optimization—theory and applications || existence theorems: weak convergence and growth conditions

CHAPTER 11

Existence Theorems: Weak Convergence and Growth Conditions

11.1 Existence Theorems for Orientor Fields and Extended Problems

A. Some General Existence Theorems for Lagrange and Bolza Type Problems

Let A be a subset of the tx-space W+ \ and let A(t) denote its sections, that is, A(t) = [x E Rn I (t, x) E A]. 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 ° be the set M ° =

[(t,x,z)l(t,X)EA,ZEQ(t,x)]cR 1 +zn, and let Fo(t,x,z) be a given real valued function defined on Mo. Let B be a given subset of the t 1x 1 tzxz-space Rzn+z, and let g(tl,X b tZ,xZ) be a real valued function defined on B. Let Q = {x} denote a nonempty collection of AC functions x(t) = (xl, ... , xn), tl S t S t z, such that

x(t) E A(t), x'(t) E Q(t, x(t)) for t E [t 1, tz] (a.e.), (11.1.1) ,

e[x] = (tl,X(tl),tZ,X(tz)) E B, Fo(',x('),x(')) E L1[t1,tZ].

We are concerned with the existence of the minimum in Q ofthe functional

(11.1.2)

Any AC function x satisfying (11.1.1) will be called an admissible trajectory, or briefly a trajectory. For every (t, x) E A we denote as usual by Q(t, x) the set Q(t,x) = [(zO,z)lzO ~ Fo(t,x,z), z E Q(t,x)] C W+ 1.

367

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

368 Chapter 11 Existence Theorems: Weak Convergence and Growth Conditions

Concerning A, M 0, F 0 we repeat here the alternate assumptions of Section 10.8. For most applications it is sufficient to assume

C. A closed, M 0 closed, F o(t, x, z) continuous on Mo.

In this case we may think of F 0 as extended to the whole space R 1 + 2n by taking F 0 = + 00 in R1+ 2n - Mo. In the classical calculus of variations we usually have Q(t, x) = Rn for all (t,x) E A and Mo = A x Rn. However, for the proof that follows the following rather general assumption suffices:

C*. Fo(t,x,z) is a given extended function in R 1 + 2n and we assume that for every finite interval [to, T] and e > 0 there is a compact set K c [to, T] such that (a) meas ([to, T] - K) < e, (b) the extended function F o(t, x, z) restricted to K X R2n is B-measurable, and (c) for almost all I E [to, T] the extended function F 0(7, x, z) of (x, z) has values finite or + 00, and is lower semicontinuous in R2".

Under hypothesis (C*) we may simply assume that F 0 is a given extended function in R1+2", whose values for almost all lEG are finite or + 00, and we denote by Ao the set of all IE R such that A(I) = [x E R"I FoCf, x,z) ¥= ± 00] =1= 0. For any (7, x) take Q(f, x) = [z E Rrl F o(t, x, z) =1= + 00], and Q(7,x) = epi Fo(I,x,z). Now A is any set of points (t,x) whose sections, for almost all t E Ao, are the sets A(t).

In any case it is clear that A; B, F 0 must be so related that the class of all AC functions x(t) satisfying (11.1.1) is not empty. Note that under condition (C) as well as under condition (C*) (parts (a) and (b) suffice), then Fo(t, x(t), z(t)) is measurable for any two measurable functions x(t), z(t). However F o(t, x(t), z(t)) may have the value + 00 in a set of positive measure. Only if x(t) E A(t), z(t) E Q(t, x(t)) a.e., then F o(t, x(t), z(t)) is finite a.e ..

The existence theorems of the present chapter will be based on the mode of convergence for AC trajectories x k , x which in Section 2.14 we called "the weak convergence of the derivatives", that is, X k --+ x in the p-metric, or uniformly, and x~ --+ x' weakly in L1 (mode (b) of Section 2.14) ..

We may be interested in the absolute minimum of lEx] in the class Q of all AC trajectories x satisfying (11.1.1). Alternatively, and as in Sections 9.2, 9.3, we may want to minimize l[ x] in a smaller class Q of such AC functions, and in this case we need to know that Q has a suitable closedness property. We say that Q is r w-closed provided: if xk(t), tlk ~ t ~ t2k , k = 1,2, ... , are AC functions satisfying (11.1.1), all in the class Q, if x(t), t1 ~ t ~ t2, is an AC function satisfying (11.1.1), and if X k --+ x in the weak convergence of the derivatives (that is, X k --+ x in the p-metric and x~ --+ x' weakly in L 1), then x is in the class Q. The same usual classes of trajectories we have mentioned in Sections 9.2, 9.3 in connection with ru-closure all have also the present r w-closedness property.

In the existence theorems below we shall need alternate global "growth hypotheses". They are the usual ones we have already mentioned, but we state them again in the form they are needed here.

11.1 Existence Theorems for Orientor Fields and Extended Problems 369

(y1) there is a scalar function cf>m, 0 ::;; , < + 00, bounded below, such that cf>(')/' - + 00 as , - + 00, and F o(t, x, z) ~ cf>(lzi) for all (t, x, z) E M 0;

(y2) for any e > 0 there is a locally integrable scalar function t/I it) ~ 0 such that Izl::;; t/lit) + eFo(t,x,z) for all (t,x,z) E Mo;

(y3) for every n-vector p E Rn there is a locally integrable function cf>it) ~ 0 such that F o(t, x, z) ~ (p, z) - cf>it) for all (t, x, z) E Mo.

These are the conditions we anticipated in Section 2.20A, and we encountered in Section 10.4 both in their global and local forms. Condition (y1) is the Tonelli-Nagumo condition. Condition (y2) is a slight generalization of (y1), and was discussed in Section 10.4. Condition (y3) is actually equivalent to (y2) as we proved in Section 10.4. The examples in Section 2.20B and those in Section 11.3 illustrate these conditions.

ll.1.i (AN EXISTENCE THEOREM BASED ON WEAK CoNVERGENCE OF THE

DERIVATIVES). Let A be bounded, B closed, and let condition (C) or (C*) be satisfied. Assume that for almost all t and all x E A(t) the extended function F o(t, x, z) be convex in z (in R" and hence the sets Q(t, x) and (2(t, x) are convex). Let g be a lower semicontinuous function on B. Assume that anyone of the growth conditions (y1), (y2), (y3) is satisfied. Let D be any nonempty r w-closed class of AC functions x(t) = (Xl, ... ,xn), tl ::;; t ::;; t2 , satisfying (11.1.1). Then the functional l[ x] in (11.1.2) has an absolute minimum in D.

It is enough we limit ourselves to the nonempty part DM ofD of all elements XED with lEx] ::;; M for some M.

For A not bounded see Section 11.2.

Proof. Here A is bounded, and thus, for every element x(t), tl ::;; t::;; t2, of D we have - M ::;; tl ::;; t2 ::;; M, Ix(t)1 ::;; M for all t E [tl' t2] and some fixed M. Under condition (y1), cf> is bounded below; hence Fo(t,x(t),x'(t» ~ -M for all t E [tl' t 2 ] and some constant M. Under condition (y2) with e = 1 we have Fo(t,x(t),x'(t» ~ -t/ll(t), where t/l1 ~ 0 is L-integrable in [ -M,M]. Under condition (y3) with p = 0 we have again F 0 ~ - cf>o(t). Thus condition (L 1) of Section 10.8 holds, and J 1 [x] = f:~ Fo(t,x(t),x'(t»dt ~ -Ml for all elements x of D and some constant M 1. Since e[x] = (t 1,X(t1),t2,X(t2) lies in B n (cl A x cl A), a compact set, and g is lower semicontinuous, we have J 2 [x] = g(e[x]) ~ -M2 for some constant M 2. Hence lEx] = J 1 + J 2 ~ - M 1 - M 2 is bounded below in D. If i = inf[l[ x], xED], then - 00 < i < + 00. Let xk(t), tlk ::;; t::;; t2k, k = 1,2, ... , be a minimizing sequence of elements xk E D, that is, l[ xk] _ i as k - + 00, with

xit) E A(t), x~(t) E Q(t, Xk(t», t E [tlk, t2k] (a.e.), k = 1, 2, ....

We can assume that i::;; l[xk] = J 1[Xk] + J 2[Xk]::;; i + 1. By a suitable selection we can even assume that J 1 [xk] - i 1, J 2[Xk] - i2 as k - 00, with i1 + i2 = i. where i1, i2 are suitable numbers (neither of which need be the infimum of J 1 or J 2)'


Under condition (yl) we have (p(lx~(t)l) S; FO(t,Xk(t),X~(t)), tlk s; t s; t2k ; under condition (y2) for any G > 0 we have Ix~(t)1 s; "'it) + BF o(t, xk(t), x~(t», tlk S; t S; t2k ; under condition (y3) for any PER" we have F o(t, xk(t), x~(t» ~ (p, x~(t» - rPp(t), t lk S; t S; t2k> k = 1, 2, . . .. In any case, by virtue of (lO.4.i,ii,iii) respectively, the equibounded sequence [x k ] is equicontinuous and equiabsolutely continuous, and the sequence of derivatives [x~] is equiabsolutely integrable. By Ascoli's theorem (9.1.ii) there is a subsequence, say still [k], such that [x k] converges uniformly to some continuous function x(t), t1 S; t S; t2, namely, in the p-metric of Section 2.14 with tlk -+ t1, t2k -+ t2. By (10.2.iii) x is AC in [t 1o t2J. By (lO.3.i), (b) => (a), we can choose the subsequence in such a way that [x~] is weakly convergent in L1 to some L-integrable function ~(t).

