Hamilton-Jacobi Type Equation for Problems of Control of Functional Differential Systems1

Copyright @ IFAC Control Applications of Optimization, St. Petersburg, Russia, 2000

HAMILTON-JACOBI TYPE EQUATION FOR PROBLEMS OF CONTROL OF FUNCTIONAL DIFFERENTIAL SYSTEMS l Nikolai Yu. Lukoyanov Institute of Mathematics and Mechanics of Ural Branch of the Russian Academy of Sciences, S.Kovalevskaya str., 16, Ekaterinburg, 620219, Russia, e-mail: [email protected]

Abstract: For control problems under lack of information, which are formalized as differential games of systems with aftereffect, an equation of Hamilton-Jacobi type is obtained for the value functional. A definition of its minimax solution (MS) is given. The existence, uniqueness, and well-posedness of MS are proved. A method of constructing extremal to MS strategies is developed. It is shown that they are optimal for the control problem considered, and the value functional coincides with MS. Copyright © 2000IFAC Keywords: Control theory, Differential games, Delay.

1. INTRODUCTION

As a development of this research, (Subbotin, 1980, 1991) gave the notion of generalized minimax solution (MS) to H-J equations, and proved that MS exists, is unique and well-posed; MS is equivalent to the viscosity solution independently defined by (Crandall and Lions, 1983); MS to the B-1 equation coincides with the value function of the corresponding control problem.

It is known, first order PDEs of Hamilton-Jacobi (H-J) type are closely connected to optimal control problems and differential games of ODE-systems. If the value function (the optimal value of the control problem (Bellman, 1957) or the value of the differential game (Isaacs, 1965» is smooth, it satisfies an equation of H-J type called (Bellman, 1957; Isaacs, 1965) the (B-1) equation. The difficulty is that as a rule the value function is not smooth, and the B-1 equation has no solution in classical sense. Nevertheless, at the points of smoothness the value function satisfies the BI equation and can be treated as its generalized solution. It was shown in differential game theory (see, e.g., Krasovskii and Subbotin, 1974), the value function possesses u- and v-stability properties fulfillment of which is necessary and sufficient.

The present paper is devoted to development of the theory of MSs for functional equations of H-J type with (Kim, 1999) co-invariant derivatives in view of applications in problems of control of FDE-systems. A problem of control under lack of information is considered. In Section 2, according to the game-theoretic approach (see Krasovskii, 1985; Osipov, 1971), the problem is formalized as a differential game of systems with aftereffect. The value of the game is here a functional, and strategies are functions, of histories of motions. In Section 3, a functional equation with co-invariant derivatives of H-J type is put into

1 Partially supported by the Russian Foundation for Basic Research (projects no. 96-15-96245, and no. 99-01-00144)

591

3,) there exists a number re, > 0 such that, for all (t,x[t.[·]t],u,v) E G x U x V, the inequality holds

the correspondence to the game. Alongside with the known difficulties for H-J equations (see, e.g., Crandall and Lions, 1983; Subbotin, 1991) this equation possesses peculiarities connected with aftereffect. Similarly to Subbotin (1991) in Section 4, a definition of its MS (that is already a functional) is given; the existence, uniqueness, and well-posedness of this solution are proved. In Section 5, extremal to MS strategies are defined; and it is shown that they are optimal for the control problem considered, and the value functional coincides with MS.

IIf(t,x[t.[·]t],u,v)1I

~ re,(1 + t.~T:::;t max IIx[r]II).

Let an initial position gO = (t°,xO[t.[.]tO]) E G, to < T, be known, and Borel-measurable control realizations u[·] = {u[t] E U, to ~ t < T} and v[·] = {v[t] E V,tO ~ t < T} respectively of the first and the second players be admissible. Under assumptions 1, )-3, ), for the initial position gO and any pair of admissible realizations u[·] and v[·], there exists a unique realization of motion of system (1), i.e., a function x[·] E C. that coincides with XO [t. [·]to] on [t., to], and, for t E [to, T], satisfies the equality

2. DIFFERENTIAL GAME OF SYSTEMS WITH AFTEREFFECT

Consider the FDE-system

f t

dx[t]/dt = f(t,x[t.[·]t], u, v), x E Rn, t. ~ to ~ t ~ T, u E U c R\v EVe Rm.

x[t] = x[tO]

f(r, x[t.[·]r]' u[r], v [r])dr.

to

The triple {x[·], u[·], v[·]} is called the realization of the control process. Let the quality of the control process be estimated by the functional

Here x is the phase vector; u and v are control actions of the first and the second players respectively; t., to and T are given moments of time (to < T); U and V are known compacta; x[t.[·]t] = {x[r],t. ~ r ~ t} is the history of motion that has realized up to the moment t.

