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CHAPTER 12

Existence Theorems: The Case of an Exceptional Set of No Growth

12.1 The Case of No Growth at the Points ofa Slender Set. Lower Closure Theorems

Any of the growth conditions (')11), (')12), (')13) in (11.1.i) can be remarkably reduced. Indeed, we may assume that on the points (t, x) of a "slender" subset S of A no growth condition holds. We shall see that this will lead to a notable enlargement of the class of problems for which we can prove the existence of an optimal solution.

Given a fixed set S of the tx-space, x = (Xl, ... ,x"), and any set rx of the t-axis, we shall denote by Pi = Si(ex) the set of all real numbers ~ such that for some (t, x) E S we have t E ex, Xi = ~, i = 1, ... , n. We shall say that Si(ex) is the image of ex on the xi-axis relative to the set S. Note that when S is the graph ofa curve x = g(t), t1 ::;; t::;; t2, in the tx-space, or Xi = gi(t), t1 ::;; t::;; t2, i = 1, ... , n, then Si(ex) = gi(ex) is exactly the image of ex on the xi-axis by means of the component gi of g.

A subset S of the tx-space R1 +", X = (Xl, ... , x"), is said to be slender if the following property holds: (S) For every set ex of measure zero on the t-axis, the sets Si(ex) also have measure zero on the xi-axis, i = 1, ... , n. In other words, S is slender if meas ex = ° implies meas Si(ex) = 0, i = 1, ... , n.

Any finite set S is slender. Any set S contained in a countable family of straight lines parallel to the t-axis is slender. Now consider any set S contained on countably many curves C in R 1 +" of the type C:x = x(t), tE 1, where x(t), t E 1, is any AC n-vector function on an interval 1 of the t-axis. Any such set S also is slender.

Also, if F is the product in R" of sets of measure zero Fi on the xi-axis, i = 1, ... , n, and S is contained in the family of straight lines parallel to the

403

L. Cesari, Optimization—Theory and Applications© Springer-Verlag New York Inc. 1983

404 Chapter 12 Existence Theorems: The Case of an Exceptional Set of No Growth

t-axis x = c, C E F (that is, Xi = ci , Ci E F;, i = 1, ... , n, c = (c 1, ... , cn) with c E F = F1 X .,. x Fn, then F is slender.

Here are examples of functions F o(t, X, x') which satisfy the local growth condition (')'1) (Section 11.1) at all points (f,x) E A = R1+n but the points (I, x) of an exceptional set S. For each example the set S is stated.

(a) n = 1, F ° = (t2 + X2)X,2, S = [(0,0)] slender; (b) n = 1, F ° = IX2 - t21x,2, S = [(t, x) I x = ± t, t E R] slender; (c) n = 1, Fo = x2 sin2(x- 1)x'2 if x ¥- 0, Fo = ° if x = 0, S = [(t,x)lx = 0,

x = (kn)-l, k integer, t E R] slender; (d) n = 1, F o = Ixlx'2, S = [(t,O), t E R] slender; (e) n = 2, F ° = It2 - x 2 - y21(x,2 + y,2), S = [(t, x, y)1 t = ± (x2 + y2)1/2,

(x, y) E R2] not slender; (f) n=2, Fo=[lx-tl+ly-t21](X,2+y'2), S=[(t,x,y)lx=t,y=t2,tER]

slender.

Of course, for F 0=(1 + X'2)1/2, F ° =x(1 + X'2)1/2, F ° =(x- a) 1/2(1 + X'2)1/2 all points (t,x) are exceptional. These are slow growth integrands, for which we refer to Chapter 14, Section 14.4, Examples 1,3, and Counterexamples 2,3.

12.l.i (A LOWER CLOSURE THEOREM WITH A SLENDER SET OF EXCEPTIONAL

POINTS). Let A be compact in the tx-space R 1+n, x = (Xl, ... ,xn), for every (t, x) E A, let Q(t, x) be a set of points (ZO, z) E R1 +n, Z = (Zl, ... , zn), and let Q(t, x) denote the projection of Q(t, x) on the z-space. Let us assume that (a) if (ZO, z) E Q(t, x) and zO' :2: zO, then (ZO', z) E Q(t, x}. Let S be a closed slender subset of A, and let us assume that (b) for every point (t, x) E S there is a neighborhood N~(t,x) and real numbers v> 0, rand b = (b 1, ... , bn), such that (t,x) E N~(I,x) n A, (zo,z) E Q(t,x) implies z°:2: r + b· z + vlzl; (c) for every point (T, x) E A - S there is a neighborhood N b(T, x) and, for every 8 > 0, an L-integrable function IjJ At), I - b :S; t :S; I + b, such that (t, x) E N b(f, x) n A, (zo, z) E Q(t, x) implies Izl:s; 1jJ,(t) + 8Zo. Let 1Jk(t), xk(t), tlk:S; t:S; t2b k = 1, 2, ... , be a sequence of functions, 1Jk(t) scalar L-integrable, Xk(t) = (Xl, ... , Xn) AC in [t lk,t2k], such that

(t, xk(t)) E A, (1Jk(t), x~(t)) E Q(t, xk(t)), t 1k :S; t:S; t2k , (a.e.), k = 1, 2, ... ,

lim inf i tlk 1Jk(t) dt < + 00. k--+oo J'lk

Then, the trajectories xk are equicontinuous and have equibounded lengths. Also, there is a subsequence [ks] such that Xks converges in the p-metric toward a continuous and AC function x(t), t 1 :S; t:S; t2.