By (10.8.ii) we have W) = x'(t), x(t) E A(t), x'(t) E Q(t, x(t», t E [t 1, t2] (a.e.), and J 1[x] = s:~ Fo(t,x(t), x'(t»dt S; i1. Since e[xk] -+ e[x] in R2+2n as k -+ 00, by the lower semicontinuity of g we derive that J 2[ x] = g(e[ x]) S;

lim J 2[ xk] = i2 • Thus,

l[x] = J 1[x] + J 2[x] S; i1 + i2 = i.

On the other hand Q is r w-closed, hence x E Q and l[ x] ~ i. By comparison we have l[x] = i and (l1.1.i) is thereby proved. D

Note that in the present situation the sets Q(t, x) = epi F o(t, x, z) are closed and convex, and certainly for almost all T the sets Q(T, x) have property (K) with respect to x by (8.5.v) and (8.S.iii). Note that, moreover, for almost all T the sets Q(Y, x) have property (Q) with respect to x as a consequence of the growth hypotheses and of theorem (1O.5.i). Indeed, this is evident under condition (yl). Under condition (y2) we note that for fixed Y, then in the relation Izl S; "'iT) + BFo(Y,x,z), "'iT) is actually a constant, and by (lO.4.v) this condition is equivalent to (y1) with respect to x only. Condition (y3) is equivalent to (y2). Thus, our sets Q(Y, x) have property (Q) with respect to x (for almost every T), and then the above recourse to (1O.8.ii), and in the last analysis to (10.7.i), is to be understood in the sense that we really need only the simpler versions of (1O.7.i) and (1O.8.i,ii) mentioned at the beginning of the proof of (10.7.i) and in Remark 4 of Section 10.8.

Remark 1. Concerning Theorem (11.1.i), we have already mentioned examples of r wclosed classes Q. Here we can add that we also obtain a r w-closed class by a restriction of the form

i '2 C[x] = H(t,x(t),x'(t))dt:=; M "

for some constant M, and an integrand H satisfying the conditions of (1O.8.i), so that C[x] is lower semicontinuous in the same topology we have been using in the proof of Theorem (l1.1.i). Often C[ x] is called a comparison functional. If H already satisfies one of the growth conditions (yl), (y2), (y3) much less needs to be required onto, as we shall see below. This happens for instance in the case of the familiar restriction

,'21x'IP dt :=; M for p > 1. J"


Remark 2. If the convexity condition in (l1.1.i) is not satisfied, let us prove the existence of an optimal generalized solution. By a generalized solution (cf. Section 1.14) we mean a solution of the new problem in which as usual relations (11.1.1-2) are replaced by

J[x,p,v] = g(tt>x(td,tz,x(tz» + 1:2 Jl Ps(t)Fo(t,x(t),z(S)(t»dt,

X'(t) = L Ps(t)z(S)(t), L ps(t) = 1, Ps(t) 2:: 0, t E [t1> tz] (a.e.),

(t,x(t» E A, (tl,X(t1),tz,x(t z) E B,

x(t) AC, Ps(t), z(s)(t) measurable, L Ps(·)F 0(·, x(·), z(S)(.» E L.

Actually this is a problem of optimal control, and the natural problem of the type (11.1.1-2) to associate to it is the following one

H[x] = g(tt> x(td,tz, x(tz)) + {t2 T(t,x(t),x'(t))dt, x AC, Jt[ (t, x(t» E A, (t t> x(t d,tz, x(tz)) E B, T(-, xC), x'(·» E L,

T(t, x, z) = inf L PsF o(t, x, z(S», z = L Psz(S), L Ps = 1, Ps 2:: o.

Now let us consider the convex sets

Q*(t, x) = [(zO,z)lzO 2:: L PsFo(t,x,z(S», z = L Psz(S), L Ps = 1, Ps 2:: oJ. s s s

Let us prove that, for almost all t, the sets Q*(t, x) have properties (K) and (Q) with respect to x. Indeed, if F 0 satisfies condition (yl) then this is a consequence of Remark 4 of Section 10.5. If F 0 satisfies condition (y2), then for any given I, in the relation z ::; ljJ it) + f.F o(I, x, z), ljJ ,(I) is a constant, and as above by (lO.4.v) this condition is equivalent to (y1) with respect to x only. Condition (y3) is equivalent to (y2). Thus, for almost all I, the sets Q*(t, x) have properties (K) and (Q) with respect to x in A(I). As a consequence, for almost all I, the sets Q*(t, x) are closed, and this implies that inf can be replaced by min in the definition of T, and the two problems relative to J and H are equivalent. Finally, for almost all I, and by (8.S.iii) and (8.S.v), T(I, x, z) is lower semicontinuous in x, z. Now (11.1.i) applies to the functional H, and the existence of an optimal generalized solution is proved.

Remark 3. It is easy to see that the same proof of (Il.l.i) above also proves that, under the same hypotheses of (Il.l.i), the class Q M of all AC functions x(t) satisfying (11.1.1) with lex] $ M for any fixed M is closed and compact in the topology of weak conver-gence of the derivatives. That is, if Xk E Q, l[xk] $ M, k = 1,2, ... , then there is a subsequence, say still [k], such that xk ..... x in the p-metric, and x~ ..... x' weakly in Ll ,

where x is AC, x E Q, and lex] $ M.


In the proof of (ll.l.i) the growth condition (yl) or (y2) or (y3) has several roles: it guarantees that condition (L 1) of Section 10.8 holds, it guarantees the boundedness below of I[ x], and it guarantees the relative compactness of the class Q = {x} (in the topology of the weak convergence of the derivatives. Thus even if we assume straightforwardly that Q is any given class possessing the weak compactness property above, we must .still see to it that I[ x] is bounded below. Indirectly, we have to guarantee the existence of functions A, Ak as in (10.7.3). Actually, we have already stated in Section 10.8 to this effect, the alternate conditions (Li) in terms of Fo. We shall use the conditions (Li) again in this Section. For other alternate conditions see the exercises below.

1l.1.ii (AN EXISTENCE THEOREM BASED ON WEAK CONVERGENCE OF THE

DERIVATIVES). Let A be bounded, B closed, and let condition (C) or (C*) be satisfied. Assume that for almost all t and all x E A(t) the extended function F o(t, x, z) is convex in z. Assume that one of the alternate conditions (Li) holds. Let g( t 1, Xl' t 2, x 2) be a lower semicontinuous scalar function on B. Let Q be any nonempty r w-closed class {x} of AC functions x(t) = (Xl, . .. ,xn ), tl ::;; t::;; t2 ,

satisfying (11.1.1), and assume that the class of derivatives {x'(t), tl ::;; t::;; t 2 }

is equiabsolutely integrable. Then the functional I[ x] in (11.1.1) has an absolute minimum in Q.

It is enough we verify the above requirements for the nonempty class Q M of all x E 0 satisfying I[ x] ::;; M for some M.

The requirement on the class {x'} is certainly verified if for instance we know that for some p > 1 and D > 0 we have s:~ IX'(t)iP dt ::;; D for all elements x of the class Q or OM. This is a consequence of (10.3.i), (c) => (b) with cfJ(O = (P. Analogously, we satisfy the requirements on the class x' by a restriction of the form

it2 C[x] = H(t,x(t),x'(t»dt::;; M

tl

for some constant M and where H satisfies the assumptions of (l1.1j), in particular one ofthe growth conditions (y1), (y2), (y3). Again by (10.8ji) C[ x] is lower semicontinuous, and the classs so obtained is r w-closed.

Theorem (l1.1j) is contained in (lUji). Indeed, under either condition (y1), or (y2), or (y3), there is some integrable function t/!(t) ~ 0 such that Fo(t,x(t),x'(t» ~ -t/!(t) for all x E Q as we have shown in the proofof(l1.1.i) and condition (L 1) holds. On the other hand, under condition (y1) there is some function cfJ(O bounded below with cfJ(O/( -+ + 00 as (-+ + 00 and F o(t, x, z) ~ cfJ(lzl). Then for every x E ° we have

M ~ it2 Fo(t,x(t),x'(t»dt ~ it2 cfJ(lx'(t)i)dt, Jtl Jtl

and the equiabsolute integrability of the class {x'} follows from (1O.1.i). Under


condition (y2) and l1(t) = F(t, x(t), x'(t)) we have J:~ l1(t) dt = J:~ F 0 dt s; M, and the equiabsolute integrability of {x'} follows from (10.1.ii). Under (y3) the analogous conclusion follows from (10.1.iii).