'Y

= 'Y({x[·],u[·],v[·]}) = a(x[to[·]T]) + T

+

Let C. = C([t., T], Rn) be the space of continuous functions x[·] = {x[t] ERn, t. ~ t ~ T}, and similarly Co = C([to, T], Rn), C. = C([t., T], RnxR). Denote by G the set of pairs 9 = (t, x[t. [.]t]), where to ~ t ~ T, and x[t.[·]t] is the part from t. to t of a function in C • . Define a metrics p on G

itor

h(t, x[t.[·]t], u[t], v[t])dt,

(2)

where a : Co t-t R is continuous, and h : G x U x xV t-t R satisfies the assumptions h)-3 h ) that are similar to 1,)-3,) (substitute in conditions 1, )-3,) the symbol h instead of 1). Suppose also that, for any 9 = (t, x[t. [.]t]) E G and s E Rn, r E R, the equality is valid

p(gl,g2) = max{p·(gl,g2),P·(g2,gd}, gl = (t1,X(1)[t.[·]td) E G, g2 = (t2,X(2)[t.[·]t2]) E G, where P·(gH1,g2-i) = max min (... ),

min max[(s, f(g,u,v)) + rh(g,u,v)] = uEU vEV = max min[(s, f(g, u, v)) + r h(g, u, v)]. vEV uEU

(3)

Here and below (., .) is the scalar product of vectors.

t. ~~:::;ti+l t. ~1'/~t2_i

(.. .) = max{l~ -7]1, Ilx(H1)[~]_ x(2-i)[7]1I1},

i = 0,1,

The aim of the first player is to minimize the functional 'Y. The aim of the second player is to maximize 'Y.

11·11 denotes the Euclidean norm of a vector. Below properties of continuity in (t, x[t.[·]t]) are treated with respect to the metrics p.

These problems are connected into the differential game (1), (2) of systems with aftereffect. Strategies of the first and the second players are treated respectively as arbitrary functions (g, c) t-t U(g, c) E U and (g,c) t-t V(g,c) E V where 9 = (t, x[t. [.]t]) E G, t E [to, T], and c > 0 is some parameter of accuracy (see Krasovskii, 1985). Take a strategy U (.), a value of the parameter c > 0, and a partition

In (1) the function f : G x U x V t-t Rn satisfies the following assumptions:

1,) it is continuous; 2,) there exists a number A, > 0 such that, for all u E U, v E V and (t, x'[t.[·]t]) E G, (t,x"[t.[.]t]) E G, the estimate holds

IIf(t, x'[t.[·]t], u, v) - f(t, x"[t.[·]t], u, v) 11 ~ ~

+

(1)

A, max IIx'[r]- x"[r]ll;

llo = {ti: t1 = to, 0

t.~T:::;t

< tH1 - ti

~ 0,

i=I, ... ,N; tN+l=T} 592

(4)

of the time interval [to, T] . This triple defines a step-by-step control low of the first player {U( ·),c,Ll o} that forms realization u[·] according to the rule:

u[t] =

U(ti,X[t.[·]ti]'c), ti :s t < ti+i, i = 1, ... ,N;

Definition 1. A functional cp : G I--t R is coinvariantly differentiable at the point g. with respect to Lip(g·) (ci-differentiable at g.) if there exist a number Otcp(g·) and an n-vector Vcp(g·) such that, for any function y[.] E Lip(g·), the equality holds

(5)

cp(t· + {, y[t.[·]t· + W - cp(g.) = Otcp(g·){+ + (Vcp(g·),y[t· + {]- x·[t·]) + 0l/(O, { E [0, T - t·],

and so, together with some admissible control realization v[·] of the second player, it generates from the initial position gO = (to, xO[t.[·]tO]) the unique realization of the control process

where 0l/({) depends on the choice of y[.] E Lip(g·) (ol/(O/{ I--t 0 as {I--t +0).