In addition, if i = lim inf s:~~ 1Jk(t) dt, and we know that for almost all IE [t 1,t2] the sets Q(t,x) have property (Q) with respect to (t,x) at (I,x(I)), then there is a real valued function 1J(t), t 1 :S; t :S; t2, such that (t, x(t)) E A, (1J(t), x'(t)) E Q(t, x(t)), t1 :S; t:S; t2 (a.e.), and - 00 < s:~ 1J(t) dt :S; i < + 00.

12.1 The Case of No Growth at the Points of a Slender Set. Lower Closure Theorems. 405

In the proof of (12.1.i) below we shall denote by Q* the class of all pairs '1 (t), x(t), t 1 :-:; t:-:; tz, '1(t) scalar L-integrable, x(t) = (xl, ... ,xn) AC in [t 1,tZ] with (t,x(t)) E A,('1(t),X'(t))E Q(t,x(t)), tE [tbt2] (a.e.).

For the proof of this theorem we need a simple preparatory lemma. To this effect, let g(t), a :-:; t :-:; b, denote any real valued continuous function, let m and M be the minimum and maximum of g in [a, b ], and let V be the total variation of g in [a, b], 0 :-:; V :-:; + 00. For every y real, let N(y) denote the number of distinct points t E [a, b] where g(t) = y, so that 0:-:; N(y):-:; + 00

for all y, N(y) = 0 for y < m and for y > M, N(y) ;?: 1 for m :-:; y :-:; M.

12.1.ii (BANAC:H). The function N(y) is measurable, and for its Lebesgue integral (finite or + 00) we have the identity

+ 00 ;?: f-+: N(y)dt = S: N(y)dy = V;?: M - m;?: O.

For proofs of this statement we refer to Banach [1], Saks [I, p. 280], and Cesari [25]. We shall not need this statement in such generality. What we need is the following related statement which we shall prove directly below.

12.l.iii. If the real valued continuous function g(t), a :-:; t:-:; b, is AC, g(a) = c, g(b)=d, c<d, if G is any open subset of [c,d], and if E=g-I(G)= [t E [a,b] Ig(t) E G] and E' = [a,b] - E, then h' Ig'(t)ldt;?: d - c - meas G.

Proof. First let us assume that G is the union of finitely many disjoint intervals (ci,dJ, i = 1, ... ,N. It is not restrictive, in this case, to assume that

c = do :-:; C 1 < d1 :-:; Cz < ... :-:; CN < dN :-:; cN + 1 = d.

Let us consider the N + 1 intervals [di ,C i+l], i = 0,1, ... , N, disregarding those, if any, which are reduced to single points. Also, note that E = [t E [a, b] I g(t) E G] is a set, open in [a, b], whose components, are disjoint intervals (ai' b), j = 1,2, ... , in [a, b]' Now, for each i = 0, 1, ... , N, and interval [dj, ci + 1]' there is at least one interval [Pi' qJ c [a, b] such that g(t) spans exactly [d i, ci + 1] as t spans [Pi' qi] (not necessarily monotonically). There are at most finitely many of such intervals, and we choose one of them we denote [Pi' qJ The intervals (Pi' qJ are disjoint, and if E* = Uf= 0 (Pi' qJ, then E* n E = 0, or [a,b] - E => E*. Thus,

{ _ Ig'(t) I dt;?: {* Ig'(t)1 dt;?: I I (qi g'(t) dtl Jla,bl E JE i=O Jp,

N

= L Ig(qJ - g(Pi)1 = d - c - meas G. i=O

Now let G be an arbitrary open subset of [c, d], and let gi, i = 1, 2, ... , denote the disjoint open intervals which are the components of G. Let E = [t E [a,b] Ig(t) E G], and let [ej,j = 1,2,.,.] denote the components of E. For every m let us consider the open sets Gm = gl U g2 U .. , U gm and


Em = [t E [a, b] I g(t) E Gm], noting that each Em is made up of a collection of the components ej. By the previous estimates we have

r Ig'(t)1 dt ~ d - c - meas Gm • J[a,bj-Em

Now Gm c G, Em C E, [a,b] - Em::;) [a,b] - E, and as m~ +00, we have Gm i G, [a,b] - Em! [a,b] - E, and hence

r Ig'(t)1 dt ~ d - c - meas G. J[a,bj-E o

Proof of (12.l.i).

(a) We shall denote by Nh(I, x) the open ball of center (I, x) in W+ 1 and radius h > 0. Since SeA, S closed, A compact, then S is compact. Let (J = 2Jn+I + 1. For every (t, x) E S let (j > 0, v> 0, r, b = (b 1 , ••• ,bn ) be the numbers such that (t, x) E N all, x) n A, (Zo, z) E Q(t, x) implies Zo ~ r + b . z + vlzl. Actually, we shall consider the smaller neighborhoods N AI, x), which still form an open cover of S, and finitely many of them, therefore, cover S, say N b,(ti' Xi), i = 1, ... , M. Let (j = min (ji' We divide the tx-space R1 +n into cubes of side length (j (and therefore of diameter In+!(j), by means of the hyperplanes t = M, Xi = li(j, i = 1, ... ,n, h, Ii = 0, ± 1, ± 2, .... Finitely many of these cubes have points in common with A, and they cover A; let us denote them by Q, 1= (/1' ... ,In). Anyone of these cubes has side length (j and diameter In+!(j. Some of these cubes, say Qhb may have points in common with balls Nb,(ti,X;), but then b s bi, bi + In+!b < (Jb i ,

and then Qhl is completely contained in the larger ball N abi(t;, xJ We associate to such Qhl the expression ~hl = ri + bi • Z + vilzl relative to N ab.(t;, Xi), and we denote it by ~hl = rh1 + bh1 . z + vhllzl.