Proof of (l1.1.ii). Let M be a number such that OM is not empty. It is not restrictive to search for the minimum of l[x] in OM' By (1O.3.i), (b) => (a), the class {x'} is relatively sequentially weakly compact in Lh and by (1O.2.i) the class {x} is equiabsolutely continuous. Since A is bounded, the same class {x} is also equibounded, that is, -Mo S; t1 < t2 S; M o, Ix(t)1 S; Mo. Then, OM is relatively sequentially weakly compact in the "topology of the weak convergence of the derivatives (mode (b) of Section 2.14). Again, g is lower semicontinuous in the compact set B n (cl A x cl A), hence J 2 = g(e[ x]) is bounded below in OM' Here I = J 1 + J 2 as before, and we do not know yet whether J 1 = s:~ F 0 dt is bounded below in OM' Let i = inf[I[ x], x E OM], - 00 S; i < + 00. Let Xk(t), tlk S; t S; t2k, k = 1,2, ... , Xk E OM, be a minimizing sequence, that is, l[xk] ~ i as k ~ 00. Here [xk] is a subset of OM; hence there is a subsequence, say still [k], such that x~ ~ e weakly in L 1, and Xk ~ x in the p-metric, and thus tlk ~ t1, t2k ~ t2. As we have seen in the proof of (11.1.i), x is AC in [t1' t2] and x'(t) = W) a.e. in [t1' t2]. Thus, the sequence of L 1-norms Ilx~11 is bounded. We have seen in the proof of (10.8.i) that in this case each of the assumptions L 1, L2, L 3 , L4 guarantees thatJ lk = J:~~ Fo(t,xk(t),x~(t))dt is bounded below, say J 1[x] ~ -M l' As in the proofof(11.1.i) we also have J2[x] = g(e[x]) ~ -M2. Thus,i ~ -M1 -M 2 is finite. As in the proof of (11.1.i) by a suitable selection we may assume that J 1 [Xk] ~ i1, J 2[Xk] ~ i2 as k ~ 00, both i1, i2 finite with i1 + i2 = i. The proof is now the same as for (11.1.i). 0

Remark 4. If the convexity assumption in (ll.1.ii) is not satisfied, then as for (11.l.i) we can guarantee the existence of generalized solutions if the sets R(t, x) = co (2(t, x) have property K with respect to x, where Q(t,x) = epi Fo(t,x,z).

Statement (l1.1.ii) for classical integrals with integrands which are continuous in A x Rn has a simpler form:

1l.1.iii (AN EXISTENCE THEOREM BASED ON WEAK CONVERGENCE OF THE

DERIVATIVES). Let A be compact, B closed, and let F o(t, x, z) be continuous on A x Rn, and convex in z for every (t,x)EA. Let g(t1,X1,t2,X2) be lower semicontinuous on B. Let 0 = {x} be any nonempty r w-closed class of AC functions x(t) = (xl, ... ,xn ), t1 S; t S; t2, satisfying (11.1.1), and suppose that the class {x'} is equiabsolutely integrable. Then the functional lEx] in (11.1.2) has an absolute minimum in O.

It is enough we verify the above requirements for the nonempty class OM

of all x E 0 satisfying l[x] S; M for some M. This is a corollary of (l1.1.ii) and (L4)'


B. An Existence Theorem for Mayer Problems

For F 0 = ° the problem (11.1.1-2) reduces to a Mayer problem. Now we are concerned with the existence of the minimum in the class Q of the functional lex] = g(e[x]) under the constraints

x(t) E A(t), x'(t) E Q(t,x(t)) for t E [t1' t 2] (a.e.),

e[x] = (t 1,X(t1), t2 ,X(t2 )) E B. (11.1.3)

Instead of (11.l.ii) we have the simpler statement:

1l.1.iv (AN EXISTENCE THEOREM FOR MAYER PROBLEMS). Let A be bounded, B closed, and glower semicontinuous on B. Let us assume that for almost all Y the sets A(Y) are closed and that the sets Q(Y, x) are all closed and convex and bave property (K) on the closed set A(Y). Let Q = {x} be any nonempty rw-closed class of AC functions x(t) = (Xl, ... ,x"), t 1 :s; t:s; t 2 , satisfying (11.1.3), and assume that the class of derivatives {x'} is equiabsolutely integrable. Then the functional lex] = g(e[x]) has an absolute minimum in Q.

It is enough we verify the above requirements for the nonempty class QM of all x E Q satisfying l[ x] :s; M for some M. This is a corollary of (l1.l.ii) and (L1)'

C. The Linear Integrals

Let us consider now a linear problem

(11.1.4) l[x] =g(t1,X(t1), t 2 ,X(t2 ))+ 1:2 [Ao(t,X)+it1 Ai(t,X)x'i]dt,

(t,x(t)) E A, (t1,X(t 1), t2 ,X(t2 )) E B,

where A is a subset of [to, T] x R", to, T finite, Q(t, x) = R", M = A x R". Let H(t, x, z) = Ao(t, x) + 2:7= 1 Ai(t, X)Zi.

1l.1.v (AN EXISTENCE THEOREM FOR LINEAR INTEGRALS). Let A be a subset of [to, T] x R", and assume that all Ai(t, x), i = 0, 1, ... , n, satisfy the condition (CL) or (CL *) of Section 1O.8B. Assume that for every N > ° there are a function ¢(t) ~ 0, ¢ E L 1[to, T] and a constant c> ° such that IAo(t,x)1 :s; ¢(t), IAi(t, x)1 :S C for all (t, x) E A with Ixl :S N. Let Q = {x} be any nonempty r w-closed class of AC functions x(t) = (Xl, .. . ,x"), t1 :S t:S t 2 , with graph in A, such that each trajectory x E Q has at least a point (t*,x(t*)) in a given compact set P, and such that the class of derivatives {x'} is equiabsolutely integrable. Let B be closed, and glower semicontinuous and bounded below on B. Then the functional (11.1.4) has both an absolute minimum and an absolute maximum in Q.


In particular the same conclusion holds if, for some p > 1 and any N > 0 there are functions </J, t/I 2:: 0, </J E L 1, t/I E L q , lip + 11q = 1, such that IAo(t, x)1 s </J(t), IAi(t, x)1 s t/I(t) for all (t, x) E A with Ixl s N, and if the class {x'} is relatively sequentially weakly compact in Lp.

Note that if for some p > 1 and D we have J:~ Ix'ip dt s D for all xED, then the class {x'} is certainly relatively sequentially weakly compact in Lp and in L 1, and equiabsolutely integrable.

Proof. By (1O.2.ii) the total variations Vex; t1, t2] of the elements xED are equibounded, say Vex, t 1, t2] s N 1. Since P is bounded, then Ix(t*)1 s M 1

and Ix(t)1 s M 1 + N 1 for all xED. Thus all trajectories xED lie in the compact set S = [to, T] x [Ixl s M 1 + N 1]. If </J and C are the corresponding elements, then IAo(t, x)1 s </J(t), IAi(t, x)1 s c, i = 1, ... , n, for all (t, x) E S. Then IH(t, x, z)1 s </J(t) + nqzl for all (t, x) E Sand z ERn, and condition (L2) holds. The theorem follows now from (11.l.ii).

Here is an alternate proof of the main statement. Let i = inf[ I[ x] I xED], and let xk(t), to s tlk S t S t2k S T, k = 1,2, ... , be a minimizing sequence, so that I[xk] -+ i and (tt,xk(tt) E P for some tlk S tt S t2k. First, there is a subsequence, say still [k], such thatt lk -+ t1, t2k -+ t2, (tt, xk(tt)) -+ (t*, x*) E P. Also x~ is equiabsolutely integrable, hence by (10.3.i) there are a function ~ E L1 and a further subsequence, say still [k], such that x;' -+ ~ weakly in L 1 • Now {xk} is absolutely equicontinuous, and also equibounded since P is bounded and the functions X k have equibounded total variations. Thus, Ilx;'111 s j.l., and there are </J E L1 and constant C such that IAo(t,xk(t»1 s </J(t), IAi(t,Xk(t»1 s C. Hence IIH(t,Xk(t),X~(t))lll s 11</J11t + nCj.l.. Thus, i is finite. From xk(t) = xk(tt) + J:t x;'(-r) dr we derive that x(t) = x* + J:. ~(r) dr, hence x'(t) = ~(t) a.e .. Finally, by (1O.8.iii) we derive that H(t, xk(t), x~(t» -+ H(t, x(t),x'(t» weakly in L 1, and I[xk] -+ I[x] = i.

