5)

Note that uV(t)and v,,(t) depend continuously upon 7, disregarding sets of measure zero. Consequently the functionals Q(t,do,a,,), @(t,$0, v,,), and T(c#O, u, ;$O, v,,) are continuous in 7 a3 well as in t. If T(c$O,a, ; $0, v,,) = T o for some vector 7, the vector 71 and determine the same optimal strategy {+O, uo;$O, vo}. We now prove the following theorem. Theorem 11.4. There exist two positive numbers y and S such that Q(t,+O, u,,) and @(t,$O, v,,) are boundary points of the sets C( TO)for all t in T o - y < t T o and for all 7 in / / 7 - 7 O / / < 6 , provided Q(t, +O, u,,) E C ( T o ) and @(t,$ O , TI,,) E C(To).

<

Suppose that @(t,$O, v,,) is an interior point of the set C(T0) for some t and some 7. Hence, by Property (iii), Section 11.2, there exists an E > 0 such that N,(@(t,$O, v,,)) C C ( T ) for all T in TO - E < T To, where N , ( @ ) is an E neighborhood of @(t,$O, v,,). Consequently (I I S.6) I! @ ( t , dJo, v,) - A"!I 2 Proof.

<

€7

since Ao lies on the boundary of C( TO).From the continuity of @(t,$O, 8), at t = T o and 7 = 7 O and since Ao = @(To,~,!JO, z+), we can find two positive numbers y and 6 such that

11 @(t,4 0 , vv) - A" /I < €

( 1 1.5.7)

274,

XI. OPTIMAL PURSUIT STRATEGY

for all t in T o - y <' t < To and for all 7 in / / 7 - yo 11 < 6. Therefore @(t,$ O , ZJJ cannot be an interior point of C ( T o )for T o - y < t TO and /I 7 - qo j, <' 6, because, in the contrary case, the inequality ( 1 1.5.6) must be satisfied, which contradicts the inequality (1 1.5.7). Since @(t,$O, uq) E C ( T o ) ,@(t,$O, v,,) is a boundary point of C(To). It can similarly be shown that Q(t, +O, u,) lies on the boundary of the set C(To). Consider now the time T,, = T(+O,u, ; $O, v,,) at which Q(t, do, us) encounters @(t,$O, v,,). As we mentioned above, T,, is continuous for almost all 7. Accordingly, for almost all r ) in 11 r ) - r ) O 11 < 6, we have T o - y < T,, TO. Suppose now that y and 6 satisfy the conditions of Theorem 11.4. Then, Q ( t ,+O, u,,) and @(t,$O, v,) lie on the boundary S(To)of the set C( TO) and coincide for t = T , . Let So be the portion of S( To)described by the points Q ( t , + O , u,,) and @(t,$O, v,) for To - y < t T o and j / -~ i/ < 6. Consider the convex set H o of all vectors 7, orthogonal to the support hyperplanes Il to S o and directed into the half-spaces (defined by these hyperplanes n),which do not contain the convex body C(To).Clearly qo E HO. Thus,

<

<

<

77 . Q(t,4,

and for all 7 E HO.

Q(t,

+, u )

4 < 17

77 . @ ( t .#, v) E

'

( 1 1.5.8)

Q(T, , 4 O > u,)

< 77 . @(T,,,+O,

( 1 1.5.9)

u,)

C( TO), @(t,$, v) E C( TO), To - y


< To, and

Define the function w&t) by zu,,(t) == W(@(f,#", v,),

uq(t),t ) --

v(Q(t,4',u,,),u,,(t),t ) ,

(1 1.5.10)

where V and W are given by (1 1.4.13) and (1 1.4.14). As in the proof of Theorem 11.3, we can easily show that (w,,(t), 77)

G0

for all 7 E HO. Consider now the function F(t, 7)defined by

( 1 1.5.1 1)

I 1.5. Optimal Controls in a Particular Case

275

From the continuity of functions Fi(t, v), i = 1, 2, 3, for all t (3max{to , to*))and for almost all 7, we can easily see that the function

F(t, 7)is continuous in t and in 7,disregarding a set of measure zero. We have also F( To,7)= 0, for 7 E HO. ( 1 1.5.1 4) Let H be the set of all vectors 7 for which F(e, 7) < 0,

(1 I 5 1 5 )

where e = max{to, to*}, and denote by H' the subset of H whose elements r] verify the inequality

F ( P ,7)< 0.