(b) We shall now refine the partition {Qhl}' Note that the cubes Qhl' which all together cover A, when projected on the t-axis, are contained in a minimal interval [p(j, q(j] with p < q integers. For every to E [pb, qb], {to} has measure zero on the t-axis; hence Si( {to}) has measure zero, i = 1, ... , n, and can be covered by an open set Fi of measure sli for any ° < Ii < 15/2. If H(to) denotes the hyperspace t = to, and F = F1 X ... x Fn c W is the Cartesian product of these open subsets, then F ° = [(to, x) I x E F], is an open subset of H(to), and (H(to) - F 0) n A is compact and free of points of S. In other words, (H(to) - F 0) n A c A - S. Since S is closed, and (H(to) -F 0) n A compact and free of points of S, the minimum distance of the two sets Sand (H(to) - F 0) n A is positive. Thus, there is a Po > ° such that (to - Po, to + Po) x (H(to) - F 0) n A also is free of points of S. In other words, the set of points of S in the slab (to - Po, to + Po) x Rn is contained in (to - Po, to + Po) n F. For each (to, x) E (H(to) - F 0) n A, and given N > 0, there are p, ° < p S Po, and an L-integrable function r/lo(t) ~ 0, to - pst s to + p, such that (t, x) E N 3p(to, x) n A, (Zo, z) E Q(t, x) implies Izl s r/lo(t) + N- 1zO. The compact set (H(to) - Fo) n A can be covered, therefore, by finitely many of these balls, Nps(to,xs), S = 1, ... , M 2 , and we

12.1 The Case of No Growth at the Points of a Slender Set. Lower Closure Theorems 407

take p=min[p"s= 1, ... , M 2] and also t/!N(t) = max [t/!o,(t), s= 1, ... , M], where t/!o,(t) is the function t/!o relative to N p.(to, x,).

In this manner we have associated an open interval of the form (t- p, t + p) with each t E [pb, qb], thereby obtaining an open covering of [pb, qb]. Thus, finitely many of them cover [pb, qb]. By a suitable contraction, these finitely many intervals can be used to define a partition P:pb = to < tl < ... < tR = qb of [pb, qb], and it may be assumed without loss of generality that the points sb for p :::; s :::; q, s integer, are all used in the partition. Now we can refine the partition of A into parts Qht by means of the hyperplanes t = t j , j = 1, ... , R. The new parts are intervals, which we still denote by Qjt. We shall call them cubes, for the sake of simplicity. Their sides parallel to the xi-axis have all length b; their sides parallel to the t-axis have lengths tj - tj - 1 :::; b.

Summarizing, the following type of partition has been obtained. Given J1 > 0, N integer, there are expressions ~jt as above and a partition of the tx-space into cubes Qjt as above, whose edges in the xi-direction have length b > 0 independent of J1 and N, such that (a) Zo ~ - Nt/!N(t) + Nizi for all (t, x) E A, (ZO, z) E Q(t, x), and (t, x) in any of the cubes Qjt of the slab (tj_l, tj) x W minus the set Hj = (tj-I,t) x F 0; and (b) ZO ~ ~jt for all

° - . (t, x) E A, (z ,z) E Q(t, x), and (t, x) E Hj = (tj- l , t) x F 0' In thIS second case then the relation ZO ~ ~jt holds also in anyone of the 3n - 1 cubes not in Hj of the same section (t, x) E A, tj_ 1 :::; t:::; tj. The projection of Hj on each of the xi-axes has measure :::;J1. Note that the constants b, rjt, bjt, Vjt are independent of J1 and N. Let r = maxhtl, bo = maxlbjll, v = min Vjt, and take o < J1 :::; b/2 and N > 2bo(l + 4.Jn+l).

(c) Let '1(t), x(t), a:::; t:::; b, be any element of Q*. Let C/x = x(t), tj_ 1 :::;

t:::; tj, denote the part of C:x = x(t), a:::; t:::; b (if any) defined in [tj-I,tj]. Divide Cj into more subarcs Cjl' ... , Cn'} as follows: the first end of Cjl is x(tj_ l) [or x(a) if tj - l < a < tj]; the second end point is either the first point where Cj leaves the 3n - 1 cubes in the section [(t, x) E A, tj- l :::; t < tJ adjacent to the cube containing x(tj _ I ), or x(tj ) if Cj does not leave these 3n

cubes (or x(b) if tj _ 1 < b < t}. Continuing in this manner, Cj is broken up into arcs Cjs> S = 1, ... , T j • This process must terminate after a finite number of steps, since each arc Cjs> except the first and the last one, has length ~ b.