For the case p > 1 the proof is the same but we can take the subsequence in such a way that x;' -+ ~ weakly in Lp and ~ E Lp. Now there is some t/I E Lq such that IB(t, xk(t»1 s t/I(t), and again by (1O.8.iii), H(t, xk(t), x;'(t» -+ H(t, x(t), x'(t» weakly in L 1• 0

D. Existence Theorems with Comparison Functionals and Isoperimetric Problems

Problems with constraints of the form C[x] S D (i.e., with a comparison functional), and of the form C[x] = D (isoperimetric problems) can be written in terms of optimal control as we have already shown in Sections 1.5 and 4.8, and as such we shall discuss them again in Section 11.4. However we shall prove here a few statements directly. Therefore let us consider here


functionals and constraints of the form

(11.1.5)

I[x] = g(t1,X(t1),t2 ,X(t2)) + (r2 Fo(t,x(t),x'(t))dt, Jr!

C[ x] = (r2 H(t, x(t), x'(t)) dt, x(t) = (xl, ... ,xn ), Jr! x(t) E A(t), x'(t) E Q(t, x(t)) for t E [tl' t2] (a. e.),

e[x] = (t 1,X(t1),t2 ,X(t2)) E B, Fo(·,x(·),x'(·)) ELI'

H(·,x(·),x'(·)) E Ll

where F 0 and H satisfy either condition (C) or (C*).

1l.1.vi (AN EXISTENCE THEOREM WITH A COMPARISON FUNCTIONAL). Let A be bounded, B closed, and glower semicontinuous on B. Assume that both functions F o(t, x, z), H(t, x, z) satisfy condition (C) or (C*), that are both convex in z, that one of them satisfies any of the growth conditions (yl), (y2), (y3), and that the other satisfies any of the conditions (L j ). Let Q be any nonempty r w

closed class of AC functions x(t) = (xl, ... ,xn), tl ::s; t::s; t2 , satisfying x(t) E

A(t), e[ x] E B, and both F o( ., x( .), x'( .)), H( ., x( .), x'( . )) are L-integrable. Let D be a constant such that the class QD of all x E Q with C[ x] ::s; D is not empty. Then I[x] has an absolute minimum in QD.

Also, let N be a constant such that the class Q~ of all x E Q with I[x] ::s; N is not empty. Then C[ x] has an absolute minimum in Q~.

Proof. Let i = Inf[I[x], x E QD] andj = Inf[C[x], XE Qa For the first part of the statement we take a minimizing sequence [xk] for I[ x] in Q D, that is, with I[ Xk] -+ i, C[ Xk] ::s; D. For the second part we take a minimizing sequence Xk for C[ x] in Q~, that is, with I[ Xk] ::s; N, C[ Xk] -+ j. In either case, because of the growth conditions, the sequence of derivatives [x~] is equiabsolutely integrable. As in (l1.1.i) there is an AC function x(t), tl ::s; t::s; t2 , and a subsequence, say still [Xk], such that Xk -+ x in the p-metric, and x~ -+ x' weakly in L 1• Now both sequences I[xk] and C[Xk] are bounded below, one because of the growth conditions, and one because of the properties (L j ), and thus both i andj are finite. Now the argument for (11.l.i) applies. In the first case we conclude that I[ x] ::s; i, C[ x] ::s; D; in the second case we conclude that I[x]::S;N, C[x]::S;j. Since Q is rw-closed we conclude that xEQ. In the first case we conclude that I[ x] ::s; i, and by (1O.8.i) we derive that C[ x] ::s; D. In the second case we conclude that C[ x] ::s; j; and by (1O.8.i) we derive that I[ x] ::s; N. Since Q is r w-closed, we conclude that x E Q. Thus, in the first case, x E Q D and I [x] ~ i, hence I [x] = i, while in the second case, x E Q~ and C[x] ~j, hence C[x] = j. Both parts of(1l.1.vi) are thereby proved. 0

For a moment let I[x] and C[x] be any two functionals in a class Q of admissible trajectories, and assume that, for all D in a certain range, I[ x] has always an absolute minimum in the class KD of all x E Q with C[ x] = D.


1l.1.vii. If Min[I[ x] Ix E Q, C[ x] = D] = iD and iD is a strictly increasing function of D, then Max[ C[ x] Ix E Q, I[ x] = iD] = D, and the maximizing elements Xo E Q are the same with I( xo] = iD, C[ xo] = D. An analogous statement holds by exchanging minima and maxima and the sense of the inequalities. Proof. First we see that for any minimizing element Xo E KD for I we have I[ xo] = in, C[ xo] = D, and Xo belongs to the class K' of all elements x E Q

with I[x] = iD' If this class would contain an element x with C[x] = 15 > D, then in the class K jj of all x E Q with C[ x] = 15, the minimum of I[ x] is given by an element Xo E K jj with I[xo] = ijj ::; iD, while I[xo] = iD, C[xo] = 15 shows that the minimum of I[x] in KD must be <iD' a contradiction. 0

Now we prove an existence theorem for isoperimetric problems with F 0

a general extended function and H a linear function in x'.

1l.1.viii (AN EXISTENCE THEOREM FOR ISOPERIMETRIC PROBLEMS). Let A be bounded, B closed, and glower semicontinuous on B. Assume that F o(t, x, z) satisfies condition (C) or (C*), that F 0 satisfies any of the growth conditions (yl), (y2), (y3), and Fo is convex in z. Let C[x] =S:~Hdt with H(t,x,X') = Ao(t, x) + L~ AJt, x)xli and all Ao, Ai continuous on the closed A. [Alternatively, let us assume that H is B-measurable and that for almost all Y the functions Ao(Y, x), AJt, x) are continuous on the set A(Y), and that IAo(t, x)1 ::; ¢(t), IAJt, x)1 ::; Ci for all (t, x) and suitable constants Ci and L-integrable function ¢(t).] Let Q be any nonempty r w-closed class of AC functions x(t) = (Xl, ... ,xn ),

tl ::; t::; t2 , satisfying the requirements (11.1.5). Let D be a constant such that the class K D of all x E Q with C[ x] = D is not empty. Then I[ x ] has an absolute minimum in K D.

If A is unbounded but contained in a slab [to::; t ::; T] x W, then theorem (1 Ll.viii) still holds provided we know that every trajectory x E KD has at least a point (t*,x(t*)) on a given compact set P.

Actually, in the latter situation, we may relax the requirements above as follows.

First, we may only assume that H is B-measurable, that for almost all t

the functions Ao(t, x), Ai(t, x) are continuous with respect to x on the set A(t), and moreover that for any constant N > 0 there are constants Ci and an L-integrable function ¢(t) such that IAo(t, x)1 ::; ¢(t), IAi(t, x)1 ::; Ci for all (t, x) E A with Ixl ::; N.

Alternatively, we may only assume that F 0 satisfies one of the conditions (L;) (instead of (yl)-(y3)), but we need to know that the class {x'} of the derivatives of the elements x E KD is equiabsolutely integrable.

Finally, we can only assume that F 0 satisfies one ofthe conditions (L i), that for every N > 0 there are functions ¢(t), tfii(t), ¢ ELl' tfii E Lq, lip + llq = 1, p> 1, such that IAo(t, x)1 ::; ¢(t), IAi(t, x)1 ::; tfi;(t) for all (t, x) E A with Ixl ::; N, and that the class {x'} is relatively sequentially weakly compact in Lp.


Proof of (11.1.viii). For i = inf[I[x], x E K D] and any minimizing sequence xk(t), tlk S t S t 2k , k = 1,2, ... ,for/Ex] inKD' we have I [xk] -+ i, C[xk] = D. Under the main assumption of the theorem, then by the process used for the proof of(11.1.i) we derive that i is finite and that there are a subsequence, say still [k], and an AC function x(t), t1 S t S t 2, such that Xk -+ x in the p-metric, x~ -+ x' weakly in L 1, and that l[ x] s i. Since C[ x] is a continuous functional as we proved at the end of Section 10.8, we conclude that C[ x] = D. Hence, by the closedness property of D, we derive that xED, and then x E K D , lEx] ~ i, and by comparison lEx] = i.

If A is unbounded but contained in a slab as stated, then the sequence x;' is equiabsolutely integrable, hence the total variations V[ Xk] are equibounded. Since the trajectories Xk have a point on the compact set P, the same trajectories Xk are also equibounded, say /xk(t)/ s N for some Nand all t and k. Again C[x] is a continuous functional and C[x] = D. An analogous argument holds in the other cases. 0

It is possible to invert the role of F 0 and H in (11.1.viii) under some mild assumptions on the class Q, on H and on Fo.

We say that an element x(t), t1 :::;; t:::;; t2 , of a given class Q has property (n) at a point (I, x), x = x(f), provided t1 < I < t2 , (I, x) is in the interior of A, and there is (j > 0 with Nd(I,X) c A such that if we replace any arc Ao:X = x(t), IX:::;; t:::;; P, contained in Nd(I,X), by any other arc A.:X = A(t), IX:::;; t:::;; P, A(IX) = X(IX), A(P) = x(P), also contained in Nd(I, x), then the new trajectory x(t), t1 :::;; t:::;; t2 , belongs to Q.