( 1 1.5.16)

By the inequalities (1 1.5.8)-( 11.5.10), H o C H . Clearly,

F(T0,7)> 0

for

7EH

--

H'.

(11.5.17)

Suppose now that F(t, 7) is strictly increasing at t = TO for every H'. From (11.5.14), (11.5.15), and ( 1 1.5.17), we see that there exists a unique T(7)such that

r] E

F ( T ( 7 ) ,7)= 0

(11.5.18)

for t in a neighborhood of To and 7 in a neighborhood of qo E H1. Clearly. if 7 E HO, T(q) = TO, and if 7 E H - H', T(7) < T o according to (1 1.5.17). 'rhus, we have the following theorem, which is an extension of a Theorem due to L. W. Neustadt [ I ] and M. N. O@ztOreli'[3]. Theorem 11.5. Suppose that the control regions R, and R,, are of the form (1 1.5.1) and the matrices N,(T, t ) , N2(7,t ) , A ( t ) and B ( t ) are analytic. Let To be the pursuit time for the systems X and 2. Then the unique control functions uo(t) and vo(t) are given by (1 1.5.2) and (1 1.5.3) with some vector 7 in some set H'. If for every 7 E H' the function F(t, 7) defined by ( I 1.5.12) is strictly increasing with t at t = TO, then for 7 in a neighborhood of H o and t in a neighborhood of T o the vectors 7 E H 1 maximize the time for which F(t, 7) = 0.

This theorem is very close to D. L. Kelendzheridze's main result when all the delays in X and 2 vanish. I t should be noticed that Theorem 11.5 only gives a necessary condition for an optimal strategy.

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[2] O n the theory of linear equations with a lagging argument and periodic coefficients. Dokl. A k a d . , i a u k S S S R 128, 882--885 (1959). [3] General solution of a linrar differential equation with perturbed argument. NuuEn. Dokl. VysS. Skoly, Fiz.-Mat. Nauki No. 1, 30-37 ( I 959). Zverkin, A . M., Karnenskii, (; A , , and Norkin, S. R. [ I ] 1;ormulation of t h e intial value problem for differential equation with negative tirne-lag. Uspvhi Mat. -\’auk 15, No. 6, 133-136 (1960). Zverkin, A. M., Kamenskii, <;. .4., Norkin, S. B., and Els’gol’ts, L. E. [ I ] Differential equations with a perturbed argument. Ruis. Math. Surveys 17, No. 2, 61-146 (1962). Zverkina, T. S. [ I ] Xpprouimate solutions of differential equations with perturbed argument and of differential equations with discontinuous right hand sides. Kept. No. I of t h e Seminar of IIitT. Eqs. with Prrturbed Argument, LIniv. D r u z h b y Narodov, pp. 7693, 1962. Zwinggi, I<, [ I ] %urn Problem d e r I‘rneuerung. RIdtter Vrrsicherungrnath. 2, 73-82 (I93 1).

Author Index Agranovich, M. S., 276 Aizerman, M. A., 276 Alaoglu, L., 276 Alander, M., 276 Alimov, Y. I., 276 Allais, M., 276 Andre, J., 138, 276 Andrew, G. M., 276 Andronov, A. A., 8, 277 Anosov, D . V., 277 Anselone, P. M., 277 Ansoff, H. I., 8, 277 Aris, R., 219, 277 Balakrishnan, A. V., 277 Bandy, M. L., 312 Barba, A. V., 277 Rarbashin, E. A , , 276, 277 Bass, R. W., 277 Bat', M. I., 277 Bateman, H., 8 , 277 Bauer, G. N., 278 Bautin, N. N., 278 Beke, E.. 278 Bellman, R., 8 , 92, 101. 112, 147, 148, 170, 219, 221, 278, 279 Beltz, M. N., 291 Benes, 1 '. E., 279 Bennett, A. A,, 279 Berkovitz, L. B., 279 Bhabha, H. J., 279 Bially, H., 279 Bilharz, H., 279 Blackwell, D., 196, 202, 279 Blaz, J., 279 Blokh, Z. Sh., 279 Hochner, S., 279 Bogner, I., 279 Bogoliobov, N., 279 Boiarinov, V. S., 280 Boltyanskii, V. G., 8, 240, 255, 264, 280, 302

