Optimization—Theory and Applications || Existence Theorems: Problems of Slow Growth

CHAPTER 14

Existence Theorems: Problems of Slow Growth

We discuss here existence theorems for the usual integrals J[ x] = S:; F o(t, x(t), x'(t)) dt as in Section 11.1, but where F o(t, x, x') does not satisfy any of the growth conditions we have considered in Chapters 11, 12, 13. Well known problems are of this kind (cf. Section 3.12). There are a number of methods to cope with these problems; we mention here one based on their reduction to equivalent "parametric problems" (Sections 14.1-2). In Section 14.3 we state a number of existence theorems for the usual integrals J[ x], and in Section 14.4 we present many examples of problems for which the existence theorems in Section 14.3 hold.

14.1 Parametric Curves and Integrals

A. Parametric Curves

The concept of a parametric curve (£ in Rn occurs when we agree to consider a suitable equivalence concept between n-vector continuous maps x = xC,), a:::;; r:::;; b, and y = y(O'), C :::;; 0' :::;; d, x = (xl, ... ,xn), y = (yl, ... ,yn). A parametric curve (£ is then a class of equivalent maps. The concept of equivalence will leave unchanged the sense in which the curve is traveled, and thus we shall speak of oriented parametric curves.

The concept of Lebesgue equivalence is a natural one and must be mentioned: Two continuous maps x and y as above are said to be Lebesgue equivalent if there is a strictly increasing continuous map 0' = h(r), a :::;; r :::;; b, h(a) = c, h(b) = d(or homeomorphism) such that y(h(r)) = x(r) for all a:::;; r :::;; b. For technical reasons only a slightly more general concept is needed, namely the concept of Frechet equivalence. Two continuous maps x and y as above are said to be Frechet equivalent, or F -equivalent, if for every /; > 0 there is some homeomorphism h:O' = h(r), a:::;; r :::;; b, h(a) = c, h(b) = d, such that Iy(h(r)) - x(r)l:::;; /; for all a:::;; r:::;; b. If we represent this relation by writing x - y, it is

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

14.1 Parametric Curves and Integrals 431

easily seen that (a) x ~ x; (b) x ~ y implies y ~ x; (c) x ~ y, y ~ z implies x ~ z. Then a class of F-equivalent maps is called a parametric curve <r, or a Frechet curve (briefly, an F -curve), and every element ofthe class is said to be a representation of <r, in symbols, <r:x = x(r), a:::;; r :::;; b. The main concepts associated with a Frechet curve <r must be F -invariant, that is, must be shared by each of its representations.

For any given F-curve <r:x = x(r), a:::;; r :::;; b, the set [(£] = [x] = [x E Rnlx = x(r), a :::;; r :::;; b] is said to be the set covered by <r in Rn. Then x ~ y implies [x] = [y], and this is certainly a compact subset of Rn.

The Jordan length L[<r] of a Frechet curve G: is then defined as a total variation,

N

L[G:] = Vex] = sup L Ix(ri) - x(ri-1)1, i=l

where sup is taken with respect to all subdivisions a = r 0 < r 1 < ... < r N = b of [a, b J. If x ~ y, one can see easily that V[ x] = V[y], and thus L[G:] = V[ x] is independent of the representation ofG:. An F-curve G::x = x(r), a:::;; r:::;; b, is said to be rectifiable if L[G:] < + 00. A rectifiable curve G:: x = x(r), a :::;; r :::;; b, covers a set [G:], or [x], of measure zero in Rn. The following further properties of rectifiable F-curves G: are relevant:

14.1.i. (a) G::x = x(r) = (xl, ... xn), a:::;; r:::;; b, is rectifiable if and only if the n continuous functions Xi(t), i = 1, ... n, are of bounded variation (BV) (Jordan, 1889).

(b) If G::x = x(t) = (xl, ... ,xn), a:::;; t:::;; b, is rectifiable, then x(t) is BV in [a,b], the n derivatives x'(t) = (X,l, ... ,x'") exist a.e. in [a, b] and are L-integrable in [a, b], and L[G:] = L[x];?: S:lx'(t)ldt (Lebesgue integral of the Euclidean norm of x') (Tonelli, 1912).

(c) The equality holds if and only if x is AC, that is, the n functions Xi are AC in [a, b] (Tonelli, 1912).

(d) If G: is rectifiable, it always possesses AC representations. In particular, the arc length parameter s, 0 :::;; s :::;; L, yields a unique AC representation x = X(s),O :::;; s :::;; L, with IX'(s)I = 1 a.e. in [O,L].