For n = 1, H = Ao(t,x) + A 1(t,x)x', we assume that Ao, A1 are of class Cl, and that every element x of Q has at least one point (I, x), x = x(f), possessing property (n) and at which Aox #- Alt.

Finally, we need some assumptions of F o. Indeed for n = 1 we need to know that for all (t, x) E N d(I, x) the convex set Q(t, x) coincides with Rand F 0 is continuous in Nd(t,X) x R.

ll.l.ix. Under the conditions of (11.1.viii), for n = 1 and the above assumptions, if N is such that the class K/; of all x E Q with l[ x] = N is not empty, then C[ x] has an absolute minimum in K/;.

Proof. As usualletj denote the infimum of C[ x] in KN and, as in the proof of(11.1.vi), let [xk] be a minimizing sequence for C[ x], that is, Xl E K/;, C[ Xk] - j, l[ Xl] = N. By the same argument as in (11.1.i) there is an AC function x(t) = (xl, ... ,x"), t1 :::;; t:::;; t2 ,

and a subsequence, say still [Xl], such that X k - x in the p-metric, x~ - x' weakly in L10 and lEx] :::;; N. By the continuity of C[x] as in Section 10.8, we also have C[x] = j. If lEx] = N, then x E K/; and the proof is complete. We shall prove that lEx] = N. To this effect, we assume that l[ x] < N and we construct another trajectory x E Q, with l[ x] = Nand C[ x] < j, a contradiction.

For n = 1, let us assume that say, Aox > Alt at (I, x). We can take (j > 0 so small that Aox - AlT > 0 in Nd(I,X). Let Ao:X = x(t), IX:::;; t:::;; p, be an arc of x contained in Nd(I,x) and such that X'(IX) and x'(P) exist and are finite, and let A.:X = A(t), IX:::;; t:::;; p, be a polygonal line also contained in Nd(I,x) with A(IX) = X(IX), A(P) = x(P), and completely above Ao. Actually, we may think of the slopes of A to be ±k with k large.

11.2 Elimination of the Hypothesis that A Is Bounded in Theorems (l1.l.i-iv) 379

Because of the growth hypotheses on F 0' l[ A] can be made as large as we want by taking k large. On the other hand, by taking fJ as close to ex as needed, we may give to leA] any value ~1[Ao]. Thus we can always arrange that 1[A] = N - lex]. If,x is the trajectory x with the arc A replacing Ao, then lex] = N. On the other hand, by Green's theorem, if L denotes the region bounded by Ao and - A, then

Lo H dt - L H dt = Lx Aodt + Al dx = ff(A ox - Alr)dtdx > O. I

Hence

a contradiction. If Aox < Air the argument is the same with the arc A completely below the arc Ao. This proves (l1.l.ix) for n = 1. D

For n> 1, H = Ao(t,x) + Ii Ai(t,x)X'i, we assume that Ao, ... ,An are of class C l

in A, so that, ifV(t, x) denotes the (n + l)-vector V(t, x) = (Ao, . .. , An) and n' = n(n + 1)/2, then the usual n' -vector curl V, constructed with the first order partial derivatives of Ao, ... , An, is a continuous function of (t, x) in A. Then we need the same assumption (n) and the assumption that curl V of- 0 at some point (7, x) of the trajectory, that is, some of its components are not zero. Then the same theorem (l1.l.ix) still holds under mild and generic further assumptions which for the sake of brevity we do not state here. We refer for the precise statement and proof to P. Pucci [1]. For n > 1 the proof is rather technical and differs from the previous one in many respects.

11.2 Elimination of the Hypothesis that A Is Bounded in Theorems (ll.l.i-iv)

The hypothesis that A is bounded in the theorems (l1.1.i-iii) can be easily removed as we have done in Chapter 9. All we have to do is to guarantee that a minimizing sequence can be contained in some bounded subset Ao of A. After that the statements and the proofs are the same. Often all this results at a glance from the data and the geometrical configuration of the particular problem under consideration. However, we shall list here some general conditions for this to occur.

A. If the set A is not bounded, but A is contained in a fixed slab [to S t S T, x ERn] of Rn + I, then Theorem (1L1.i) is still valid if we know, for instance, that

(hi) 9 is bounded below on B, say 9 ~ - M I, and (C 1) Q is a given nonempty r w-closed class of AC trajectories x each of which has at

least one point (t*, x(t*)) on a given compact subset P of A (t* may depend on the trajectory).

For instance, the curves C:x = x(t), tl S t S tz, may have the first end point (or the second end point) either fixed or on some fixed compact set P of A.

It is enough to consider only those elements x E Q with lex] s M for some M; hence f:: Fodt s M + MI' Then, under conditions (C 1) and (yl), we have 4>(() ~ -v, v ~ 0, a constant, and there is some N ~ 0 such that 4>(0 ~ , for all , ~ N. If x E Q, if


E* is the subset of [tl,tZ] where Ix'(t)lz N, and if E = [tl>tz] - E*, then

F oft, x(t), x'(t» + v z 0

for all t, F oft, x(t), x'(t» z Ix'(t) I

for t E E*, Ix'(t)1 s N for tEE, and

M + MI z (12[Fo(t,x(t),x'(t» + v]dt - (12 vdt Jt1 Jtt

z IE' [F oft, x(t), x'(t» + v] dt - v(T - to)

z IE> Ix'(t) I dt - v(To - to) + IE [lx'(t)1 - N] dt

= {'2Ix'(t)1 dt - (v + N)(T - to), JI.

or J:; Ix'(t)1 dt s M + (v + N)(T - to), a fixed number. Thus, the curves c:x = x(t), tiS t S tz, under consideration have total variation below a fixed number, and since they contain a point of the bounded set P, they are contained in some fixed cylinder [(t,x)lto s t s T, Ixl s Mol

Under condition (CI) and (y2), taking 6 = 1, we have Izl s tjJ I (t) + F 0, or

M + M I Z (12 F oft, x(t), x'(t» dt z ('2 [lx'(t)I- tjJ I (t)] dt z f,'2Ix'(t)1 dt - (T tjJl dt, Jtt Jtt t1 Jto and again the curves under consideration have length below a fixed number.

The same conclusion holds under conditions (CI) and (y3). If A is neither bounded nor contained in any slab as above, then theorem (ll.l.i) is

still valid if we know for instance that (hi) and (C I ) hold, and in addition that

Cz. There are constants Ji > 0, Ro z 0 such that F oft, x, z) z Ji > 0 for all (t, x, z) with Itlz Ro·

Indeed, any part of the curve C:x = x(t) lying in the slab [ -Ro S t s Ro, x E R"] has an integral bounded below, say z -MI for some constant MI' We assume Ro large enough so that P is completely contained in the slab. Again let lEx] sM. Now, if RI = Ro + Ji-I(MI + M), then any of the curves C above must be contained in the slab [ -RI s t SRI, X E R"], since otherwise, such a curve would contain at least an arc with -RI s t s -Ro, or Ro s t sRI> and then lEx] z Ji(R I - Ro) - MI > M.

B. Concerning Theorem (1l.1.ii), if A is not bounded but contained in a slab [to s t S T] x R", to, T finite, we may again assume that (hd and (CI) hold, and that the r w·closed class Q = {x(t), tl S t S tz} of AC trajectories is such that the class {x'(t), tl s t S tz} is equiabsolutely integrable. Indeed, by (lO.2.ii), the total variations VEx; t I, tz] are equibounded, say V[ x] s N I' Since any trajectory x in Q has at least one point (t*,x(t*» E P in a given compact set P, then Ix(t*)1 s MI and Ix(t)1 s MI + N I ,

tl S t S tz, for every element x of Q. It may be of interest to know that the same conclusion can be derived from assump

tions (hd, (CI), and the following one:

C3• There are constants c > 0, R z 0 and a locally integrable function tjJ(t) z 0, t E R, such that F oft, x, z) z - tjJ(t) + clzl for all (t, x, z) with Ixlz R.

11.3 Examples 381

Indeed, we may take R so large that P is completely contained in the cylinder A = [(t,x)lto :::;; t:::;; T, Ixl:::;; R]. Now let x be an element of Q with lex] :::;; M. The parts of C inside A contribute to the value of l[ x] an amount certainly above some constant -Jl. Now take the cylinder Al = [(t,x)lto :::;; t:::;; T, Ixl:::;; R + R l]. If E* denotes the set of all t E [tl> t2 ] with Ix(t)1 ~ R, then

JE" Ix'(t)1 dt :::;; c- l JE" [F o(t, x(t), x'(t» + t/!(t)] dt

:::;; c- l [lex] + JI+ IT t/!(t)dt] Jlo :::;; c-l(M + JI + M 2),

a fixed constant. Thus, C is completely inside Al if Rl is larger then the above constant. If A is not bounded nor contained in any slab as above, then (11.1.ii) is still valid

provided we know, for instance, that (hl), (C l), (C2), (C3) hold. Concerning Theorem (11.l.iii), the same remarks above hold as for (11.l.ii). C. Concerning Theorem (11.1.iv) for Mayer problems, then for A closed and con

tained in a slab as above, the following condition may be used:

For A not compact or contained in any slab as above, we may use the condition:

11.3 Examples

Many simple examples concerning Lagrange problems ofthe calculus ofvariations have been anticipated in Section 2.20B and they all concerned Theorem (l1.1.i) with 9 = ° and F 0 continuous in A x RO. Here are a few more examples concerning Theorems (11.l.i-iv) without restrictions.

