SIAM J. CONTROL AND OPTIMIZATIONVol. 26, No. 3, May 1988

1988 Society for Industrial and Applied Mathematics

014

ERRATUM AND ADDENDUM: THE EXISTENCE OF VALUEAND SADDLE POINT IN GAMES OF FIXED DURATION*

LEONARD D. BERKOVITZ

Martin Brokate has pointed out an error in the proof of the crucial Lemma8.3. The point defined on page 186 depends on the strategy Fr, and thus the assertionthat the right-hand side of (8.7) equals W-(tl, gl) is incorrect. Below, we shall presenta proof of Lemma 8.3 suggested to us by K. Haji-Ghassemi. Before presenting thisproof we shall take this opportunity to improve our definition of strategy.

Let [a, b] denote the set of measurable functions u on [a, b] such that u(t)e Yalmost everywhere. Let [a, b] denote the set of measurable functions v on [a, b]such that v(t) Z almost everywhere.

A strategy F for Player i is a choice of a sequence II {Ii,} of partitions of to, T]and a choice of a sequence of maps Fn {Fri,n}, where the Fn,n are to be definedbelow. Thus F (Fn, II). For typographic smplicity, we will suppress the dependenceon II in the notation and write F for Fn and {Fn} for {F,n}. We restrict the choice ofsequences of partitions to those such that IIrI.II 0, as n-, c. Let the partition pointsof IIn be to < t < < tp T. Each map Fn is a collection of maps Fn,, , i’n.p asfollows. The map Fn, selects an element in [to, h). For 2 -<__j --< p, the map Fn/is amap from [to, tj_) ’[to, tj_) to ld[tj_, tj).

A strategy A for Player II is a choice of sequence of partitions II {IIn} of to, T]such that III 11-,0 as n-,oo and a choice of sequence of maps {An}. Each An is acollection of maps An.,’" ", An,q as follows. If IIn has partition points to So < s <

< sq T, then An, selects a function v in [So, s). For 2 =<j _-__ q, An/is a map from[to, sj_) x [to, sj_) to Lr[s_, s).

As before, each pair (Fn, A,) determines an outcome (un, vn), and we proceed asbefore. The change in definition does not involve any major modifications in thearguments of the paper, and in fact simplifies some of the constructions. We note here,however, that in defining the extremal strategies Fe and Ae the associated sequencesof partitions {lien} and {fien} must have the number of partitions at the nth stage equalto n. Thusp=q=n.

We now turn to the proof of Lemma 8.3. The first paragraph of the proof stands,and the rest of the proof is replaced by the following.

Let (u) {x" xl (t, , , u, )" an arbitrary relaxed control}. ThenC(vo) ck. Let Fn denote the constant strategy over [, t] with value u. By Lemma6.1, (u)={x" x=o[t, , :, F,, A], A arbitrary over [, hi}. By Lemma 6.2,is compact.

Let v inf { W-(h, x)" x q)(u)}. Since W- is continuous and q)(u) is compact,there exists an .1 q)(u) such that v W-(h, ). Moreover, since q)(u) fq C(vo)if c vl Vo, then c > 0.

For every x(u), there exists a F(x) on [h, T] such that for all A on [h, T]and all motions o[ t, x, F(xl), A], we have

(1) g([ T, t,, x,, F(x,), A]) >-- v--.From Lemma 6.5, with z’= and 0 the identity map, and from the continuity of g we

* SIAM J. Control. Optim., 23 (1985), pp. 172-196.

" Department of Mathematics, Purdue University, West Lafayette, Indiana 47906.

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ERRATA 741

get that for every x ebb(u) there exists a 6(x)> 0 such that if Ix-xl< 6(x), then

(2) g(p[ T, t,, x, F(x,), A*]) >. v, a/2 >. v0+2

for all A* over [q, T].The open balls B(x, 6(x)) in R" with center at x and radius 6(x)>0 cover

(u). Since (u) is compact, there exists a finite set of points ,,.., .k in (u)such that the balls B(i, 6(i)), i= 1,..., k cover (u).

The remainder of the proof is devoted to the definition of a strategy on [r, T]such that for all strategies A over r, T] and all motions o[ r, s, F, A], we have

(3) g(0[ T, n , r, a]) => Oo+.This, of course, will lead to a contradiction of the assumption that (r, sc)e C(vo), andthe lemma will be proved.