Booton, R. C., 280 Border, R. F., 280 315

Borisovir, Ju. G . , 280 Horn, M., 280 Bothwell, F. E., 280 Bovsheverov, V. M., 280 Bremrner, H., 102, 310 Briggs, B. R., 291 Brownell, F. H., 280 Hruijn, N. G., 280 Bruwier, L., 280 Bryson, A. E., Jr., 281 Budak, B. M., 281 Bueckner, H. F., 277 Bulkagov, B. V., 281 Burger, E., 281 Burke, M. H., 312 Bushaw, D. W., 281 Bykov, Ya. V., 281 Callender, A , , 8, 281 Capprin, V. M., 281 Carmichael, R. D., 281 Cartwright, M., 281 Chakrabarthy, S. K., 279 Chang, Hsueh-niing, 281 Chebotarev, N. G., 281 Cheng, Sin-I., 282 Chin, Yuan-shun, 282 Choksy, N. H., 8, 282 Chyung, D., 282 Cistyakov, N.I., 8, 282 Collatz, L., 282 Colombo, G., 282 Cooke, K. L., 8, 92, 101, 112, 147, 148, 278, 282 Covezzoli, P., 282 Crocco, L., 282 Cunningharn, W. J., 282 Danskin, J. M., 8, 282 Danskin, J. M., Jr., 278 Dantinne, N., 282 Datko, R., 283 Dehousse, L., 283 Denham, W. F., 28/

316

AUTHOR INDEX

Desoer, C. A , , 283 Dickson, I>. G., 283 I k o n , 4. C . , 283 Dlotko, I)., 283 Dolgolenko, Y u . A., 283 I l o s s , S., 283 I h y f u s , S., 170, 219, 278, 281 Driver, R. I)., 7, 13, 28, 29, 81, 83, 283 Drozdovich, V. N., 3 / 2 Ihbois-Violette, 1’. I,., 283 Duhoviekii, A . J a . , 283 Dunford, N., 25, 150, 193, 284 I h o r e t z k i , A , , 195, 197, 284 I.:gglcston, H.G., 177, 284 Egorov, J u . V., 284 Ehrrnann. H., 284 Kl’sgol’ts, I,.E., 8 , 284, 299, 314 Engcl, P. A . , 30/ Engen, W . K., 280, 300 I-sipovich, 1.:. &I., 284 Fall>, P. L., 285 Fel’dbaurn, A. A , , 285 Feller, W., 285 I:tnyes, 285 Fett, G. H., 287 Fiallio. (;. M. , 296

’r.,

Filippo\, A . Fite, B., 285

F., 192, 193, 285

FJVIdSt>ltj,J. E., 285 Fliimant, I’., 285 Fleishman, n. A., 285 Fodruk, V. I., 285 Ford, T., 286 Franklin, J., 286 FrGchct, >I., 113, 286 Friant, I). R., 287 Frid, 1. A., 286 Frisch, I<., 286 F U J l i , s., 286 l7uIler, A. T., 286 Fuller, u‘. R., 286 (;abasov. It., 286 Gaisarjan, S. S., 286 (;anikrelidze, R. V., 8 , 209, 240, 255, 264, 286, 302 C;sntmak)ier, F. R . , 276 (;erasinlo\-. S. G.,286

Gerkhen-Gubanov, G . V., 287 Germaidze, V. E., 287 GlIcksberg, I., 8, 278, 279 Gnornskii, I,. S., 287 G o m e z , R . I>., 287 Goodwin, K.,287 Gorhunov, 4. D., 281 Gorelik, G.,8 , 287 Grafi, D., 287 G r e c n , J . W., 287 Gross, O., 8 , 279 Gruzintsev, G.,287 Gukov, V. I., 287 Gul, I. M ., 287 (;urnowski, I., 287 G u n d e r , I). F., 287 Gurevich, B. L,., 287 Hadwiger, H., 287 Hagin, E. J., 287 H a h n , W., 8 , 288 Halanay, A., 8, 288 Hale, J. K., I , 288, 289 Halilov, 2. I., 289 Halkin, H., 28Y Halmos, P. K., 150, 196, 202, 289 Hammersley, J . M., 289 Hanny, J., 28Y Haratiivili, G. I,., 8, 240, 255, 289 I-lartree, I). R., 8 , 281 Harvey, C . A , , 289 Hasse, H., 289 Hayes, N . I>,, 289 Hazelhroek, P., 289 Heins, A . E., 289 Herlestam, T., 290 Hermes, H., 193, 195, 205, 207, 290 Herzog, F., 290 Higgins, T. J . , 294 Hilb, E., 2YO Hillc, E., 44, 290 Ho, Y . C . , 290 Hoheisel, G . , 290 Hole, N., 290 Holrne, H., 286 I-Iolt, C. C., 290 H u l l , T. E., 290 Izaw;~,K . , 290 Izumi, S., 290