Given two F-curves G:1:x = x1(r), a:::;; r:::;; b, and G:2:x = x2(a), c:::;; a:::;; d, it is useful to have a concept of distance d(G:1,G:2) between G:1 and G:2. The F-distance is defined by d(G:1, G:2) = d(XloX2) = inf[lx2(h(r» - x1(r)l, a:::;; r :::;; b], where inf is taken over all homeomorphisms a = h(r), a :::;; r :::;; b, h(a) = c, h(b) = d. It is immediately seen that if Xl ~ Y1' x2 ~ Y2' then d(X lo X2) = d(Y1' Y2) (that is, d depends only on the two Fcurves G:1, G:2), and that d(X1' X2) = 0 if and only if Xl ~ X2. It can also be proved that (a) d(G: 1, G:2) ;?: 0, and = sign holds if and only if G:1 = G:2; i.e., Xl - X2. (b) d(G:1, G:2) = d(G:2,G:1); (c) d(G:1oG: 3):::;; d(G:1,G:2) + d(G:2,G:3). Thus, the F-curves in Rn form a metric space with distance function d. A sequence of F-curves G:k , k = 1,2, ... , in Rn is said to be convergent to an F-curve G: provided d(G:koG:) --> 0 as k --> 00. It is relevant to note here that G:k --> G: if and only if there are representations, say G:k:x = xk(r), 0:::;; r :::;; 1, G::x = x(r), 0:::;; r :::;; 1, with Xk --> x uniformly in [0,1J.

The Jordan length is lower semicontinuous with respect to this type of convergence, that is:

14.1.ii. If G:, G:k, k = 1,2, ... , are F-curves in Rn and d(G:k, G:) --> 0 as k --> 00, then L[G:] :::;; lim inf L[G:kJ.

432 Chapter 14 Existence Theorems: Problems of Slow Growth

The following compactness theorem for F-curves is most useful:

14.1.iii. (HILBERT). If K is any closed bounded subset of Rn, and {(t} any family of Fcurves lying in K and with equibounded Jordan lengths (e.g., [(t] c K, L[(t] :5: M for some M and all (t E {(t}), then the family {(t} is relatively sequentially compact with respect to the F-distance d. In other words, any sequence (tk> k = 1,2, ... , of elements of {(t} possesses a subsequence [(tk.] which is convergent in the d-metric toward some Fcurve (t, and L[(t] :5: lim infs L[(tk.] :5: M.

Proof. By representing the curves by means of their arc length parameter we have a family {X(s),O:5: s:5: L}, L = L[(t], which is equibounded and equicontinuous, namely Lipschitzian of constant one. The statement is now a corollary of Ascoli's Theorem in the form (9.l.ii). 0

Remark 1. (Existence of a geodesic between two points on a manifold). If M is a continuous manifold in Rn we say that a path curve (t is on M if all its points are on M and (t has a continuous representation in terms of the local representation parameters of M. If there are rectifiable curves joining two points P, Q on a closed continuous manifold M, then there is also a curve joining P, Q on M and of minimum length. For a sketch of proof, let Q denote the class of all rectifiable curves joining P and Q on M and let i be the infimum of the Jordan lengths of the curves in Q. Let (tk> k = 1, 2, ... , be a minimizing sequence, that is, L[(tk] ..... i, and ~k E Q. By (14.l.iii) there is a path curve (to in Rn, and a subsequence, say still [k], such that d((tk, (to) ..... 0 as k ..... 00, and by (14.l.ii) L[(to] :5: i. Now the points of (to are on M since M is closed, and because of the closure of Q with respect to the distance function d, we have (to E Q. Then L[(to] ;;;:: i. and by comparison also L[(to] = i. (See III for details).

Remark 2. The concept of parametric closed curve (to can be introduced as above by identifying logically the end points a and b of the intervals [a, b] taken into consideration (which implies x(a) = x(b». Then [a,b] becomes equivalent to a circumference, and the class of orientation preserving homeomorphisms h is now much larger.) The same concepts can be introduced as before (closed F-curves, distance of two closed Fcurves), and the same theorems hold.

B. Parametric Integrals

The concept of integral over a rectifiable F-curve (t:x = x(t) = (xl, ... ,xn ), a :5: t:5: b, in any of its AC representations x, can be defined as usual as

(14.1.1) I[(t] = I[ x] = s: fo(x, x') dt,

and this integral is independent of the chosen AC representation if and only if (a) fo does not depend on t, and (b) fo is positive homogeneous of degree one in x', that is, fo(x, kx') = kfo(x, x') for all k ;;;:: O. Thus, let A 1 be a closed subset of Rn, and let fo(x, x') be a continuous function on Al X R" satisfying (a) and (b); then the formula above defines I[(t] for any F-curve (t with [(t] c Al and for any of the AC representations of (t. Then, I[(t] is called a parametric integral and fo(x, x') a parametric integrand.


For instance, for n = 2, and by writing x, y, x', y' instead of Xl, x 2, X'I, X'2, the following functions fo all are parametric integrands:

(14.1.2)

fo = (X,2 + y'2)1/2,

fo = (2X,2 + 3y'2)1/2 _ (X,2 + y'2)1/2,

fo = (1 + x2 + y2)[2(x'2 + y'2)1/2 -lx'lJ,

j~ = (1 + x2 + y2)(X'2 + y'2)1/2 + xy' + x'y,

j~ = 2(x2 + i)-1/2(X,2 + y'2)1/2 + (x 2 + y2)-I(Xy' - x'y),

j~ = 2(1 + x2 + y2)3/2(X'2 + y'2)1/2 + x3x' + ly'.

Note that, because of the positive homogeneity of fo in x', the usual sets Q(x) = [(zo, z), Zo ~ fo(x, x'), Z E R"] are here cones with vertex the origin in Rn.