{x[·], u[·],v[·]lgO,U(·),c,Llo,v[·]}. Define the guaranteed result for the strategy U (.) as

ru(gO,u)

= limsup lim sup sup" E.j.O

0.j.0

/:;.6

v[ ·]

The values Otcp(g·) and V cp(g.) are called respectively the ci-derivative in t and ci-gradient of functional cp at point g.. A functional cp is called cidifferentiable if it is ci-differentiable at any point 9 = (t,x[t.[·]t]) E G, t < T.

(6)

, = ,( {x[·], u[·], v[·] 1 gO, U(-), c, Ll o, v[·]});

and the optimal guaranteed result of the first player as r~ =r~(gO)

= i~~ru(gO,U( . )) .

More details on co-invariant derivatives of functionals can be found by Kim (1999).

(7)

For the differential game (I), (2) define the Hamiltonian

The guaranteed result r v (gO , V (. )) for a strategy V (.) and the optimal guaranteed result r~ of the second player are defined similarly (substitute in (6), (7) instead of symbols u, U, v, sup, limsup, inf respectively v, V, u, inf, liminf, sup) . Directly from definitions of r~ and r~ the inequality is deduced r~ 2: r~. (8) IT in (8) the equality holds, the differential game (I), (2) has the value

H(g, s)

I(g, u, v)) + h(g, u, v)],

vEV uEU

= (t,x[t.[·]t])

9

E G,s ERn,

(9)

and consider the functional equation of H-J type in ci-derivatives

Otcp(g) + H(g, Vcp(g)) = 0, = (t, x[t.[.]t]) E G, t < T,

9

rO (to,xO[t.[ .]t°J) = r~ = r~ .

(10)

with the condition at the right-hand end

IT, besides that, there exist strategies U°(-) and VO (-) such that the following equalities are valid r~ r~

= max min (s,

cp(T, x[t.[·]T])

= r u(tO , XO[t.[·]tO], UO( .)), = r v (to , XO[t.[·]t°]' V°(-)),

= u(x[to[·]T]),

x[·] E C. .

(11)

Under assumptions 1,)-3,), Ih)-3h)' and (3), the Hamiltonian (9) will satisfy the following conditions:

then they constitute the saddle point of the game (I), (2), and are optimal respectively for the first and the second players.

IH) for any sE Rn, the functionalgl--t H(g,s) is continuous on G; 2H) for any 9 = (t,x[t.[·]t]) E G and s ERn, there exists a limit

3. HAMILTON-JACOBI TYPE EQUATION WITH CO-INVARIANT DERIVATIVES

lim rH(g, sir) r.j.O

The notion of co-invariant derivatives was introduced by Kim (1999, pp. 12-22) for functionals cp(t, x[t. [·]t]) defined on piecewise continuous functions x[t.[·]t] . Below similar derivatives are considered for functionals cp : G I--t R.

= HO(g, s),

and, for any s E Rn, the functional 9 is continuous on G;

I--t

HO (g, s)

3H) there exists a number >. > 0 such that, for all (t,x'[t.[·]t]) E G, (t,x"[t.[·]t]) E G and (s,r) E Rn X R, IIsll 2 + r2 = I, r > 0, the estimate holds

Let g. = (t·,x·[t.[·]t·]) E G, t· < T, and Lip(g·) be the set of functions y[.] E C. each of which coincides with x·[t.[·]t·] on [t., t·] and satisfies the Lipschitz condition on [t·, T] with a coefficient (of its own).

rlH(t, x'[t.[·]t], sir) - H(t, x"[t.[·]t], slr)1 :s >. max IIx'[T] - x"[T]II ; t. ~T~t

593

:s

4H) for any 9 = (t,x[t.[·jt]) (s",r") E B+, where B+

= {(s,r) E Rn

X

E G,

R: IIsll 2

(s',r')

The class of pairs {Q, F· (.)} ({P, F .(.)}) that satisfy 1K)-4K) will be denoted by K·(H) (K.(H)).

E B+,

+ r2 :s 1,r > O},

Remark 1. Under assumptions 1H )-4H), conditions 1K)-4K) hold for P = Q = Rn X R,

the inequality holds

:s

Ir' H(g, s'lr') - r" H(g, S" Ir")l

:s L(g)(lIs' -

= {(f,h) E F(g):

F.(g,s,r)

= {(f,h) E F(g) : (s, f) + rh :s H(g, s, rn,

(s, f) + r h

s"1l2 + (r' - r")2)l/2,

where L(t, x[t.[·jt]) is a continuous functional that satisfies the estimate

L(t,x[t.[·jt])

F·(g,s,r)

:s re

(1 +

(t,x[t.[·jt]) E G, re

where

max IIX[Tjll) ,

= const > O. L(g) is from the condition 4H), 9 thus, K·(H) "10, K.(H) "10.