AUTHOR INDEX

Jacquez, J. A., 279 Jain, S. C., 290 James, R. W., 291 Jones, A . L., 291 Jones, D. S., 7, 29/ Jury, E. I., 29/ Kadymov, Ya. B., 29/ Kakutani, S., 291 Kalaba, R . , 8, 221, 279 Kalecki, M., 29/ Kalinin, V. N., 291 Kamenskii, G. A , , 8, 2 9 / , 3 / 4 Kamke, E., 65, 292 KanB, S., 292 KaraSik, G. J., 292 Karlin, S., 177, 292 Kazda, L. F., 279 Kelendzheriedze, D. L., 260, 263, 275, 292 Kelman, R. B., 292 Kipianiak, V., 8 , 292 Kirillova, F. M., 286 Kirkaldy, J. S., 292 Kiselev, I. K., 292 Klimpt, W., 292 Klimusev, A . I . , 292 Klyamko, E. I., 292 Kolosovskii, M. Z . , 292 Koopmans, T., 292 Kosceev, P.S., 8, 292 Kosmodemianskii, V. A . , 292 Kostitzin, V. A , , 293 Kramer, J . D. R., Jr., 293 Krasnosel’skii, M. A., 2Y3 Krasovskii, N. N., 7, 8, 277, 293 Kreindler, E., 293 Krishnan, Sir K. S . , 290 Krivenkov, Yu. P., 293 Krumhansl, J . A., 8, 277 Ku, Y . H., 293 Kuczma, M., 283 Kuerti, G . , 294 Kulikovski, R., 294 Kupfmuller, K., 294 I.abazin, V. G., 294 I,acroix, S. F., 294 Lagrange, J. Id., 294 I,akshniikantham, V., 7, 14, 17, 18, 64, 69, 70, 71, 72, 73, 74, 294 I a n g , M., 294

31 7

Langer, R. E., 294 Lanzkron, R . , 294 LaSalle, J. P., 205, 243, 245, 251, 294 Lax, P. D., 294 Lee, E. B., 184, 282, 294, 295 Leonhard, A , , 295 Leonov, N. N., 280 Leontev, A. F., 295 Leontief, W., 295 Lerner, A. Ya., 295 Levin, J. J., 295 Liapunov, A., 195, 202,295 Liberman, L. H., 295 Liberman, L. Kh., 295 Lin, Chun, 295 Lin, Yung-ching, 282, 295 Livartovskii, I. V., 295 Ljusternik, L. A , , 44, 295 Lochs, G., 296 Loeb, J., 296 Lotka, A. J., 296 Lucking, D. F., 296 Lurie, A. I., 276, 296 Lyan Wang, 296 MacColl, L. A., 296 McKierman, M. A., 297 Macmillan, R . H., 296 Maezawa, S., 296 Magnus, K., 296 Mahler, K.,296 Maier, A. G., 8, 277 Mambriani, A,, 296 Maravall, D., 296 Markus, I,., 184, 2 9 / , 295 Martynov, P. V., 297 Meerov, M. V., 297 Meiman, N. N., 2 8 / , 297 Merriams, C. M., I 1 I , 219, 297 Meschler, P. A , , 297 Mikolajka, Z . , 297 Miljutin, A . A . , 283 Minorsky, N., 8, 287 Miranda, C., 298 Miranker, W. L., 298 Mirolyubov, 4 . A , , 298 Mishchenko, E. F., 8, 240, 255, 264, 302 Mitereff, S. D., 298 Mitropolskii, Y., 279 Mitsumaki, T., 298