One might think that lower semicontinuity theorems and existence theorems for the minimum of parametric integrals could all be particular cases of those we have proved in Chapters 8-13 for the nonparametric case. This is true only for the simplest theorems. Because of a different emphasis and a great many technicalities, new proofs are needed for new results which have no parallel in the nonparametric case. We refer to [III], which includes also a new approach to the unexpectedly rich class of "parametric problems of optimal control". Here we list only a few existence theorems for the minimum of I[[].

C. Existence Theorems for Parametric Integrals

Here Al denotes a subset of the x-space Rn, Bo a subset of the xlx2-space R2n, and Vo the unit sphere in the z-space, or Vo = [z E Rn Ilzl = 1]. We consider the problem of the minimum ofJ[[] in the class Q of all rectifiable F-curves [:x = x(t), a::; t::; b, [[] cAl>

and satisfying given boundary conditions which, for the parametric case, are ofthe form (x(a),x(b)) E Bo. Alternatively, we may want to minimize I[[] in any d-closed class Q

of rectifiable F -curves [. The following existence theorem is rather typical of the parametric case.

14.1.iv. If Al is compact, Bo is closed, fo(x,x') is continuous on Al X Rn and convex in x', and there is a real valued monotone nondecreasing function cjJ((), - 00 < , < + 00,

such that

(14.1.3) L[[] ::; cjJ(I[[])

for all F-curves [ lying in AI, then I[[] has an absolute minimum in the class Q of all rectifiable F -curves [ in A I .

Here is a short proof of(14.1.iv), which is only a repetition ofproofU for the Filippov theorem (9.3.i). First, it is not restrictive to assume that I[[] ::; M I for all [ in Q and some constant MI' Then, by (14.1.i), also L[[]::; cjJ(I[[])::; cjJ(MI) = M 2 • Let Wo denote the closed unit ball in R", or Wo = [z E Rn Ilzl ::; 1]. Then fo(x, x') is continuous in the compact set Al x Wo, hence bounded there, say Ifol ::; M. For any [ in Q and the arc length representation of [:x = X(s), 0::; s::; L, L = L[[] ::; M 2, we have 1X'(s)1 <= 1 a.e., and hence II[[]I ::; LM ::; M 2M. Thus, i = infn I[[] is finite. Let [k:X =

Xk(S),O::; S ::; Lb Lk = L[[k] ::; M 2, k = 1,2, ... , be any minimizing sequence, so that I[[k] -+ i as k -+ 00. Also, Xk(s) E AI> a compact set, and IX~(s)1 ::; 1, so X~(s) E Wo, a


fixed compact convex set. The functions X k are equibounded and Lipschitzian of constant one, and hence equicontinuous. By Ascoli's theorem (9.l,ii) there is a subsequence, say still [k], such that Xk -> Xo in the p-metric, or uniformly, and xo(s), 0::;; s ::;; Lo ::;; M2, is Lipschitz of constant one and hence AC, and Ixo(s)I ::;; 1, so xo(s) E Wo a.e. in [0, L]. By the same argument as in proof II ofthe Filippov's theorem (9.3,i), we have the lower semicontinuity property J[ xo] ::;; lim infk J[ Xk]. In other words, for the F-curve defined by (l;: x = xo(s), 0 ::;; s ::;; L o, we have J[(l;o] = J[ xo] ::;; i. Since L[(l;o] ::;; lim infk L[(l;k] ::;; M I, (l;o belongs to Q; hence J[(l;o] ~ i, and I[(l;o] = i and (14.1.iv) is proved.

Many existence theorems for the parametric case are actually criteria for (14.1.3) to hold. Here are some of these criteria «a) to (g), which are due to Tonelli and McShane, and we shall prove them in III):

(a) fo(x,x') > 0 for all x E Al and x' E Vo. (b) There is a fixed vector b = (bI, ... ,b") such that fo(x, x') + b . x' > 0 for all (x, x') E

Al X Vo. (c) There is a continuous n-vector gradient function G(x) = (GI, ... , G"), x E AI, such

that fo(x,x') + G(x)' x' > 0 for all (x,x') E Al x Vo. (Here G is called a gradient function ifthere is a functionF(x) of class Cl inAI such that Gi = 8F/8xi, i = 1, ... , n,

in AI)'

For instance, all the examples in (14.1.2) satisfy (a). Analogously

fo = (X,2 + y,2)1/2 + 5x'

satisfies (b) with b = ( - 5,0), and the following examples satisfy (c) with the functions F and G which are indicated:

G = (y,x), F = xy,

fo = (X,2 + y'2)1/2 - (x2 + y2)-I(xy' - x'y), G = (x2 + y2)-1( - y,x),

fo = 2(X,2 + y,2)1/2 -lx'l- (x3x' + y3y'), G = (x3, y3), F = 4 -1(X4 + y4).

In the last but one example, Al could be any simply connected region in R2 not containing the origin, and then F = arc tan y/x).