1. Take n = 1, F 0 = ° for Ix'i :::;; 1, F 0 = X,2 for Ix'i > 1, g(X2) = 4 iflx21 < 2, g(X2) = ° iflx21 ~ 2, tl = 0, Xl = 0, t2 = 1, X2 undetermined. Here A = [0,1] x R, B = (0,0, 1) x R, 9 is lower semicontinuous, F o(t, x, x') satisfies growth condition (gI) and, for every I, F o(t, x,x') is lower semicontinuous in (x,x'). Conditions (h l ) and fCl) hold. By (11.1.i), l[ x] has an absolute minimum in the class Q of all trajectories under the constraints.

2. Take n = 1, F 0 = t l/2(1 - t)l/2X,2, g(X2) = ° for - 00 < x2 :::;; 1, g(X2) = 1 for 1 < x2 < + 00, tl = 0, Xl = 2, t2 = 1, X2 undetermined. Here A = [0,1] x R, B =

(0,2,1) x R, 9 is lower semicontinuous. Let us prove that F 0 satisfies growth condition (g2). To this effect, given e > 0, take t/!.(t) = e- l t- l /2(1_ t)-1/2, 0< t < 1, an Lintegrable function in (0,1). Then, for Izl :::;; e- l t- l /2(1 - t)-l/2 we have Izl :::;; t/!.(t) :::;; t/!.(t) + Fo; for Izl > e- 1C l/2(1_ t)-1/2 we have etl/2(1_ t)1/2Izl > 1 and Izl < etl/2(1 - t)l/2Z2, hence Izi < eFo:::;; t/!.(t) + eFo. Also, for every I, Fo(t,x,z) is a continuous function of (x, z). Let Q denote the class of all AC trajectories satisfying the above constrains and also satisfying the further constraint Ix(1/2)1 :::;; 1. The class Q is r w-closed. By (11.1.i), l[x] has an absolute minimum in the class Q.


3. Take n = 2, F 0 = sin t + (2 + cos tx)x'Z + (2 + sin ty)y'z if 0 :s; t:s; 1, F 0 = 1 + x'z + y'z if t > 1, t1 = 0, Xl = 0, Y1 = 0, (tz, Xz, yz) on the locus T = [(t - 1)(xz + yZ) = 1, t > 1], g(tz, xz, yz) = (tz - 1) -1, A = [0, + (0) X RZ, B = (0,0,0) x r. Conditions (g1), (hd and (C 1 ) hold. By (11.1.i), I[ x] has an absolute minimum in the class Q of all trajectories under the constraints.

4. Take n = 1, Fo = It - xl- 1/Zx'z if It - xl:s; 1, and x' ~ -t-I, Fo = + 00 otherwise. In particular Fo = +00 for x = t, O:s; t:s; 1. Take g = 0, t1 = 0, Xl = 0, tz = 1, Xz = 1. Here A = [(t,x)IO:s; t:s; 1, Ix - tl :s; 1], B = (0,0,1,1). For every 7 E [0,1] the extended function F oCf, x, z) is lower semicontinuous in (x, z), and for every (7, x) E A the extended function F oCf, x, z) is convex in z. Here It - xl :s; 1, hence It - xi-liZ ~ 1, and F 0 ~ x'z. Thus, (g1) holds with cp(O = 'z. Conditions (h1) and (C 1) hold. The class Q of all trajectories satisfying the data is not empty. Indeed, let us prove that x(t) =

t + et(1 - t), 0 :s; t :s; 1, for any 0 < e :s; 1 belongs to Q. First Ix(t) - tl :s; 1 and thus (t,x(t)) E A for all 0 :s; t:s; 1. Moreover,x'(t) = 1 + e - 2et > _t- 1 forO < t:s; 1; hence x'(t) E Q(t,x(t)) since Q(t,x) = [zlz ~ -t- 1]. Finally,

It - x(t)I-1 /2x ,Z(t) = e - 1/Zt -1 /Z(1 - t)-l /Z(1 + e - 2et)Z

is L-integrable in (0, 1). By (11.1.i), I[ x] has an absolute minimum in Q.

5. Minimum ofJ:~ (x' - 2t + I)Z dt under the constraints t 1 = 0, x(t 1) = 1, (t z, x(tz)) on the locus T= [t2 +X2 =4, t~O], and J:~ tlx'i dt:s; 1/2. Here n= I,A= [t~O, t2 +xz:s;4], Fo = (x' - 2t + l)z, H = tlx'l, Fo satisfies ()II), H satisfies (L1). The minimum exists by (11.1.vi).

6. Minimum of J:~ (tzlx'i + x') dt under the constraints t1 = -1, x(t 1) = -1, tz undetermined, x(tz) = 1, and J:~ (1 + x'z) dt :s; 5. Here H = 1 + x'z ~ 1; hence tz :s; 4, and we can take A = [ -1, 4] x [-1,1]. Then F 0 = tZlx'l + x' satisfies (Lz), and H = 1 + x'z satisfies ()II). The minimum exists by (1 1. Lvi).

7. Minimum of f:; x'Z dt under the constraints t1 = -1, x(t1) = 0, tz = 1, x(tz) = 1, and J:~ (tx + (sgn t + x)x') dt = 2. Here n = 1, F 0 = x'z, H = tx + (sgn t + x)x', Ao = tx, A1 = sgn t + x, IAol :s; lxi, A1 :s; 1 + IxI- The minimum exists by (11.1.viii).

8. Minimum of J:~ xZdt under the constraints t1, X(t1), tz, x(tz) fixed, t1 < tz, and J:; x'z dt = C for a given C > (x(t z) - X(t1))Z(tZ - t1)-1. Here n = 1, Fo = xZ, Ao = xZ, A1 = 0, Aox - A1! = 2x "# 0 for all x "# 0, H = x'z satisfies ()II). The minimum exists by (l1.l.ix). Here the same integral has also a maximum under the same constraints.

Exercises

Show that some of the Existence Theorems of Sections 11.1-2 apply: 1. Take n = 1, g = O. For t:s; 0 take Fo(t,x,x') = (1 + t2 + xZ)x'z if x':s; 0, Fo = +00

if x' > 0; for t > 0 take F 0 = (1 + tZ + XZ)X'2 if x' ~ 0, F 0 = + 00 if x' < O. Here Q(t, x) are the sets Q(t, x) = [ - 00 < z :s; 0] if t :s; 0; Q(t, x) = [0 :s; z < + 00] if t > O. Let Q denote the class of all admissible trajectories with t1, Xl> t2, Xz fixed, t1 < 0 < t2,

X1"# xz· 2. Take n = 1, Fo = (1 + t + x)x', g = 0, t1 < tz, Xl' Xz fixed. Let Q denote the class of

all AC functions x(t), t1 :s; t:s; t2, with J:; x'z dt :s; D where D is any number larger than (xz - xdz(tz - t1)-1.

3. Take n = 2, F 0 = t- 1/Z + (1 + t + x)x' + (1 - t - y)y' a linear integrand, g = 0, t1 = 0, Xl = 1, Y1 = 1, tz = 1, (x z, yz) on the locus T = [x2 + yZ = 4]. Let Q denote the class of all admissible trajectories with J~ (x'Z + y'z) dt :s; 4.

11.4 Problems of Optimal Control with Unbounded Strategies 383

4. Take n = 2, cp(t) = sgn t, F 0 = t- 1/Z sin(x + y) + cp(t)(2x + y)x' + cp(t)(x + 2y)y', 9 = 0, tl = -1, Xl = 1, Yl = 1, t z = 1, (xz, yz) undetermined. Let Q denote the class all admissible trajectories with g(x'Z + y,z)dt ~ 2.

11.4 Existence Theorems for Problems of Optimal Control with Unbounded Strategies

A. Existence Theorems for Lagrange and Bolza Problems of Optimal Control

Essentially, most of the existence theorems for Lagrange and Bolza problems of optimal control presented here are corollaries of the existence theorems of Section 11.1 for extended free problems. However, new remarks are needed in connection with a different emphasis and different possible applications.