318

AUTHOR INDEX

Mizohata, S., 298 Muravev, P. A,, 298 Myasnikov, N . N., 298 Myshkis, A. D., 8, 298, 299 Nadile, A , , 299 Nagy, Sz., 304 Narnazov, G. K., 299 Nasr, S. K., 283 Naurnovich, A . F., 299 Neiniark, Yu. I . , 299, 300 Nemyckii, V. V., 300 Nersesyan, A . B., 300 Neufeld, J., 300 Neustadt, L. W., 183, 191, 206, 215, 255, 275, 300 Nichols, N . B., 8, 3 / 3 Nickel, K., 300 NIsolle, L., 300 Nohel, J. A,, 295, 300 Norkin, S . B., 8, 300, 3 / 4 Novosel’tsev, V. N., 301 Obrechkoff, N., 301 OBuztoreli, M. N., 8, 92, 138, 147, 206, 240, 243, 260, 264, 275, 301 Oldenhourg, R. C . , 30/ Oppelt, W., 30/ Ostrowski, A , , 30/ Pai, M. A . , 30/ Pal, R., 30/ Perello, C., 289 Perron, O., 302 Phillips, I I . S., 44, 290 Pinnep, E., 8, 101, 112, 302 Pipes, A . L., 302 Pollaczrk, F., 302 Polossuchina, O., 302 Pontryagin, I,. S., 8, 240, 255, 264, 302 Popken, J., 302 Popov, E. P., 302 I’opovici, C., 302 Porter, A , , 8 , 28/ Prinz, I). G . , 30.l Raclis, M. R., 303 Radu, I?., 303 Rarnakrishnan, A , , 303 Rapoport, 7’. N., 309

Raymond, F. H., 303 Razumikhin, B. S., 7, 303 Reid, A. T., 303 Reinhardt, E., 303 Rekhliskii, Z. I., 303 Repin, Yu. M., 303 Rhodes, E. C . , 303 Richter, H., 303 Riesz, F., 77, 86, 304 Rith, J. F., 304 Ritt, H. R., 302 Rjabov, Yu. A , , 304 Roberts, S . M., 170, 219, 304 Robins, E., 304 Robinson, L. B., 304 Rocard, Y., 304 Rodionov, A. M., 304 Romanovskii, P. I., 7, 304 Roshkov, V. I., 305 Roston, S., 304 Roxin, E. O., 183, 184, 304, 305 Rozenwasser, E. N., 305 Rozonoer, L. I., 305 Rubanik, V. P., 305 Ruchti, W., 287 Rudoi, P. K., 305 Ryabtsev, I. I . , 305 Saakov, E. O., 305 Salukidze, M. E., 305 Samuelson, P. A., 305 Sansone, G., 305 Sarachik, P. E., 293 Sarkova, N . V., 305 Sartorius, H., SO/ Schmidt, E., 306 Schrodinger, E., 306 Schiirer, F., 306 Schwartz, J . T., 25, 150, 193, 284 Schwarz, H., 306 Seibert, P., 138, 276 Selberg, H., 306 Serehriakova, N. N., 306 Shapovalova, E. I., 299 Sherman, S., 306 Shirnanov, S . N., 8, 299, 306, 307 Shimemura, E., 240, 307 Shimuzi, T., 307 Shlopak, A . S., 299 Shostak, K. Ya., 307

AUTHOR INDEX

Shtykan, A. B., 307 Shu-chuang, T. K., 307 Siewert, R. M., 307 Silberstein, L., 307 Silnikov, L. P., 300 Simon, H. A., 290 Simon, H. I., 307 Slobin, H . L., 278 Sobol, I. M., 307 Sobolev, W. I., 44, 295 Sokolov, A. A., 307 Soldatov, M. A., 307 Solodownikow, W. W., 308 Solow, .R., 308 Soper, H. E., 308 Speigel, M. R., 308 Spinadel, V. W., 305 Stein, D., 308 Steinberg, T. S., 308 Stevenson, A. G., 281 Stokes, A., 308 Strakhov, V. P., 308 Strygin, V. V., 308 Sugiyarna, S., 308 Sutherland, A. B., 308 Svirskii, 0. V., 308 Szarski, J., 308 Szatrowski, Z., 308 Tabueva, V. A., 277 Tacklind, S., 309 Taylor, A. E., 309 Theiss, E., 309 Teodorchik, K. F., 309 Tetel’baum, S. N., 309 Tikhonov, A. N., 7, 309 Tinbergen, J., 309 Titchrnarch, E., 309 Tonelli, L., 7, 309 T o u , J. T . , 219, 309 Touchard, J., 309 Troitskii, V. A., 309 Tsapyrin, V. N., 309 Tswang, Kh. G., 309 Tsypkin, Ya. Z . , 8, 310 Turritin, H. L., 310 Tustin, A,, 310 Valeev, K. G., 310 Valiron, G., 3 / 0