Let A j, be the set of all t in the domain of Cj, where x(t) E H j; let A), be the complement of A j, in this domain. LetAjs = SA}s Ix'(t)1 dt, A). = S Ajs Ix'(t)1 dt. We have

1= (b '1(t)dt = 'L(i . + i '. )'1(t)dt Ja A Js AJS

~ 'L i . [rjs + bjs . x'(t) + Vjslx'(t) I] dt AJs

+ 'L tjs [ -Nt/! N(t) + Nlx'(t)1J dt

~ - r(b - a) - 'Lbo jt}S x'(t) dtj + VA + N A.' - S: Nt/! N(t) dt,


where A = LAjs> A.' = LA} •. On the other hand

I(Lis + LJs) x'(t) dtl ::; 2c5Jiz+1,

If X'(t)dtl- f '. Ix'(t)1 dt ::; 2c5Jiz+1, Ajs A Js

ILis X'(t)dtl ::; 2c5Jiz+1 + A}.

for all j and s. Moreover A}. ~ c5 - J1. ~ c5 - c5/2 = c5/2 for s = 1, ... , Tj - 1, A}Ti ~ O. Let D denote the diameter of A. Then

Finally,

-boLILis X'(t)dtl ~ jtl ~~11 [-bo(A}. + 2c5Jiz+1)]

R

- bo L (A}Ti + 2c5Jiz+1) j= 1

R Ti-1 ~ -bo L L [A}. + 4Jiz+1A}.]

j= 1 .= 1

R

- 2boRc5J1l+l - bo L AJTi j= 1

I ~ -rD - 2boRc5Jiz+1 + VA

+ [N - bo(1 + 4J1l+1)]A.' - f qij NI/IN(t)dt Jpij

and ifvo = minE V, 2 -1 N] and I = J~ I'/(t) dt ::; M 0, we also have

Mo~I~ -rD-2boRc5Jiz+1- fqij NI/IN(t)dt+vo fblx'(t)ldt. Jpij Ja

Thus, given any constant M 0, for any pair 1'/, x in Qo with J~ I'/(t) dt ::; M 0,

the trajectory x has uniformly bounded total variation V[ x], and thus uniformly bounded length. Let L be a bound for the lengths of the trajectories x ofthe collection Q~ of the pairs 1'/, x in Q* with J~ I'/(t)dt::; Mo.

Note that I = J~ I'/(t)dt is also bounded below in Q~. This can be derived from the last inequality, which indeed yields

I ~ -rD - 2boRc5Jiz+1- fq(j NI/IN(t)dt. Jp(j

Actually, a stronger statement can be proved: For every element I'/(t), x(t), a::; t::; b, ofQ~ and measurable subsetE of [a, b] we have JE I'/(t)dt ~ - rD -

12.1 The Case of No Growth at the Points of a Slender Set. Lower Closure Theorems 409

boL - Jq~ Nl/IN(t)dt. Indeed, if El = [t E E\(t,x(t)) E H j for somej] and E2 = E ~ E 1, then 1J(t) ~ - Nl/I N(t) for t E E2, and 1J(t) ~ - r - bo\x'(t)\ for t EEl' and

f 1J(t)dt ~ f (-Nl/IN(t))dt + f (-r - bo\x'(t)\)dt JE JE2 JE!

~ -rD - boL - t: Nl/IN(t)dt.

Let Z denote this last number. Thus, in particular I = J~ 1J(t) dt ~ Z for every pair 1J, x in Q('j.

(d) Let us consider again the family Q('j of all pairs 1J(t), x(t), a :::; t:::; b, contained in Q* with J~ 1J(t)dt:::; Mo, and let us prove that the trajectories {x(t)} are equicontinuous. If they are not, then there is an B > 0 such that for every positive integer k there is some pair 1Jk(t), xk(t), ak :::; t :::; bk, in the class Q('j and two points tkl , tk2 E [ak, bkJ such that 0 < tk2 - tkl < k- l , \Xk(tk2 ) - Xk(tkl )\ > B, and Ik = J~~ 1Jk(t)dt:::; Mo. Let us suppose, without loss of generality, that tkl ~ to, tk2 ~ to, Xk(tkl ) ~ Xl' Xk(tk2 ) ~ X2 as k ~ 00. Then \x2 - Xl\ ~ B. The sets Si({tO}) have measure zero. HeliCe, they can be covered by open sets Fi of measure :::; tt, for any 0 < tt < B/4n. Let F denote the set F = [(to, x) \ Xi E Fi, i = 1, ... , n]. Then F is open in the hyperplane H(to):t = to. Let N ~ (4n/B)[M o + 1 + \2Z - 2rb - boL - I\J, where r, b, and bo are the constants defined above. The set (H(to) - F) n A is compact, and for every point (to, x) E (H(to) - F) n A there is some p > 0 and Lintegrable function l/IN(t) such that (t, x) EN p(to, x) n A, (ZO, z) E Q(t, x) implies \z\:::; l/IN(t) + N-lzO. A finite number of these neighborhoods cover (H(to) - F) n A. Let p be the minimum p for such finite covering. It is not restrictive to assume p :::; b. Divide the curve C:x = Xk(t), ak:::; t:::; bk, into three parts Ckl , Ck2 ' Ck3 according as ak :::; t:::; tkl , tkl :::; t:::; tk2 , tk2 :::; t:::; bk. Divide the interval [tkl ,tk2 ] into two subsets, say E2 = [tIX(t)EF], El =

[tkl' tk2J - E2· Then, for some ko and all k ~ ko, we have Itkl - tol :::; p, Itk2 - tol :::; p, and Ixk(tkl ) - xk(tdl ~ B/2. For k ~ ko we also have

~ 2Z + (fE! + fE2)1Jk(t)dt

~2Z+ f (-Nl/IN(t)+N\x~(t)\)dt+ f [-r-bo\x~(t)\Jdt JE! JE2

HereE2 is an open subset of [tkl' tkJ and we know that \x(tkd - x(td\ ~ B/2. Thus, for at least one component, say Xl, we also have \Xl(tkl ) - xl(td\ ~ B/2n, with B/2n > tt ~ meas Fl' Thus, by (12.l.iii),

f IX'(t)ldt ~ f \x'l(t)\dt ~ 2B - meas Fl > ~ - tt. JE! JE! n 2n


Thus,

Ik = fbk 11k{t}dt ~ 2Z - 2rb - boL + N(~ - J1) - f l k2 NrfiN(t)dt. Jak 2n Jlkl

Since tk2 - tkl --.0, we can take ko so that the last integral is s 1, and because of the choice of J1 and N, we have el2n - J1 ~ el4n, and

Ik = L:k 11k(t)dt ~ 2Z - 2rb - boL - 1

+ [Mo + 1 + 12Z - 2rb - boL - IIJ ~ Mo + 1,

a contradiction. Thus, for the pairs 1], x in Q* with S~ 1] dt s M 0, the trajectories x are equicontinuous.