A point Xo E R" is said to be a zero for fo(x, x') if fo(xo, x') = 0 for some x' E R", x' # O. For instance both

have a zero at (0,0); the first one vanishes there for all (x',y'), the second one for all (x',O) with x' ~ O. The functions

fo = x2y2(X'Z + y,Z)I/Z, fo = (X,2 + y,2)I/Z - y'(1 + X2yl)-1

have a zero at every point (x, y) ofthe x- and ofthe y-axes; the first one vanishes there for all (x', y'), the second one for all (0, y') with y' ~ O. Thefunctionfo = (X,2 + y,2)1/2 - x' has a zero at every point of the xy-plane, since fo vanishes for every (x', 0) with x' ~ O.

We are now in a position to add to the criteria (a)-(c) above for (14.1.3) to hold, the following one:

(d) fo(x,x') ~ 0 in Al x R", and there is a fixed unit vector v = (vI, ... , v"), Ivl = 1, and a number {) > 0 such that fo(x, x') > 0 for v' x' ~ {), (x, x') E Al x Vo.

This condition simply states that fo may vanish at any point x E A, but only in directions x' forming an angle 0::;; f1 ::;; n/2 - {)' with the fixed direction v and


cos(nj2 - 0') = o. The following functions satisfy (d), namely, they vanish at most in the one direction which is indicated:

fo = (1 + x2 + y2)[(X,2 + y'2)1/2 - x'], j~ = 0 at all (x, y) in the sole direction (1,0); fo = (1 + Ixl + lyl)[(X'2 + y'2)1/2 - (X,)1/3(X,2 + y'2)1/3], fo = 0 at all (x, y) in the

sole direction (1,0); fo = (X,2 + y,2)l/2 - (1 + x 2y2)-1 x', fo = 0 at all points (x, 0) and (0, y) in the sole

direction (1,0).

Two further criteria are as follows:

(e) fo(x,x');::: 0 in A1 x R"; fo possesses a finite number of zeros in Ao, say Xi' i =

1, ... , N; and for each j there is a vector Vi = (vL ... ,v~), IVil = 1, and number Oi > 0 such that fo(xb x') > 0 for Vi . x' ;::: Db Ix'i = 1.

(f) fo(x,x');::: 0 in Al x R", fo possesses a set Z of zeros, which are all contained on the finite union of simple continuous curves in Ao, which may have points in common, but form no closed curve in R". Moreover, for every x E Z there is also a vector V,

Ivi = 1, and a number 0 > 0 (both of which may depend on x) such that j~(x, x') > 0 for V • x' > 0, Ix'i = 1.

For instance fo = (X,2 + y'2)1/2 - 2x(1 + x2 + y2)-lX'

satisfies (e) with only two zeros: (1,0) in the direction (1,0), and ( -1, 0) in the direction (-1,0). For instance

fo = Ix'i + 1y'1- 2x(1 + X2)-lX' - 2y(1 + y2)-ly'

satisfies (e) with only four zeros (1,0), (-1,0), (0,1), (0, -1), and fo vanishes there in the corresponding directions (1,0), (-1,0), (0, 1), (0, -1).

For instance,fo = (X,2 + y'2)1/2 - (1 + x 2y2)-1X', already mentioned above, satisfies (f). The restriction on the zeros stated in (e) and (f) can be removed provided fo behaves

suitably around such zeros. Here is one criterion:

(g) fo(x, x') ;::: 0 in Al x R",fo possesses a finite number of zeros in Al, all interior to A 1, say Xi, j = 1, ... ,N, where fo(x;, x') = 0 for all x' E R". However, each point x;, j = 1, ... , N, has the following property: in a neighborhood N of Xi' we have clx - xilY ::;; fo(x, x') ::;; qx - xil Y for all x E N, x' E R", Ix' I = 1, and constants 0 < c < C < 00, y > O. For instance, all functions below satisfy (g) with Al as indicated:

fo = (x2 + y2)(X,2 + y'2)1/2, A1 = [(x, Y)I-1 ::;; x, y ::;; 1],

fo = (a2x 2 + b2y2)(X'2 + y,2)l/2, a, b > 0, A1 = [(x, Y)I-1 ::;; x, y::;; 1],

fo = (x2 + y2)1/3(X'2 + y'2)1/2, A1 = [(x, Y)I-1 ::;; x, y ::;; 1],

fo = [(x2 - 1)2 + y2](X'2 + y,2)l/2, A1 = [(x, Y)I-2::;; x, Y ::;; 2], fo = [(x2 - 1)2 + (y2 - 1)2J1/3(x'2 + y,2)l/2, Al = [(x, y)l- 2::;; x, y ::;; 2]

fo = (x2 + y2)[(1 + x 2 + y2)(2x,2 + 3y'2)1/2 _ (X,2 + y'2)l/2],

A1 = [(x,y)l- 1::;; x,y::;; 1]

fo = ((x2 - 1)2 + (y2 _ 1f + (Z2 _ 1)2)1/4(x'2 + y,2 + Z,2)l/2,

n = 3, A1 = [(x,y,z)lx2 + y2 + Z2::;; 4].

For proofs of criteria (a)-(g) and details see, e.g., [III].


14.2 Transformation of Nonparametric into Parametric Integrals

A. The Parametric Integral .3[<rJAssociated to I[ x ]

We present here existence theorems for free nonparametric problems concerning the usual integral

(14.2.1) 1t2 l[x] = F o(t, x(t), x'(t)) dt

tl

as in Sections 11.1-3, but where F o(t, x, x') does not satisfy any of the growth properties (g) of Sections 11.1-3.