3 19

van der Corput, J. G., 310 van der Pol, B., 102, 310 van der Waerden, B. L., 289 van We&, 1. T., 310 Vasil’eva, A. B., 310 Vaulot, A., A., 310 Vazsonyi, A., 311 Velikii, A. P., 31 I Verblunsky, S., 311 Verzhbinskii, M. L., 311 Vikker, D. A , , 311 Vogel, T., 31 1 Volosin, H., 8, 311 Volterra, E., 31 I Volterra, V., 7, 44, 46, 80, 81, 311, 312 von Karrnan, T . , 280 Vorobyev, Yu. V., 312 Wachi, A., 312 Wald, A., 195, 197, 284 Wang, Lian, 282 Wang, P. K. C., 312 Wang, R., 312 Warga, J., 207, 209, 21 I , 312 Wick, G. C., 312 Wilder, C. E., 312 Wilkins, J. E., 312 Williams, K. P., 312 Wilson, E. B., 312 Wing, J . , 283 Wloka, J., 312 Wolf, A. A., 293 Wolfe, W. A., 290, 312 Wolfowitz, J., 195, 197, 284 Worchester, J., 312 Wright, E. M., 8, 312, 313 Yates, B. G., 313 Yoshizawa, T., 313 Yu, Ming-chin, 313

Zaanen, A. C . , 86, 313 Zarernba, S. K., 313 Zhang, Bing-gen, 313 Ziegler, J. G., 8, 313 Zima, K., 279, 313 Zubov, W. I., 8, 313 Zverkin, A. M., 8, 313, 314 Zverkina, T. S., 314 Zwinggi, E., 314

Subject Index A

Continuation of solutions, 137 Continuity of a functional, 1 I Continuous, kernel, 78 part, 78 Control, 3 adaptive, 176 admissible, 5, 167, 241, 261 averaging, 173, 235 function, 167 parameter, I74 region, 4, 167 signal, 3 terminal, 173, 183, 233, 236 vector, 3 Controller, 167 cost, 5 optimal, 239 Convergence, strong, 152 weak, 152 Convex, 26, 167, 177 combination, I78 cone, 179 hull, 178 strictly, 178

Absolute measure, 199 Absolutely continuous, 59 Adaptive control, 176 Admissible, 167, 241 control, 5, 167, 241, 261 initial function, 5, 168, 261 pair, 5, 168, 261 policy, 5 , 168, 261 trajectory, 210 Adjoint equations, 92 Advance of initial instant, 13 Aftereffect, 7 Approximation, 114, 127 successive, 36 Atom, atomless, 197 Attainable set, 183 Averaging control, 173, 235

B Ranach space, 25, 151 Bang-bang control, I74 principle, 205 Bore1 field, 156 Houndedness, 63, 108, 131, 143 of a functional. 48. 77

D

C

Decomposition n-vector, 196 Delay, 3 switching, 136 Delay-differential equation, 4 Deterministic, 176 Dependence on initial data, 38, 95 on parameters, 41, 95 on right-hand side, 42 Difference-differentia1 equations, 6 Differentiation of a functional, 45, 48 with respect to initial instant, 59 with respect to parameters, 51 Dirac’s function, 89 Distance, 10

Capture, point, 267 time, 241 Characteristic function, 196 Closed convex hull, 178 Closed-cycle, 80, 81 Compact, 25, 167 Complementation, I96 Complete, 26, 191 weakly, 186 Conjugate, 151 Constraint, 168 integral, 169, 172 320