(e) Let 1]k(t), xk(t), ak S t S bb k = 1,2, ... , be a sequence of pairs from Q* with S~~ 1]k(t) dt s Mo. Then the sequence Xk is equicontinuous and the total variations V[ XkJ are bounded. By a suitable extraction there is a subsequence which is convergent in the p-metric to a continuous function x(t), a S t S b. Let us prove that x is AC. Suppose that x is not AC. Let s, ° s s s 1 = 1( C), denote the usual arc length parameter for the curve c: x = x(t), a S t S b, thought of as a path curve in the tx-space Rn+ 1. Note that given any measurable set E c [a, b J the usual Lebesgue measure lEI of E is the infimum of the number Li ([3i - oc;) for any countable covering (OCi' [3i), i = 1, 2, ... , of E. Analogously, we can define another measure (length measure) I(E) by taking the infimum of the numbers Li (S([3i) - S(OCi» for all the same open coverings of E. Obviously, lEI s I(E). If x is not AC, then there is some set E of Lebesgue measure zero on [a, b] which has positive length measure, or lEI = 0, I(E) = A > 0. Now the n sets Si(E), i = 1, ... , n, have all zero Lebesgue measure. If P = [(t, x) I tEE, x = x(t)], then P n S has projection of zero Lebesgue measure on each coordinate axis. As a consequence, there is some subset E' of E with IE'I = 0, I(E') > A12, and (t,x(t» ¢ S for tEE', or P' n S = 0 with P' = {(t,x)lt E E', x = x(t)}. Let2p be the distance of the two sets P' and S. Let N = (2IA)(IMol + 1 + IZ - Ii). Then there is an L-integrable function rfiN(t) ~ ° such that for (t,x) at a distance sp from P' and (ZO,z) E Q(t,x) we have Izi s rfiN(t) + N- 1zO. Since E' is compact, IE'I = 0, it may be covered by a finite set of open intervals (OCj' [3), j = 1, ... , R, such that if F = Uf=l (OCj,[3) we have SF NrfiN(t)dt < 1, and x maps F into the p-neighborhood of P'. Let ko be such that SF Ixi(t)1 dt > AI2 for all k ~ ko. Finally

I = fbk 1]k(t) dt ~ Z + f 1]k(t) dt Jak JF

~ Z + IF (- NrfiN(t) dt + Nlx~(t)l) dt

~ Z - 1 + NAI2

~ Z - 1 + (IMol + 1 + IZ - 11) ~ IMol + 1 ~ Mo + 1,

a contradiction. We have proved that x is AC.

12.2 Existence Theorems for Extended Free Problems with an Exceptional Slender Set 411

We have proved the first part of(12.l.i). The second part is a corollary of (8.8.i) and Remark 2 of Section 8.8. D

12.2 Existence Theorems for Extended Free Problems with an Exceptional Slender Set

We are interested here in existence theorems for the minimum of Bolza and Lagrange problems

(12.2.1) lEx] = g(tl ,X(tl ),t2,X(t2)) + f r2 Fo(t, x(t),x'(t)) dt Jr!

under the usual constraints

(t, x(t)) E A, x'V) E Q(t, x(t)),

where Q(t,x) are given subsets of Rn, B C R2+2n, A c Rn+l. Then, for every (t, x) E A we denote by Q(t, x) the set Q(t, x) = [(ZO, z) I ZO ~ F o(t, x, z), z E

Q(t, x)]. The existence theorems of this section can be thought of as variants of (11.1.i) where the global growth conditions (}'1) or (}'2) or (}'3) are replaced by the local growth condition (gl) of Section 10.4, which is assumed to hold at all points (t, x) E A but those of a slender subset S of A. At the points (7, x) E S we shall need a much milder condition. On the other hand, the present theorems will be based on uniform convergence oftrajectories (mode (a) of Section 2.14), and in this situation it appears convenient to assume F 0 continuous in its arguments.

The local condition which will be assumed at the points of S is only a geometrical transcription of condition (b) in (12.l.i):

JJ. We say that the local condition (13) is satisfied at the point (7, x) of A provided that there are a neighborhood N .If, x), a vector b = (b l , ..• ,bn ) E

Rn, and numbers r real and v > 0 such that (t, x) E N ~(7,x) n A, Z E Q(t, x) implies Fo(t,x, z) ~ r + b· z + vlzl.