First let us note that, ifin (14.2.1) we think oft as an increasing AC function of a new variable, t = t(r), r 1 ~ r ~ r 2, with t1 = t(r1), t2 = t(r2), then for X(r) = x(t(r», per) = t'(r), we have

lex] = 1:2 Fo(t(r), x(t(r», x'(t(r»)t'(r)dr

= 1:2 Fo(t(r), X(r), X'(r)/p(r»p(r) dr,

with obvious conventions whenever per) = o. We are going to recognize the last integral as a parametric integral in Rn + 1.

Given a set A in the tx-space Rn + 1, X = (xl, ... ,xn), and a scalar function F oCt, x, u), (t, x, u) E A x Rn, U = (u1, ... , un), we introduce an auxiliary variable p > 0 and the new integrand function

(14.2.2) Go(t,x,p,U) = pFo(t,x,u/p)

with Go(t, x, 1, u) = F oCt, x, u). It is convenient to think of (t, x) as a new "space" variable, or (n + I)-vector Z = (ZO, Zl, .•• ,zn), and of (p, u) as a new "direction" variable, or (n + I)-vector w = (p, u) = (p, ut, ... ,un), or Z' = (z'O,z') = (z,O,z't, ... ,zm), so that Go(t,x,p,u) becomes Go(z,z') with ZE A c Rn + 1, Z' E Rn+ 1, Z,O > O. Note that Go is positive homogeneous of degree one in w = (p, u), or z', that is, Go(z, kz') = kGo(z, z') for all k > o. When possible, the scalar function F oCt, x, p, u) will be extended by continuity into a function Go defined for all (t,x) E A, (p,u) E Rn+1, p ~ O.

We shall assumeF o(t,x, u) to be continuous in A x Wand then Go(t, x,p, u) is certainly continuous in A x (0, + 00) x Rn. The function Go mayor may not be extendable into a continuous function in A x [0, + 00) x W. If Go admits of such a continuous extension, then

(14.2.3)

is a parametric integral such as we have considered in Section 14.1 and which is defined for rectifiable F -curves <£: in W + 1 lying in A as the value of (14.2.3) for any AC representation of<£:, namely z(r), r 1 ~ r ~ r2,z AC, or t(r), x(r),

14.2 Transformation of Nonparametric into Parametric Integrals 437

'1::::;;' ::::;; '2, t, X AC, with fer) 2: ° (that is, t monotone nondecreasing). In this situation, if x(t), t 1 ::::;; t ::::;; t2, is any AC n-vector function with (t, x(t» E A for all t E [tl' t2J, and fo(', xC), xl» E L 1[t 1, t2J, then t = t, x = x(t), tl ::::;; t::::;; t2 , is a parametric rectifiable curve-actually a particular representation of a rectifiable F-curve (£. If t = tee), x = xC'), '1 ::::;; , ::::;; '2, is any AC representation of the same curve, then we derive

i t2 ~[(£J = Go(t(,),x(,), t'(,),x'(,»d,

T!

1t2 = Go(t, x(t), 1, x'(t» dt

tl

1t2

= F o(t, x(t), x'(t» dt = I[ x]. tl

Actually it may well occur that Go cannot be extended by continuity in A x [0, + CYJ) X En, and yet (14.2.3) preserves its character of a parametric integral. The parametric integral ,3[(£J is said to be associated to the nonparametric integral I[ x]. Statements concerning the equality ,3[(£J = I[ x J will be proved in [III]. Also, we shall need statements guaranteeing that a parametric curve t = teL), x = Xc!), '1 ::::;; , ::::;; '2, possesses a representation t = t, x(t), t1 ::::;; t::::;; t2, or simply a nonparametric representation x(t), tl ::::;; t ::::;; t2 •

B. Examples

We give here a few usual integrands F o(t, x, u) and the corresponding "parametric integrands" Go(t, x, p, u):

1. n = 1, F 0 = (1 + U2 )1/2, Go = (p2 + U2 )1/2.

2. n = 2, F 0 = (1 + u2 + V2)1/2, Go = (p2 + u2 + V2 )1/2. 3. n = 2, F 0 = (1 + u2 + V2)1/4, Go = pl/2(p2 + u2 + V2)1/4 4. n = 1, Fo = u2 , Go = p- IU2.

We shall often need below the partial derivative Gop of Go with respect to p (under differentiability assumptions on F 0), or

Gop(t, x, p, u) = oG%p = (%p)[pF oCt, x, u/p)J

n

= Fo(t,x, u/p) - p-I L uiFox,,(t,x,u/p). i= 1

For instance, in Examples 1-4 above we have, respectively,

1: Gop = p(p2 + U2)-1/2; 2: Gop = p(p2 + u2 + V2)-1/2; 3: Gop = 2-1p-I/2(p2 + u2 + V2)-3/4(2p2 + u2 + V2); 4 G -2 2

: Op = -p u.


14.3 Existence Theorems for (Nonparametric) Problems of Slow Growth

We begin with some remarks concerning the parametric integrand Go(t, x,p, u), or Go(z, z'), we have just associated to the usual (nonparametric) integrand F o(t, x, x').