Let {Q,Y( ·n E K·(H), {p,F.( ·n E K.(H), E Rn X R . Consider differential inclusions with aftereffect dx[tj/dt E F·(t,x[t.[.jtj,q), dX[tj/dt E F.(t,x[t.[·jtj,p).

Denote

H( s r) = { IrIH(g, sllrl) for r "I 0 , g, , HO(g,s) for r = 0 where 9 = (t, x[t.[·jt]) E G, (s, r) E Rn HO (g, s) is from the condition 2H ).

X

Q 3 (g,q) ~ F·(g,q) C Rn P 3 (g,p) ~ F.(g,p) C Rn

X X

R, and

:s

R, R,

:s

Definition 2. A minimax solution of problem (10), (11) is a continuous functional tp : G ~ R that satisfies the condition at the right-hand end (11) and, for some {Q,Y} E K·(H), {P,F.} E K.(H), the inequalities

satisfy the following conditions:

1K) for any 9 = (t,x[t.[·jt]) E G, pEP, q E Q sets F·(g,q) and F.(g,p) are nonempty convex compacta in Rn X R; 2K) there exists a number a > 0 such that the estimate is valid

sup

inf

gO = (to,XO[t.[.]t O]) E G, t E [to,T],p E P,q E Q, x·[·j = (x·[·],z·[·j) E X"(gO,O,q I F·), x.[.] = (x.[·], z.[·]) E X • (gO ,O,p I F.), ~tp(t,x[t.[·]t],gO) = tp(t,x[t.[.]t]) + z[t] _tp(gO).

4K) for any 9 = (t, x[t.[·jt]) E G and s ERn, R, the equalities hold

pEP

(16)

where

r E

m~

The following propositions, which clarify the connection of minimax solutions with equation (10), can be proved directly from Definitions 1,2.

[(s,f)+rhj

i=(f,h)eF (g,q)

= inf _

max ~tp(t,x.[t.[·]t],gO) ~ 0,

max Ilx[Tjll),

to~~9

3K) for any pEP and q E Q multivalued mappings 9 ~ F·(g,q) and 9 ~ F.(g,p), are upper semicontinuous on G with respect to inclusion;

qeQ

(15)

(gO,t ,p) zo[.)

9 = (t,x[t.[·jtj) E G, pEP, q E Q;

H(g,s,r)=sup _

:s 0,

min ~tp(t,x·[t.[.jt],gO)

(gO ,t,q) ZO [.)

max{lIflllf E F·(g,q) U F.(g,p)}:S

:s a(l +

(13) (14)

Let gO = (to,xO[t.[·jt°j) E G, pEP, q E Q. The solution of inclusion (13) (respectively (14)) that corresponds to a pair gO and a fixed q (p) is treated as a function x[·j = (x[·], z[·]) E C. that coincides with XO[t.[·jtOj = {XO[Tj = (XO[Tj, ZO[T] = 0), t. T to} on [t., to], is absolutely continuous on [to, Tj, and for almost all t E [to, T] satisfies inclusion (13) (respectively (14». Denote the set of all these solutions by X· (gO, 0, q I F·) (X. (gO, 0, p I F.)), where 0 is the scalar function identically equal to zero on [t., tOj. Because of 1K )-3K) these sets are nonempty compacta in C •.

(12)

Let P and Q be some nonempty sets (to be definite, one can assume that P and Q are subsets of finite-dimensional spaces), and multivalued mappings

X

= (t,x[t.[.jt]);

x = (x, z)

4. MINIMAX SOLUTIONS

X

H(g, s, r)},

to~~~t

Note that, under the given conditions, a ci-differentiable functional that satisfies (10), (11) may fail to exist; thus, generalized solutions of problem (10), (11) must be considered.

G G

~

ma.!

[(s, f)

+ r hj.

i=(f,h)eF ° (g ,p)

594

Proposition 1. If a minimax solution is ci-differentiable at a point gO = (to, xO [t. [.]to]) E G, to < T , then it satisfies equation (10) at this point.

1',,(g,u) = co{(f(g,u,v),h(g , u , v» E Rn X R I v E V}, 9 = (t,x[t.[ ·]t]) E G, u E U, v E V. (17)

Proposition 2. If a continuous functional ep : G t-+ R is ci -differentiable and satisfies (10), (11), then it is a minimax solution of problem (lO) , (l1) .