32 1

SUBJECT INDEX

Disturbance vector, 3 Dynamic programming, 177, 219

E End values, 170 r-approximate, 14 Equicontinuity, 25 Essential supremum, ess. sup., 151 Euler’s polygonal method, 30, 92 Existence theotems, 5 , 24 Extended Filippov lemma, I93 Extreme point, 179

F First variation, 45, 53, 155, 156, 157, 161, 162, 224 Fixed initial function cross section, 183 point, 28 target, 170 - time cross section, 183 Functional, 4, I I analytic, 47, 5 I , I 14 bounded, 48, 77 continuous, I derivative, 45 linear, 48, 76 measurable, 60 partial derivative, 155 polynomial, 114 power series, 120 Volterra, 4

G Gronwell’s lemma. 43

1

Inhomogeneous linear eq. 77 Initial condition, 4, 154 d@a, 4 function, 4, 12 instant, 4, 56 interval, 137, 241 value, 4 Integral constraint, 169, I72 equation, 104 Lehesgue-Stieltjes, 175 Integro-difference-differential equation, 6 -differential eq., 6 Iterated kernels. 100

K

Kernels iterated, 100 matrices, 78, 92 Pincherle-Goursat, I24

L Lipschitz constant, 12 Local maximum principle, 255 Locally Lipschitzian, I I

M Maximum of a functional, 47 Maximum principle, 255 Measurable function, 196 functional, 60 right-hand side, 59 Minimum of maximum deviation, 175 Minimizing trajectory, 210 Minkowski inequality, IS I

N

H Heredity, 3, 75 finite, 75 invariable, 80 Homogeneous linear eq. 77 Holder inequality, I5 I Hyperplane, 177

n range, 197 Necessary condition for optimality, 233, 254, 255 Norm in L, spaces, 150, I52 of a linear functional, 48 of a matrix, 10 of a vector, 10

322

SUBJECT INDEX

0 Optimal initial function, 252 control, 250 cost, 239 pair, 5 policy, 5 , 172 problem, 171 pursuit problem, 262 strategy, 262 Optimization problem, 171 Original problem, 210 Osgood’s integral, 23, 29, 63, 92

P Performance criterion, 5 , 170 index, 5 Perturbed system, 73 weakly, 73 Picard’s iterant, 39 Piecewise continuous, 136 Pincherle-Goursat kernels, 124 PoincarC’s expansion theorem, 52 Principle of closed-cycles, 80 optimality, 221 Polar cone, I79 Policy space, 168 Probability n vector, 196 Pure n vector, 196 Pursuit problem, 175, 176, 260 Pursued system, 176, 261 Pursuing (pursuer) system, 176, 261

Q Quadratic performance index, 238 Quasi-continuous, 2 10

R Range of a vector measure, I94 Region of control, 4, 167 Relaxed problem, 209, 210 Resolvent kernel, 100 Restricted control, 205 Retardation, 6

Right-hand side vector, 77 measurable, 59 piecewise continuous, 136

S Saltus matrix, 78 Segment of a function, I0 Self-learning, 176 Separable kernels, 122, 125 Separation theorem, I78 Shauder’s fixed point theorem, 28 Signum, sgn., 136 Simplex, 195 Sliding regime, 209 Solution, 4, 137 in the extended sense, 60, 153 Span, 6 Stability, 5 , 63, 108, 131, 143 State, 3 variable, 3 vector, 3 Step function, 196 n range, 197 n vector, 196 Stochastic, 176 Strictly convex, 178 Strong convergence, I52 Successive approximations, 36 Supporting hyperplane, 177 Switching delay, 136 space, 137

T Target set, 170 Terminal control, 173, 183, 233, 236 values, 170 Theorem of Ascoli, 25 Time-lag, 6 T i m e optimal problem, 174, 240 Trajectory, I68 Transfer, I70 time, 174 Transpose, 9 Triangle inequality, 10

323

SUBJECT INDEX

U Uncontrolled, 168 Uniqueness, 12, 20 Unit matrix, 89 step function, 105 vector, 155 Upper semi-continuous, 191 variation, 21

V Variation, 53 Volterra functional, 4 integral equation, 99

W Weak convergence, 152 Weakly complete, 186 perturbed, 73 Weierstrass’ approximation theorem, 11 3

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