Here b . z = (b, z) denotes the inner product in Rn. Summarizing, let A be a subset of the tx-space Rl +n, and for every (t, x) E A let Q(t, x) be a given subset of the z-space Rn, x = (Xl, ... ,xn), z = (Zl, ... ,zn). Let M 0 be the set Mo = [(t,x,z)l(t,x) E A, z E Q(t, x)] C Rl +2n, let Fo(t,x,z) be a given realvalued function defined on M 0, and let us extend F 0 to all of R 1 + 2n by taking Fo = +00 on R1+2n - Mo. Let B be a given subset of the tlxlt2x2-space R 2n+2, and let g(tl,Xl,t2,X2) be a real valued function on B. Let Q be the class of all AC functions x(t) = (Xl, ... , xn), tl :::; t:::; t2, satisfying

(12.2.2) (t, x(t)) E A, x'(t) E Q(t, x(t)), t E [tb t2] (a.e.),

(tl,X(t l),t2,X(t2)) E B, Fo(',x('),x'(')) E L l [t l ,t2].


12.2.i (AN EXISTENCE THEOREM FOR EXTENDED FREE PROBLEMS WIlli A

SLENDER SET OF EXCEPTIONAL POINTS). Let A be compact, B closed, Mo closed, glower semicontinuous on B, F o(t, x, z) continuous on M 0, and assume that the sets (2(t, x) satisfy property (Q) with respect to (t, x) at every point (I, x) EA. Let S be a closed slender subset of A, and assume that (a) for every point (I, x) E A - S the local growth condition (gl) holds; (b) for every point (I, x,) E S condition (f3) holds. Then the functional I[x] in (12.2.1) has an absolute minimum in the class Q of all AC functions x(t) = (Xl, ... ,xn ),

t 1 :::; t :::; t2, satisfying (12.2.2).

Remark 1. In Theorem (12.2.i) the required condition (Q) of the sets Q(t, x) at the points (t, x) E A is, under mild assumptions, an immediate consequence of the other hypotheses. Indeed, if we assume that the extended function T(t, x, z) is lower semicontinuous in (t, x, z), and convex in z for every (t, x) E A, then the sets Q(t, x) are convex and have property (K) with respect to (t, x) in A. They have property (Q) at every point (t, x) E

A - S as a consequence of property (gl) and (1O.5.i). Concerning the points (t, x) E S, let us assume that (X) for every (t, X)E S, Z E Q(t, x), and I: > 0 there are constants r,b=(b\> ... ,bnl real, .5>0 such that T(t,x,z)<r+b'z+l:, and (t,x)EIVd(t,x) implies T(t, x, z) ;?: r + b . z for all z. Also we assume that (IX) for every (Y, xl c S, then (zo, z) E nd cl co Q(7, x,.5) implies that z E Q(7, x). Here, as usual, Q(Y, x,.5) = [UQ(t, x), (t, x) E IV d(Y, xl]. As we prove in (17.5.i), property (Q) at (Y, x) is equivalent to properties (lXl and (X) together.

Remark 2. In Theorem (12.2.i) the sets Q(t, x) are defined by Q(t, x) = epiz Fo(t,x,z). Note that having assumed that A is compact, the assumption "Mo closed and F ° continuous on Mo" corresponds to condition C of Section 11.1. Again for A compact, statement (12.2.i) holds even under the alternative assumption "F ° satisfies condition C*" of Section 11.1.

Proof of (12.2.i). At the points (I, x) E A - S condition (c) of (12.1.i) holds, since this is condition (g2), and we know from (lO.4.iv) that (gl) implies (g2). The sets (2(t, x) satisfy all conditions required in (12.1.i). The set A is compact, and thus there is some M such that (t, x) E A implies It I :::; M, Ixl :::; M. Thus, for every AC trajectory x(t), t 1 :::; t :::; t2, satisfying (12.2.2) we have - M :::; t1 < t2 :::; M, Ix(t)1 :::; M. Since Q is assumed to be nonempty, then for i =

inf I[ x], we certainly have - 00 :::; i < + 00. Note that (t b x(t 1), t2, x(t2 )) E

B n (A x A) for every x E Q, and the lower semicontinuous function g has a minimum - M 1 in the compact set B n (A x A). Let xk(t), t lk :::; t:::; t2k , k = 1, 2, ... , be any sequence of elements Xk of Q such that I[ Xk] = g(e[xk]) + J:~~ F o(t, xk( t), x~( t)) dt ~ i as k ~ 00. If h 1 and I k2 are the two terms whose sum is I[xk], we have Ik1 + Ik2 ~ i and Ikl ~ -M1. There is a subsequence, say still [k] for the sake of simplicity, such that Ikl ~ i1 ~ - M 1, Ik2 ~ i2,

i1 finite, - 00 :::; i2 < + 00, i1 + i2 = i.

As usual we take 1/k(t) = F o(t, Xk(t), x~(t)), t lk :::; t :::; t2k , so that (Ih(t), x~(t)) E

(2(t, xk(t)), t E [tlk' t2k] (a.e.). From (12.1.i) we derive that i2 too is finite, and that there are an L-integrable function 1](t), an AC function x(t), t1 :::; t:::; t2 ,