We are concerned here with the problem of the absolute minimum of the functional

(11.4.1)

with constraints, boundary conditions, and differential equations

x(t) E A(t), (t1> x(td, t2, x(t2)) E B, u(t) E U(t, x(t)),

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

fo{-, x(·), u(·)) E L 1[t 1, t2]' x(t) AC,

u(t) measurable in [t1> t2]'

where x = (xl, ... , xn), f = (f1, ... ,fn), U = (u 1, ... , u"'). Here A is a given subset of the tx-space Rn+l, B a given subset of the t1x1t2x2-space R2n+2, and for every (t, x) E A, U(t, x) is a given subset of the u-space Rm. Let g(t 1, Xl' t2 , X2) be a given real valued function on B. Let A o denote the projection of A on the t-axis. For every t E Ao let A(t) denote the corresponding section of A, or A(t) = [xl(t,x) E A] c W. Let M denote the set of all (t, x, u) with (t, x) E A, u E U(t, x). Let fo(t, x, u), f(t, x, u) = (f1, ... ,fn) be given functions defined on M, and for every (t, x) E A let Q(t, x) denote the set of all (ZO, z) E Rn+ 1 with ZO ~ fo(t, x, u), z = f(t, x, u), U E U(t, x). Then the projection Q(t,x) ofQ(t,x) on the z-space Rn is the set Q(t,x) = [zlz = f(t,x,u), u E U(t, x)], or Q(t, x) = f(t, x, U(t, x)). Below we shall assume Ao to be an interval of the t-axis, finite or infinite.

As usual, we say that a pair x(t), u(t), t1 ~ t ~ t2, is admissible for the problem (11.4.1-2) if x is AC, u is measurable, and the requirements (11.4.2) are satisfied. A function x(t), t1 ~ t ~ t2, is said to be admissible for the problem (11.4.1-2) if there is some u such that (x,u) is admissible. Given a


class Q = {(x, u)} of admissible pairs for (11.4.1-2) we may denote by Q x = {x} the class of corresponding trajectories, or Qx = {x} = {xl(x,u) E Q}. Given any class Q of admissible pairs for (11.4.1) we shall denote by i the infimum of I[ x, u] in Q.

We may be concerned with the problem of the minimum of I[x,u] in the class Q of all admissible pairs. Alternatively, and as in Sections 9.2, 9.3, 11.1, we may want to minimize I[x, u] in a smaller class Q of admissible pairs, and in this case we need to know that Q has a mild closedness property. We say that Q is row-closed provided (a) Qx is r w-closed (Section 11.1), and (b) x E Qx and (x, u) admissible for (11.4.2), implies that (x, u) E Q. In other words, the class Q is row-closed provided: if Xk(t), Uk(t), tlk:s;; t:s;; t2k , k = 1, 2, ... , are admissible pairs all in Q, if x(t), u(t), t 1 :s;; t :s;; t2, is an admissible pair, and if Xk -+ x in the weak convergence of the derivatives, (that is, Xk -+ x in the p-metric and xi, -+ x' weakly in L 1), then (x, u) belongs to Q.

Actually somewhat less than (b) is needed. Namely, in proving the existence of the minimum it would be enough to know that (a) holds, and that (b') if x E Qx and (x, u) is admissible, then either (x, u) E Q, or there is some u such that (x, u) E Q and I[ x, u] :s;; I[ x, u].

Let M ° denote the set of all (t, x, z) E R2n + 1 with (t, x) E A, Z E Q(t, x). We shall need the function T(t, x, z), - 00 :s;; T < 00, defined on M ° by taking

T(t, x, z) = inf[ ZO I (ZO, z) E Q(t, x)]

= inf[ ZO I ZO ~ fo(t, x, u), Z = f(t, x, u), u E U(t, x)].

We may extend T to all of R 2n + 1 by taking T(t, x, z) = + 00 for T(t, x, z) E

R 2n +1 _ Mo. Then Q(t,x), Q(t,x) are the empty sets for (t,x) E R"+1 - A. In the discussion below we shall reduce the problem of the absolute

minimum of the functional (11.4.1) under the constraints (11.4.2) to the problem of the absolute minimum of the functional

(11.4.3)

under the constraints

(11.4.4)

(t,x(t» E A, (t 1,X(t1),t2,X(t2» E B, x'(t) E Q(t,x(t»,

t E [t1> t2] (a.e.), T(·, x(·), x'(·)) E L 1[t1,t2],

a problem we have studied in Section 11.1. For most applications it would be enough to assume

C'. A closed, M closed, and fo, f continuous on M.

However much less is needed. For instance, if A and M are products of intervals (possibly infinite) A = It x Ix, M = It x Ix x I y, It cR, Ix c R", Iy = U c Rm, then we could simply assume that fo and fare Caratheodory functions on M, namely, measurable in t for every (x, u), and such that for almost all t, fou, x, u), fu, x, u) are continuous functions of (x, u). Then such


functions would have the Lusin property as stated in (8.3.v), namely, given E > 0, there is a closed subset K of It such that fo and f, restricted to K x Ix x Iu, are continuous and meas(lt - K) < E.

But it may well be that either A or M or both are not products of intervals, namely, the sections A(t) of A may well depend on t, and the control sets U(t, x) may well depend on t and x. In this situation we could require that the relevant set valued maps t --> A(t) and (t, x) --> U(t, x) are measurable (Section 8.3), and structure accordingly the whole argument. All this is unnecessary. All we need on A, M, fo, and f, for the proof that follows, is the following Caratheodory type property:

C'*. For every E >0 there is a closed subsetK ofAo such that meas(Ao - K)<E, the sets AK = [(t,x) E Alt E K], MK = [(t,x,u) EMit E K] are closed, and both fo(t, x, u),f(t, x, u) = (fl, ... ,f,,) are continuous on M K'

It is easy to see that in any case the function T(t, x, z) is either B-measurable, or at least it satisfies requirements (a), (b) of condition (C*) of Section 11.1. Indeed, if M is closed, and fo, f are continuous on M, then the B-measurability of Twas proved in Exercise 4 of Section 8.2e. Under the alternate assumption (C'*) for every E = S-l, S = 1,2, ... , there is a closed subset Ks of Ao such that meas(Ao - Ks) < S-l and j~,f are continuous on M K , and then Tis B-measurable on (Ks X R2n).

As mentioned in Section ILl, this is enough to guarantee that for any AC function x(t), tl ~ t ~ t2, then T(t,x(t),x'(t» is measurable in [tbt2]. If Q = {(x, un is any nonempty class of admissible pairs for the problem (11.4.l-2), then x E Q x is AC, u is measurable, and fo(t, x(t), u(t» 2 T(t, x(t), x'(t», t E [t1' t2] (a.e.), where fo is L-integrable and T is measurable. Thus T(t, x(t), x'(t» is L-integrable in the sense that its L-integral is either finite or - 00. Thus J[x] is defined in Q x and we denote by j the infimum of J[ x] in Qx' Certainly - 00 ~ j ~ i < + oc;.

Remark 1. It may well occur that for an AC trajectory x E Q x the measurable function T(t, x(t), x'(t» has actually L-integral -(J). Take fo(t,x,u) = -t- Iu2(1 + U2)-I, f(t,x,u) = sin u, U(t, x) = [uIO::; u < +(J)]forO < t::; l,xE R;fo = 0,1 = 0, U(O,x) =

{O} for t = 0, X E R. Then A = [0, 1] x R, M = M 1 U M 2 is the union of M 1 = (0, 1] x R x [Izl::; 1] and of M2 = {OJ x R x {OJ. Also, T(t,x,z) = _t- I on M I , T(t,x,z) = ° on M 2' Let Q consists of all pairs x(t), u(t), 0::; t::; 1, x AC, u measurable, n = m = 1, x(O) = 0, u(t) E U. Then x(t) = 0, u(t) = 0, 0::; t::; 1, is an admissible pair. Yet, T(t,x(t),x'(t» = _t- I is not L-integrable, that is, the AC trajectory x(t) = ° does not satisfy the requirements (11.4.4).

Remark 2. It may well occur that for an AC function x satisfying (11.4.4) there is no u such that (x,u) is admissible for (4.4.1-2) and fo(t,x(t),u(t» = T(t,x(t),x'(t». Indeed, take fo(t, x, u) = (1 + U2)-I,f(t, x, u) = sin u, U(t, x) = [uIO::; u < + (J)]. ThenMo is the set [(t,x,z)l(t,x) E R2, Izl::; IJ, and T = ° on Mo. Now for the AC function x(t) = 0, 0::; t::; 1, we have T(t, x(t), x'(t» = ° and for no measurable u we can have fo(t, x(t), u(t» = 0, x'(t) = ° = f(t, x(t), u(t» for almost all t E [0,1]'


In the existence theorems below we shall need alternate "growth hypotheses" of the scalar function j~ with respect to the vector function f =

(j~, ... ,jJ They are the usual growth conditions but expressed directly in terms of fo and f.

(gl') There is a scalar function ¢(n, 0 ~ ( < + 00, bounded below, such that ¢(W( --+ + 00 as (--+ + 00, and fo{t, x, u) :::: ¢(If(t, x, u)l) for all (t,x, u) E M.

(g2') For any c > 0 there is a locally integrable scalar function t/J ,(t) :::: 0 such that If(t, x, u)1 ~ t/J it) + cfo(t, x, u) for all (t, x, u) E M.