Here coF denotes the convex hull of a set F in RnxR. Due to 1,)-3,)' h)-3h) and (3),(9) ,(12), the inclusions are true

The existence, uniqueness and well-posedness of the minimax solution to problem (10), (11) can be proved with the help of reasonings that mainly follow the schemes of proof of the analogous statements for the minimax solution of usual (for functions of finite-dimensional argument) H-J equations (see Subbotin, 1991); here the peculiarities should be taken into account connected with the presence of aftereffect and the functional argument of the solution sought for .

( . )}

E K·(H), {U, F,,(-)} E K.(H).

(18)

Let 9 = (t, x[t.[·]t]) E G, w[·] = (w[·], z[·]) E C., > 0, and ep : G t-+ R be the minimax solution of problem (9)-(11) . Put

£

v(g, w[·]) = exp{ -2A(t - to)} X x max Ulx[r]- w[r] 112 + z2[r]], t. ~ T:St

where A = (AJ + A~)1/2, A, and Ah are from the assumptions 2,) and 2h) ;

Theorem 1. (Existence and uniqueness) Let (f : Co t-+ R be a continuous functional, and the Hamiltonian H satisfy conditions IH) -4H) in Section 2. Then there exists a unique minimax solution of problem (10), (l1). It satisfies conditions (15) and (16) for any {Q ,1'·(.)} E K·(H) and {P,F.( ·)} E K.(H).

= {w[·] E C. : v(g , w[·]) ~o(g,£) =

O(g, £)

inf

( w[ ·],z[ '])EO(9 ,€)

~O(g,£)

~

£(t - to)};

[ep(t, w[t.[.]t])

+ z [t]];

(19)

[ep(t, w[t.[·]t])

+ z[t]].

(20)

= sup

Theorem 2. (Well-posedness) Let j = 1, 2, . . .; (fo , (fj : Co t-+ R be continuous functionals; Hamiltonians Ho, H j : GxRn t-+ R satisfy assumptions IH) -4H) in Section 2; L o, L j : G t-+ R be functionals defined respectively for Ho, H j in accordance with condition 4H) (where re = reo = rej). Let (f j -+ (fo as j -+ 00 uniformly on any compactum from Co; H j (·, s) t-+ Ho(" s), L j -+ Lo as j -+ 00 for any s E Rn and compactum DeC. uniformly on G(D) = {(t , x[t.[·]t]) : t E [to,T], x[·] E D} ; and functionals epj : G t-+ R be minimax solutions of problems

( w[,] ,z[ '])EO(g ,€)

Denote by Wo(g,£) (respectively by WO(g,£» the set of partial limits wo[t] = (wo[t], zo[t]) E Rn X R (WO[t] = (WO[t], zO[t]) E Rn X R) of sequences {WOj[t] = (WOj[t],ZOj[t]),j = 1,2, .. .} ({'WJ[t] = = (WO[t] , zJ[t]),j = 1,2, ... }) that correspond to the niinimizing in (19) (the maximizing in (20» functional sequences {WOj['] = (WOj['], ZOj[']) E E O(g,£),j = 1,2, .. .} ({'WJ[.] = (wJ[.],zJ[ .]) E E O(g,£),j = 1,2, ... }). These sets are nonempty for any 9 = (t, x[t. [·]t]) E G, £ > O. The extremal strategy Ue (.) of the first player is defined on the basis of arbitrary choice:

Otep(g) + Hj(g , Vep(g)) = 0, 9 = (t,x[t.[·]t]), ep(T, x[t.[.]T]) = (fj(x[to[·]T]), (j = 1,2, ... ).

=ue ,

Ue(g,£) Ue

Then the sequence
E argmin{

uEU

m~

[(su,f) +ru h ]} =

(f,h)EF.(g,u)

= arg min {max[(su, f(g, u, v») + ruh(g, u, v)]}, uEU "EV Su = x[t]- wo[t] ,

Otep(g) + Ho(g, Vep(g» = 0, 9 = (t, x[t.[·]t]) ,
ru

= -zo[t],

(wo[t],zo[t]) E Wo(g,£).