and a subsequence which we still denote as [k], such that Xk ~ x in the

12.3 Problems of Optimal Control with an Exceptional Slender Set 413

p-metric (in particular, tlk --+ tb t2k --+ t2, and e[xk] --+ e[x]), and

(12.2.3) (1] (t), x'(t)) E Q(t, x(t)), i/2 1](t) dt s i2 < + 00,

J/I

F o(t, x(t), x'(t)) S 1](t), g(e[x]) S liminf g(e[xk]) = il k~r1J

Let us prove that F o(t, x(t), x'(t)) is L-integrable in [tl' t2]. Indeed the graph r = [(t,x(t)), tl s t s t2] of x is a compact set, and for each of its points (I, x(I)) there is a neighborhood N iI, x(I)), say a ball (J of center (I, x(I)) and some radius 2p, for which either (g1) or (f3) holds. The concentric balls (J' of radius p form a cover of r. Hence finitely many of such balls (J' (say (J'b (J~, ... ,(J',;, of radii Pl' ... , PN), cover r. Let Po = min[pi' i = 1, ... ,N]. Now it is easy to divide r into at most Narcs ri:x = x(t), t; s t s t;', each of length ';?Po, each contained in only one of the balls (Jl" .. ,(IN of radii 2Pb ... ,2PN respectively. Now in the balls (Ji corresponding to property (g1), the corresponding function ¢ is bounded below, and so is F o(t, x(t), x'(t)) for t; s t s t;'. In the balls (Ji corresponding to property (f3), then Fo(t,x(t),x'(t)) ';? -r + b· x'(t) + vlx'(t)1 for t; s t s t;', certainly an Lintegrable function. Thus, F 0 has a Lebesgue integral J:~ F 0 dt in [t1' t2] which either is finite, or + 00. The latter case, however, is excluded by relations (12.2.3), or F 0 s 1] with 1] E L 1[t b t2]. This proves that F o(t, x(t), x'(t)) is L-integrable in [tbt2]; hence x E Q, and

is lEx] = g(e[x]) + '1/2 Fodt s g(e[x]) + i'21](t)dt s il + i2 = i. J/l J/I

Since i = i 1 + i2 is finite, equality must hold throughout, or l[ x] = i. Theorem (12.2.i) is thereby proved. 0

12.3 Existence Theorems for Problems of Optimal Control with an Exceptional Slender Set

We are interested here in existence theorems for the minimum in Bolza and Lagrange problems of optimal control

(12.3.1)

(12.3.2)

l[x,u] = g(t1,X(t1),t2,X(t2)) + fo(t,x(t),u(t))dt, f, 12

II

dx/dt = f(t,x(t),u(t)), t E [tbt2] (a.e.),

u(t) E U(t,x(t)),

The existence theorems of this section can be thought ofas variants of(11.4.i) where the global growth conditions (g1'), or (g2'), or (g3') are replaced by a local condition (g1') which is assumed to hold at all points (T, x) E A except those of a slender subset S of A. At the points (I, x) of S we shall need a much milder condition (f3). Again, as in Section 12.2, the theorems are based on uniform convergence, and we assume that fo and f are continuous functions of their arguments.


Summarizing, let A be a subset of the tx-space Rl +n, for every (t, x) E A let U(t, x) be a given subset of the u-space Rrn , and let M = [(t, x, u) 1 (t, x) E A, u E U(t, x)]. Let fo(t, x, u), f(t, x, u) = (fl, ... ,In) be given functions on M, and for every (t, x) E A let Q(t, x) = [z 1 z = f(t, x, u), u E U(t, x)] c Wand Q(t, x) = [(ZO, z) 1 ZO ~ fo(t, x, u), Z = f(t, x, u), u E U(t, x)] c Rl +n. Let B be a given subset of the tlxlt2x2-space R 2n +2, and let g(tt>Xt>t2,X2) be a real valued function on B. Here are the conditions (gI') and (/3') we need:

(gI') We say that the local condition (gI') is satisfied at the point (T, x) of A provided there are a neighborhood N o(T, x), a function </>(0, 0 ~ ( < + 00, bounded below, such that </>(0/( -+ + 00 as ( -+ + 00, and such that for all (t, x) E N o(T, x) n A, u E U(t, x) we have fo(t, x, u) ~ </>(If(t, x, u)I).

(/3') We say that the local condition (f3) is satisfied at the point (I, x) of A provided there are a neighborhood NiT, x), a vector b = (b 1 , ••• ,bn ) E

W, and numbers r real and v > 0 such that for all (t, x) E N o(T, x) n A, u E U(t, x) we have fo(t, x, u) ~ - r + b . f(t, x, u) + vlf(t, x, u)l.

12.3.i (AN EXISTENCE THEOREM FOR PROBLEMS OF OPTIMAL CONTROL WITH

A SLENDER SET OF EXCEPTIONAL POINTS). Let A be compact, B closed, M closed, glower semicontinuous on B, fo(t, x, u), f(t, x, u) continuous on M, and assume that the sets Q(t, x) are convex and have property (Q) with respect to (t, x) in A. Let S be a closed slender subset of A, and assume that (a) for every point (I, x) E A - S the local growth condition (gl ') holds ; (b) for every point (T,X)ES condition (f3') holds. Then the functional I[x,u] in (12.3.1) has an absolute minimum in the class Q of all admissible pairs x(t) = (xl, ... , x"), u(t) = (u 1, • •• , urn), tl ~ t ~ t2, satisfying (12.3.2).

The proof is left as an exercise for the reader.

12.4 Examples

A.

The integrands below satisfy the conditions of (12.2.i). For each we take for A a compact subset of Rl +", Q = R", and we indicate the slender set S of the tx-space Rl +".

1. Fo = Ixlx'2 + (1 + X'2)1/2, n = 1, S = [x = 0, t E R]; 2. Fo = x2(1 - x)2Ix'lq + (1 + X'2)1/2, n = 1, q> 1, S = [(t,x)lt E R, x = 0 and

x = 1]. 3. Fo = (x2 + y2)(X'2 + y'2) + (1 + X'2 + y'2)1/2, n = 2, S = [x = 0, y = 0, t E R]. 4. F 0 = (x - t)2X'2 + (1 + X'2)1/2, n = 1, S = [x = t, t E R]. 5. Fo = (x sin(x- 1»2x,2 + (1 + X'2)1/2 for x #- 0, Fo = (1 + X'2)1/2 for x = 0, n = 1,

S = [(t,x)1 t E R, x = 0 and x = (kn)-l, k = ± 1, ±2, ... ], countably many straight lines parallel to the t-axis.