Corresponding to each of the points ,..., g.k which serve as centers of theballs in the finite open cover of (u) there exists a strategy F(g.) as in (1). LetII’(i)={IIl.,} denote the sequence of partitions of [t,, T] associated with F(:.),

1,..., k. Let II’, be the partition of [q, T] that is the common refinement of1-I,,..., II,,. Let 1-I, be the partition of [r, T] such that t is a partition point, theinterval [z, fi] is partitioned into n equal subintervals and the interval [fi, T] ispartiti^oned by II’,. We thentake II {II,} to be the sequence of partitions associatedwith F. It follows from the definition of II that JlII,[I--> 0 as n-->

For each n 1, 2,. , we now define F, (F,,, , F,.p). Let zo < r < <rp T be the partition points of II,. Let the integer r be such Zr q. For 1, , rwe define F,.i to be the mapping that always selects the function u on the interval[Ti--1, ’i)" Thus

(F,.,([ro, r_,) [Zo, ri_,)))(t) u(t), < t< ’ri_

for i=l,...,r.For i= r+ 1,..., p we define F,, as follows. Let v ’[z, q). The pair of controls

(u, v) determine a function o uniquely as the solution of the differential equation

(4)dt

f(t, x, u(t) v(t), X(Z) .If (q, o(t)) does not belong to the cover LI k= B(,i, 6(i)) of (u), we define ’,.,,

r+ 1, , p to be the map that always selects a fixed element y* in Y. If (fi, o(fi))belongs to the cover there will be a smallest index io, which we take to be 1 fortypographic simplicity, such that (q, o(t)) B(g, 6(:)). For i= r+ 1, r+2,..., p,we shall take F,. to be F,() in the following sense.

Let t r < r, < ri, < < ’p =Zp T denote the points in the partition II, thatare also points in the partition II., for F,(). Let u,.i, be the control that F,.()selects on [’r, ri,). The map ’,.r+ then selects the control u,., on [rr, rr+,). For anyother integers such that r+i<= i, the map ’,.r+ will select the control u,., on theinterval [zr+_, zr+). If u,.i denotes the control that F,.2() selects on [r,, -2) thenat any partition point of the form r,+j with i +j =< i2, the map ’,.,+j will selecton [r,,+_, ri+). Proceeding in th fashion we define all of the maps ’,,, i=r+ 1,..., p. Thus, we have defined F, (F,.,..., ’,.p). We define F {F,}.

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742 LEONARD D. BERKOVITZ

Now let A be any strategy for Player II over [-, T]. Then (’, A) will result in asequence of outcomes (un, vn), where u,(t) u(t) for -<_- t<_- t. Let Ar At(A) be thestrategy on tl, T] such that the partition of tl, T] is that induced by A on this intervaland such that At,, is the constant component strategy which always selects v, at thenth stage.

Let o[ , , F, A] be any motion in the game over [-, T] resulting from (F, A).Let o,,( -, ,, un, vn) be the relabeled subsequence of nth stage trajectories converginguniformly to [ r, , ’,A]. Since " always chooses u over [% tl] we have thatxl [tl, ’, :, [’, A] belongs to (u). Thus x B(gi, t(gi)) for some e {1,. ., k}.Let i0 be the smallest integer for which this is true. To simplify the notation, we againsuppose that io 1.

Let

Xn n(t, "r, Sin, Un, Vn): (Pn( tl, % Cn, U, Vn),

where the equality occurs because u u for -= =< t. For t-> tl

(5) c#,,(t, % , u,,, vn)= q,(t, t,, Xl,,, u,,, v,,),

where o( t, Xl,, un, vn) is the solution of (4) with u u, v v and initial conditionsx(t) x. Since x B(, ()) and Xl- x, there exists an integer N such thatfor n > N, x, B(, (.1)). Thus, for n > N, the pair (u,, v), which is the outcomeof (F,, A) on [-, T], when restricted to the interval Its, T] is the outcome ofand At.,. If we let n-->oo in (5) we can conclude that there exists a motionq[ tl, x, F(., )A(A)] such that for tl _<- _<- T

tp[ t, % r, t, tl, X1,

Since x B(g, ((gl)) it follows from (2) that (3) holds for the given A and thegiven motion. However, A and the motion were arbitrary, so (3) holds for all A andall motions q[ r, :, F, A]. The lemma is proved.

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