It can be proved that, if F o(t, x, x') is convex in x' in Rn, then for all (t, x) E A, Go(t, x, p, u) is convex in (p, u) in the open half space (0, + 00) x Rn. Also, it can be proved that if A is closed, F o(t, x, x') is continuous in A x Rn and convex in x' for every (t, x) E A, then Go(t, x, p, u) is lower semicontinuous in A x [0, + 00) x Rn (and continuous in A x (0, + 00) x Rn). Moreover, if Go(t, x, 0, u) happens to be finite everywhere and continuous in (t, x, u), then Go(t,x,p,u) is (finite everywhere and) continuous in A x [0, +00) x Rn.

Since Go(z, kz) = kGo(z, z) for all k > ° and z E A, z E (0, + 00) x Rn,

we may define G o(z, 0) to be zero, so that the homogeneity property alone holds for all k ~ 0. Furthermore, note that for F 0 convex in x', if we keep (t, x, u) fixed and we allow p > ° to approach zero, then Go(t, x, p, u), a convex function of p, must approach a finite limit or + 00. This limit, finite or + 00, will be taken as a definition of Go(t, x, 0, u), though this function, so defined, may not be continuous at p = 0. For u = 0, then Go(z, x, p, 0) = pF o(t, x, 0) approaches zero as p --+ 0+, and thus the value zero for Go at x' = ° coincides with the value we have already agreed upon on the basis of homogeneity above.

Concerning the possibility of inverting an AC function t(s), a:::;: s :::;: b, into an AC function set), c:::;: t::; d, we mention here the statement: If t(s) is an AC monotone nondecreasing function, a ::; s ::; b, c = tea), d = t(b), then the inverse function set), c ::; t ::; d, of t(s) exists and is continuous and AC in [c, d] if and only if t'(s) > ° a.e. in [a, b ].

As usual we consider the problem of the minimum of the integral

(14.3.1) 112 lex] = Fo(t,x(t),x'(t))dt

I,

in the class Q of all AC functions x(t) = (Xl, ... ,xn ), t 1 ::; t ::; t2, with (t,x(t))EA, (t1,X(t1),t2,X(t2))EB, and F(·,x(·),x'(·)) L-integrable. We say that these are the admissible trajectories. The following existence theorems, and related implications, are proved in [III].

14.3.i (EXISTENCE THEOREM FOR USUAL INTEGRALS). Let A = [to, T] X A 1, where A 1 is a compact subset of the x-space Rn; let B = B 1 X B 2, where B 1, B 2 are closed subsets of the tx-space Rn + 1 such that for every (t1> X1,t2, X2) E

B1 x B2 we have t1 < t2; let Fo(t,x,x') be of class C1 in A x Rn and convex in x' for every (t, x) E A. Let us assume that the associated parametric integrand Go(t, x, p, u) is continuous in A x [0, + 00) x Rn, and that (a) Gop(t, x, 0, u) = - 00 for (t, x, u) E A x Rn, lui = 1, and (b) there are constants M 1, M 2, b > ° such that for all (t, x) E A, Z' = (p, u) E Rn+ 1 with p ~ 0,

14.3 Existence Theorems for (Nonparametric) Problems of Slow Growth 439

IZ'I = 1, and t* with It* - tl < ~ we have IGot(t*, x, p, u)1 :s; M 1 Go(t, x, p, u) + M 2. Let us further assume that (A) there is a monotone nondecreasing function cf>m, - 00 < , < + 00, such that L[{t] :s; cf>(3[{t]) for all rectifiable parametric curves (t: t = t(s), X = X(s), O:S; s :s; L, with graph in A, t(O), X(O), t(L), X(L) E B, t(s) monotone nondecreasing, and s the arc length parameter. Then, I[ x] has an absolute minimum in the class of all admissible trajectories.

For n = 1 condition (A) is certainly satisfied if, say, for every (f, x) E A there are a vector b = (bO,b l ) E R2 and constants v> 0, ~ > 0 such that Go(t,x,p,u) + bOp + blu ~ v for all (p,u) E R2, P ~ 0, and all (t,x) E A at a distance :s; ~ from (f, x).

For n ~ 1, condition (A) is certainly satisfied if, say, Go is continuous in A x [0, (0) x R" and Go satisfies any of the conditions (a)-(g) of Section 14.lC.

For n> 1 condition (A) is certainly satisfied if (,1,1) for every (f,x) E A there are a vector b = (bO, bt, ... , bn) E Rn+ 1 and constants v > 0, ~ > 0 such that Go(t,x,p,u) +bop +.L~ bid ~ v for all (p,u) E Rn+t, p ~ 0, and all (t,x) E A at a distance :s;~ from (f,x); (,1,2) Go is bounded below; (,1,3) there is a constant J.l > 0 such that all rectifiable parametric curves {to lying in any hyperplane t = constant, or (to: t = c, x = x(s), 0 :s; s :s; Lo, with graph in A and 3[{to] = 0 have length :s; J.l.

Conditions (,1,2), (,1,3) are certainly satisfied if Go{t, x, 0, u) > 0 for all u ERn, U i= 0, and all (t, x) E A except at most for a countable set Ee on any hyperspace t = c.