(21)

The extremal strategy Ve (.) of the second player is defined on the basis of arbitrary choice:

Ve(g, £)

5. EXTREMAL STRATEGIES

= v e,

Ve E argmax{ min [(5", f) + r"h]} "E V (f,h)EF" (g ,v)

Consider multivalued mappings

?(g,v) = co{(f(g, u, v), h(g, u, v» E Rn

U

{V, F

=

= argmax{ min[(s", f(9, u, v») + r"h(g, u, v)]}, X

"EV uEU 5" = wO[t]- x[t], r"

R I u E U}, 595

= zO[t],

(WO[t], ZO[t]) E w<> (g, c).

cides with the minimax solution cp of problem (9)(11). The extremal strategies UeO and Ve( ·) constitute the saddle point of the game, and are optimal.

(22)

F" (.)

and F v (-) are from (17), the function t-+ Rn and functional h : G x U x V t-+ R are continuous; therefore, the values U e and Ve needed in (21) and (22) exist for any g = (t, x[t.[·)t]) E G, € > O. Here

f :G x U xV

Remark 2. Inequalities (15) (with Q = V, F·O = F"( .)) and (16) (with P = U, F.(·) = = F v 0) express respectively the corresponding properties of u- and v-stability (see, e.g., Krasovskii and Subbotin, 1974; Osipov, 1971) of the value functional of differential game (1), (2) .

6. MINIMAX SOLUTION AND VALUE FUNCTIONAL Take a value of the parameter € > 0, a partition ilo (see (4)), and consider the control low of the first player {Ue (-), € , il o}. Let

Remark 3. If the minimax: solution of problem (9 )-( 11) is continuously ci-differentiable, then the optimal strategies UO (.) and VO (.) can be constructed according to (21) and (22) with Su = SV = V'cp(t,x[t.[·]t]), ru = rV = 1.

{xe[ ·]'Ue[·],v[·) I gO,Ue( ·),€,ilo,v[·n be the control process that has been realized under the action of this low from a known initial position gO = (t°,xO[t.[·]tO]) E G, to < T, and for some admissible control realization v[·) of the second player.

7. CONCLUSION The given results show that the considered functional equation of H-J type with ci-derivatives can be treated as the main equation of differential games of systems with aftereffect.

Because of (18) and Theorem 1, for the minimax solution cp of problem (9)-(11) the inequality (15) holds for Q = V and F·( .) = F"( .). With the help of this and (21), it is possible to prove by induction that there exists a 8° = 8° (€, gO) > 0 such that, for any partition ilo with 8 ~ 8°, the inequalities are valid

!

REFERENCES Bellman, R. (1957) . Dynamic Programming. Princeton Univ. Press, Princeton, NJ. Crandall, M. G. and P.-L. Lions (1983). Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc., 227, 1-42. Isaacs, R. (1965). Differential Games. John Willey, New York. Kim, A. V. (1999). Functional Differential Equations: Application of i-smooth calculus. Kluwer Academic Pablishers, Dordrecht. Krasovskii, N. N. (1985) . Control of Dynamic Systems. Nauka, Moscow. (in Russian) Krasovskii, N. N. and A.!, Subbotin (1974). Positional Differential Games. Nauka, Moscow. (in Russian) Osipov, Yu. S. (1971) . Differential games of systems with aftereffect. Dokl. Akad. Nauk SSSR, 196, 779-782. Subbotin, A.!, (1980). Generalization of the main equation of differential game theory. Dokl. Akad. Nauk SSSR, 254, 293-297. Subbotin, A. !, (1991). Minimax Inequalities and Hamilton-}acobi Equations. Nauka, Moscow. (in Russian)

ti

+

h(t, xe[t.[·)t], ue[t], V [t])dt,

(23)

i=1,2, ... , N+1, where the value <1>0(-) is the same as in (19). From (19) and (23) (for i is deduced

= N + 1) the inequality

Similarly, with the help of (16) (for P = U , F.(·) = Fv(-)) and (22), one can show also that

rv(gO, Ve(-)) ~ cp(gO).

(25)

Inequalities (8) and (24),(25), imply

cp(gO) ~ ru(gO,ue (-)) ~ r~ ~ ~ r~ ~ rv(gO, VeO) ~ cp(gO). Thus, the following theorem is true. Theorem 3. Let FDE-system (1) and functional (2) satisfy conditions given in Section 1. Then, for any initial position (t°,xO[t.[·]tO)) E G, to < T, the differential game (1),(2) of systems with aftereffect has the value r°(t°, xO[t.[·]tO]). The functional (to,xO[t.[·]tO]) t-+ r°(t°,xO[t.[·)tO]) coin596