Bibliographical Notes 415

6. F 0 = [(XZ - tZ)Z + yZ](X'Z + y'Z) + (1 + X'Z + y'Z)l/Z, n = 2, S = [(t, x, y)lx = ±t, y = 0, t E R], two straight lines.

7. F 0 = Ixlx'z + (1 + X'Z)1/2 - 2x', n = 1, S = [x = 0, t E R]. 8. Fo = (x z + yZ)(x'z + y'z) + (1 + X'2 + y'2)1/Z - 3x' - 5y' - 1, n = 2, S = [x = 0,

y = 0, t E R], a straight line parallel to the t-axis.

B.

The functions 10, I below satisfy the conditions of (12.3.i). For each we take for A a compact subset of R1 +n, Q = Rn, and we indicate the slender set S ofthe tx-space R1 +n.

1. 10 = (x2 + yZ)(UZ + vZ) + (1 + u2 + VZ)1/2 + 2- 1(1 + XZ + y2)-1(U + v), 11 = u + v, Iz = u - v, n = 2, m = 2, U = R2, S = [(t,O,O), t E R].

2. 10 = [(x - t)2 + (y - t2)2](U2 + v2) + lui + lvi, 11 = u + v, Iz = u - v, n = 2, m = 2, U = RZ, S = [(t,x,y)lx = t, y = t2", t E R].

3.10 = (Ixllx - tl + Iy + ti)<lul + Ivi)2 + lu - vi, 11 = u + v, 12 = U - v, n = 2, m = 2, U = R2, S = [(t,x,y)l(x = t, y = -t) and (x = 0, y = -t), tE R].

4. 10 = Ixlu4 + (1 + U4 )1/2,f = uZ, n = 1, m = 1, u E R, S = [x = 0, t E R]. 5. 10 = Ixlu4 + (1 + U4 )1/2 - 3u2, I = uZ, n = 1, u E R, S = [x = 0, t E R], m = 1. 6. 10 = X3U2 - x2u, I = x2u, n = 1, m = 1, x ;;:: 0, u E R, S = [x = 0, t E R]. 7. 10 = (1 - X 2)ZU2 + 11 - x2l1ul, I = (1 - x 2)u, n = 1, m = 1, u E R, S = [x = -1,

x = 1, t E R].

12.5 Counterexamples

1. n = 2, Fo = xy' - yx'. All curves joining (0,0,0) and (2n, 0, 0), A = R3. All points of A are exceptional. For the sequence xk(t) = k- 1 sin k4 t, Yk(t) = k- 1 cos k4 t - k- 1, ° ;5; t ;5; 2n, we have I[ Xk' Yk] = - 2nk2 -+ - 00 as k -+ 00. The problem has no absolute minimum.

2. n = 1, F 0 = (x2 + X'2)1/2. All curves joining (0,0), (1, 1). (We have seen in Section 1.5, Example 5 that this problem has no minimum.) All points of A = RZ are exceptional.

3. n = 1, Fo = (x'y - y'x) + (x2 + y2)3(X'2 + y'2). All curves joining (0,0,0) to (2n, 0, 0). The singular set is S = [(t, 0, 0), t E R], a slender set. Condition (fJ) is not satisfied on the set S. For the sequence Xk(t) = k- 1 cos k4 t - k- 1, Yk(t) = k- 1 sin k4 t, ° ;5; t ;5; 2n, k = 1,2, ... , we have I[ Xko Yk] -+ - 00 as k -+ 00. This problem has no absolute minimum.

4. n = 1, m = l,fo = (x + U)2t, I = x + u, U = R, S = [(O,x)lx E R] is not slender. For O;5;t;5;l, x(O) = 1, x(l) =0, T(t,x,z)=min[Jo(t,x,u)lz=x+u, uER]=tz2. The problem is equivalent to I = g tx'2 dt, x(O) = 1, x(l) = 0, which we know has no minimum (Section 1.5, Example 4).

Bibliographical Notes

L. Tonelli [II; [10] and E. J. McShane [10] have proved existence theorems for classical Lagrange problems of the calculus of variations in which the growth assumption was allowed to fail in sets made up ofa finite number of points, or contained in one or finitely


many straight lines, or contained in one or finitely many smooth curves. The concept of a "slender" set of points, including all these cases, can be found in L. H. Turner [1] for Lagrange problems, and in L. Cesari, J. R. LaPalm, and D. A. Sanchez [1] for problems of optimal control. The present exposition is modeled on the latter paper. The lower closure theorem (12.i.i) is difficult to prove, because it also proves that in the present situation a minimizing sequence of trajectories is equicontinuous with integrals bounded below, possesses an AC limit, and the lower closure property holds. The existence theorem (12.2.i) is based therefore on mere uniform convergence, and this requires property (Q) with respect to (t,x). This property however can be guaranteed under mild additional assumptions in the present situation. The existence theorem (12.2.i) includes most ofthe results obtained by L. Tonelli and E. J. McShane for specific forms of the exceptional slender set. The existence theorem (12.3.i) is the expected extension of (12.2.i) to problems of optimal control.

optimization—theory and applications || existence theorems: the case of an exceptional set of no...

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