14.3.ii (EXISTENCE THEOREM FOR USUAL INTEGRALS). Let A, B = B1 X B2 as in (14.3.i), let F o(x, x') be independent of t, continuous and bounded below on A 1 X R", and convex in x' for every x E A 1. Let us assume that the associated parametric integrand Go(x, p, u) is continuous in A x [0, + 00] x R" with continuous partial derivative Gop in the same set A x [0, + 00] x Rn with (p, u) i= (0,0). Let us further assume that (a') for every x E A, u ERn, U i= 0 we have Goix,p, u) = 0 if and only if p = 0; and ifn > 1, (b') the same as (A) in (14.3.i). Then, I[ x ] has an absolute minimum in the class Q of all admissible trajectories.

The following statement concerns integrals of the form

(14.3.2) I*[x] = rt2 ("'(x))-1fo(x(t),x'(t))dt Jtl under the same general assumptions as for (14.3.i or ii).

14.3.iii. Let "'(x) ~ 0, X ERn, be a continuous function. Let Z denote the closed subset of Rn where "'(x) = 0, and let us assume that Go(x,p, u) > 0 for all x in some neighborhood U of Z, all p > 0 and u ERn. Then, the integral I*[x] has an absolute minimum in the class Q of all admissible trajectories.


Remark. The following statement concerning slow growth integrals is revealing of what actually may occur in situations not covered by theorems (14.3.i-iii). We consider here the non parametric integral with n = 1

(14.3.3)

where rJj is positive and continuously differentiable in [tl' t2]. We take A = [tb t2] x R, and we consider the case of both end points fixed: 1 = (tl' XI), 2 = (t2' X2), tl < t2. Note that rJj(t)(1 + X'2)1/2 is L-integrable in [tl' t2] for all AC real valued functions x.

14.3.iv. The integral (14.3.3) has an absolute minimum in the class Q of all AC functions x with x(td = Xb x(t2) = X2, tb t2, Xl' x2.fixed, if and only if

(14.3.4) IX2 - xII:::;; il2 m[<p2(t) - m2]-1/2dt JI. where m = mine <P(t), tl :::;; t:::;; t2].

For instance, iftl = 1, t2 = 2, <P(t) = t, m = 1, then lex] = Ii t(1 + X/2)1/2 has an absolute minimum in Q if and only if IX2 - xli:::;; Ii (t2 - 1)-1/2 dt = log(2 + 31/2) = 1.317. On the other hand, for any tl < t2, <P(t) = 1, m = 1, then l[ x] = J:~ (1 + X'2)1/2 dt has an absolute minimum for all Xl> X2 (namely, the segment s = 12, as we know), and indeed the integral in (14.3.4) is + 00

in this situation. For the proof of (14.3.iv) we refer to P. Kaiser [1,2]'

14.4 Examples

1. Fo = (1 + X'2)1/2 satisfies the conditions of (14.3.ii). Note that Fo ~ 0, Go(p,u) =

(p2 + U2)1/2 is continuous for p ~ 0, U E R; Gop = p(p2 + U2)-1/2 is continuous for all p ~ 0, U E R, (p, u) # (0,0); and Gop = 0 with u # 0 if and only if p = O. Thus, lex] has an absolute minimum in the class of all AC curves joining say, any two points 1 = (tl,XI), 2 = (t2,X2), tl < t2; or 1 to any curve r:x = g(t), t':;:;; t:;:;; t" with tl < t'; or any two sets BI compact, B2 closed, with the property that (tbxd E BI, (t2' X2) E B2 implies tl < t2. Here lex] is the length integral.

2. F 0 = (1 + X'2)1/2 - (1 + X'2)1/4, satisfies the conditions of (14.3.i). Note that F o(x') ~ 0, Go(p, u) = (p2 + U2)1/2 - pl/2(p2 + U2)1/4 ~ O(p ~ 0, U E R), Gop(O, u) = - 00.

Because of F Ot = 0, condition (b) of(14.3.i) is satisfied, and by Section 14.1qd) condition (2) of (14.3.i) is satisfied. Indeed, Go(t, x, cos e, sin e) = 1 - (cos e)I/2, -n/2 ~ e :;:;; n/2, and Go = 0 only for e = o. lex] has an absolute minimum in classes of AC curves as described in Example 1.

3. Fo = (x - a)-1/2(1 + X'2)1/2, x ~ a, satisfies the conditions of(14.3.iii) with I/I(x) =

(x - a)I/2 ~ 0 continuous and fo = (1 + X'2)1/2 satisfying those of (14.3.ii). Here F 0

corresponds to the problem of minimum time of descent (brachistochrone, Section 3.12). Thus, the problems of minimum time of descent from 1 = (tl' XI) to 2 = (t2, x2), t2 > t l ,X2 > XI' or from 1 = (tl' XI) to a curve B2 = [x = x(t), t' ~ t ~ t"], tl < t', x(t') > Xl' have always an optimal solution.

14.4 Examples 441

Counterexamples

1. This example shows that the minimum may not exist if B = BI X B2, Bb B2 closed, and (tbXbt2,X2)EB does not imply t l <t2. Indeed, for Fo=(1+x'2)1/2, n = 1, the problem of an AC curve x = x(t), tl ::; t::; t 2 , of minimum length joining (0,0) to B2 = crt, x), t ;;:: 0, x = 1] has no optimal solution.

2. In Section 3.13 we have discussed at length the integral F 0 = x(l + X'2)1/2,

x ;;:: 0, n = 1, (problem of minimum area of a surface of revolution), and we have seen that, given two fixed points 1 = (tl, xd, 2 = (t2, X2), tl < t2, XI > 0, X2 > 0, the existence or nonexistence of an absolute minimum of I[ x] for AC curves x = x(t), tl ::; t::; t2 ,

joining 1 and 2, depends on whether 2 is above or below a suitable curve x = T(t), tl ::; t < + 00 (r(tl) = 0, 0 < T(t) < r(t') for tl < t < t', T( + 00) = + 00). The corresponding parametric integral Go = X(p2 + U2)1/2 is continuous in [0, + 00) X R2 and has x = 0 as the only zero, but Go does not satisfy any of the local conditions stated in (14.3.i~iii). For instance, Gop = Xp(p2 + U2)-1/2 and Gop = 0 for allp if x = O.

3. F 0 = (x2 + X'2)1/2, n = 1. Here Go(x, p, u) = (p2X2 + U2)1/2 is continuous in R x [0, + 00) x R. For x#-O we have Gop(x, p, u) = 0 if and only if p = O. For x = 0, however, we have Gop(O,p,u) = 0 identically. In Section 1.6, Example 5 we mentioned that the problem of minimum with Fo as integrand and fixed end points (0,0), (1, 1) has no optimal solution.

4. Here is an example of a problem with n = 2 and no optimal solution, where condition (A3) of (14.3.i) is not satisfied. Take A = [(t, x, y)iO::; t::; 1, x2 + y2::; 1], F o(x, y, x', y') = 2(yx' - xy') + (1 + x2 + y2) (1 + X'2 + y'2)1/2, and consider the problem of the minimum of I[ x, y] = SF 0 dt in the class of all AC trajectories x(t), y(t), o ::; t ::; 1, joining (0, 1,0) to (1, 1,0). Here

Go(x,y,p,u,v) = 2(yu - xv) + (1 + x 2 + y2)(p2 + u2 + V2)1/2.

By the elementary inequality 1 + x2 + y2 ;;:: 2(x2 + yZ)I/2 in A, and the Schwarz inequality (in R2) we derive

F 0;;:: 2[(yx' - xy') + (x2 + y2)1/2(1 + X'2 + y'2)1/2] > 0,

Go ;;:: 2[(yu - xv) + (x2 + y2)1/2(p2 + u2 + V2)1/2] ;;:: 0,

and equality Go = 0 holds if and only if x2 + y2 = 1, P = 0, xu + yv = O. Obviously, condition (b) of (14.3.i) holds. No minimizing curve exists. First, for the infimum i we obviously have i ;;:: O. On the other hand, if we consider the sequence

o ::; t ::; 1, k = 1, 2, ... ,

joining (0, 1,0) to (1, 1,0), we have

I[ Ck] = I[ Xb Yk] = 2( - 2kn + (1 + 4k2n2)1/2)

which tends to zero as k --+ 00. Thus, i = O. For no trajectory x, y we can have I = 0, since F 0> 0, and for no curve <f we can

have 3 = 0, since this would imply t' = 0 a.e., or t = constant, while we need to join points with t I = 0, t2 = 1. Condition (A3) of (14.3.i) is not satisfied, since for curves <f:t = c, x = cos 2km:, y = sin 2km:, 0::; T ::; 1, we have 3[<f] = 0, and L[<f] = 2kn as large as we want.


Bibliographical Notes

In Sections 14.3-4 we have briefly mentioned existence theorems for problems for which no growth conditions hold of any kind, such as the length integral or the integral in the classical brachistochrone problem. We have followed the approach proposed by E.1. McShane ([5],1934), which consists in reducing the given usual (nonparametric) problem to a parametric one, showing that the parametric problem has an optimal solution, and proving, under hypotheses, that such parametric solution has an AC representation as a nonparametric curve in Rn+ 1, and actually is an optimal solution ofthe original nonparametric problem (cf. III for details and proofs).

An alternate approach has been proposed by L. Tonelli [10], and another one by E. J. McShane [7].

In III we shall present further work by C. Vinti concerning the underplaying between usual integrals and their associated parametric ones (cf. Bibliography at the end of Chapter 11). We shall also present in III recent work on parametric problems of optimal control (R. M. Goor [1, 2]). This work, with connection of our problems with questions of algebra and topology, shows also that the class of'parametric optimal control problems is much larger th;m, one would expect.

The concept of parametric, or Frechet, curves, the lower semicontinuity theorems for parametric problems with respect to Frechet distance (uniform topology), and the existence theorems will be discussed in more details elsewhere (Cesari [III]). The existence theorems for parametric problems can be traced back to L. Tonelli [II] for n = 2 and under some smoothness assumptions on the data, and to E. J. McShane [10] for any n.

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Optimization—Theory and Applications || Existence Theorems: Problems of Slow Growth