(g3') For every n-vector p ERn there is a locally integrable function ¢p(t) :::: 0 such that fo{t, x, u) :::: (p,f(t, x, u)) - ¢it) for all (t, x, u) E M.

1l.4.i (AN EXISTENCE THEOREM BASED ON WEAK CONVERGENCE OF THE

DERIVATIVES). Let A be bowzded, B closed, and let condition (C') or (C'*) be satisfied. Assume that (k) for almost all 7 E Ao the sets <2(7, x) are convex and have property (K) with respect to x only on the closed set A(7). Let g be a lower semicontinuous scalar function on B. Assume that anyone of the growth conditions (gl'), (g2'), (g3') is satisfied. Let Q be a nonempty row-closed class of admissible pairs x, u. Then the functional (11.4.1) has an absolute minimum in Q.

For A not bounded see Section 11.S. It is enough we limit ourselves to the nonempty part QM of Q of all elements (x, u) E Q with lex, u] ~ M for some M. We just recall from (8.S.vii) that, if for almost all 7, the functions fo(7, x, u), f(7, x, u) are continuous in (x, u) and either fo(7, x, u) --+ + 00, or If(7, x, u)l--+ + 00 as lul--+ + 00 locally uniformly with respect to x, then the sets <2(7, x) certainly have property (K) with respect to x.

Proof of (11.4.i). First note that conditions (gl'), (g2'), (g3') for problem (11.4.1-2) imply conditions (yl), (y2), (y3) respectively for problem (11.4.3-4).

Under hypothesis (gl') the function fo is bounded below, say fo :::: - M 1; under hypothesis (g2') we have fo 2: - t/J l(t); under hypothesis (g3') we have j~ 2: - ¢o(t); in any case fo(t, x, u) :::: - t/J(t) where t/J is a nonnegative locally integrable function. By virtue of (8.S.v) and of (8.S.iii) and subsequent Remark 2, and under condition (k) above, then for a.a. 7 E G the sets <2(7, x) are closed, hence <2(7, x) = epiz T(7, x, z), and the extended function T(Y, x, z) is lower semicontinuous in (x, z) and convex in z, or equivalently for almost all 7, the sets M(I) = [(x, zo, z) 1 x E A(I), ZO :::: fo(I, x, u), z = f(7, x, u), u E U(7, x)] are closed and convex. Since we have already proved that Tis B-measurable on the sets K X R 2n, we see that, under either condition (C') or (C'*) on the functions fo, f, the extended function T(t, x, z) satisfies the condition (C*) of Section 11.1. Also, there is at most a set of measure zero of points 7 where the function T(T, x, z) may take the value - 00 for some (x, z). Otherwise, the extended function T takes only finite values or + 00.


For any admissible pair (x, u) E Q, that is, x E Qx, we have now fo(t, x(t), u(t)) ~ T(t, x(t), x'(t)) ~ - t{!(t). Hence, the measurable function T(t, x(t), x'(t)) is L-integrable as lying between two functions having the same property. Moreover, - 00 < j ~ i < + 00. Finally, the class Qx is r w-closed. All conditions of Theorem (11.1.i) are satisfied. By (11.1.i) then there is an element x E Q x such that

(T(t,x(t),x'(t)), x'(t)) E Q(t,x(t)), t E [t1' t2] (a.e.),

lex] = g(t1,x(td,t2,X(t2)) + ft2 T(t,x(t),x'(t))dt=j~ i. Jt.

Let R+ = [0 ~ v < +00], and take O(t, x) = U(t,x) x R+, j(t,x,u,v) = (J oJ) with 10 = fo(t, x, u) + v. With these definitions and for almost all t we have

- ° 1 ° ] Q(t, x) = [(z ,z) z ~ fo(t, x, u), z = f(t, x, u), u E U(t, x)

= [(zO,z)lzO = fo(t,x,u) + v, z = f(t,x,u), U E U(t,x), v E R+]

= [(ZO, z) = J(t, x, u, v), (u, v) E O(t, x)J.

For every E = S-l, S = 1, 2, ... , we take Ks closed such that meas(Ao -K.) < S-l, AK is closed, MK is closed, andfo,fare continuous on M K. Then, J is continuous on the closed set M K X R + . By the implicit function theorem (8.2.iii) there are measurable functions us(t) E U(t, x(t)), vs(t) ~ 0, t E Ks' such that

(T(t, x(t), x'(t)), x'(t)) = j(t, x(t), us(t), vs(t)), t E Ks(a.e.),

that is,

T(t, x(t), x'(t)) = fo(t, x(t), uit)) + vs(t), x(t) = f(t, x(t), uit)).

Since T ~ fo we must have Vs = ° and T(t,x(t),x'(t)) = fo(t,x(t),us(t)), t E Ks (a.e.).lfwe take u(t) = us(t) for t E Ks - (K1 U· .. u Ks- 1), S = 1,2, ... ,then u(t) is defined a.e. in [t b t2], is measurable, u(t) E U(t,x(t)), T(t,x(t),x'(t)) = fo(t, x(t), u(t)), and x'(t) = f(t, x(t), u(t)), t E [tb t2] (a.e.). In other words, (x, u) is an admissible pair for problem (11.4.1-2) and l[ x, u] = J[x J.

Since Q is row-closed, by the part (b) of the definition of r ow-closedness, we conclude that (x, u) belongs to Q and hence l[ x, u] ~ i. Thus i ~ l[ x, u] = J[ x] = j ~ i, where iis finite, equality holds throughout, lex, u] = i and(ll.4.i) is proved. Under the alternate assumption (b') then there is a measurable u such that (x, u) E Q, lex, u] ~ lex, uJ. Thus, i ~ lex, u] ~ lex, u] = J[x] = j ~ i, and equality sign holds throughout. Theorem (11.4.i) is thereby proved. D

Remark 3. If the convexity condition in (11.4.i) is not satisfied, there still exists an optimal generalized solution. The argument is similar to the one in Remark 2 of Section 11.1A.


To state the next theorem, we shall denote, as usual, by Q a class of admissible pairs x, u, and by Qx the corresponding class of trajectories. Analogously let QM denote the subclass of all (x, u) E Q with I[ x, u] :::; M, and Qx,M the corresponding class of trajectories x.

Also we shall need certain alternative conditions similar to the conditions (Li) of Section 10.8:

(L~) There is a locally integrable real valued function t/J(t) ~ 0, t E R, such that fo(t, x, u) ~ - t/J(t) for all (t, x, u) E M.

(Lz) There are a locally integrable real valued function t/J(t) ~ 0, t E R, and a constant c>O such that fo(t,x,u)~-t/J(t)-clf(t,x,u)1 for all (t,x, u) E M.

(L~) There are a locally integrable real valued function t/J(t) ~ 0, t E R, and an n-vector valued bounded measurable function cp(t) = (cpt> ... ,CPn), t E R, such that fo(t, x, u) ~ - t/J(t) - (cp(t),f(t, x, u)) for all (t, x, u) E M.

(L~) There are constants rI. ~ f3 real and y > ° such that (a) for every (t, x) E A the set Q(t, x) contains the ball Bo = [z E Rn Ilzl :::; y] and fo(t, x, u) ~ f3 for all (t, x) E A and all u E U(t, x) with If(t, x, u)1 :::; y; (b) fo(t, x, u) :::; rI.

for all u E U(t, x) with f(t, x, u) = 0. These requirements on fo, fare certainly satisfied if M is closed, fo(t, x, u), f(t, x, u) are continuous on M, and If(t, x, u)I--+ + 00 as lul--+ + 00 uniformly for (t, x) E A.

1l.4.ii (AN EXISTENCE THEOREM BASED ON THE WEAK CoNVERGENCE OF THE

DERIVATIVES). Let A be bounded, B closed, and let condition (C') or (C/*) be satisfied. Assume that (k) for almost all I E Ao the sets QCf, x) are convex and have property (K) with respect to x only on the closed set A(I). Let g be a lower semicontinuous scalar function on B. Assume that anyone of the conditions (LD holds. Let Q = {(x, u)} be a nonempty row-closed class of admissible pairs, and assume that the class {x'} of the derivatives of the trajectories x in Q is equiabsolutely integrable. Then the functional I[ x, u] in (11.4.1) has an absolute minimum in Q.

It is enough that we limit ourselves to the nonempty part QM of Q of all elements (x, u) E Q with I[ x, u] :::; M for some M. For A not bounded see Section 11.5. A sufficient condition for requirement (k) was mentioned after statement (l1.4.i).

Proof. Each of the conditions (Li) for the problem (11.4.1-2) implies the corresponding condition (Li) of Section 11.1 for the problem (11.1.3-4). Under condition (L't) we have the same type of lower bound for fo and T we had in the proof of (11.4.i), or fo(t, x(t), u(t)) ~ T(t, x(t), x'(t)) ~ - t/J(t), where t/J ~ ° is a fixed locally integrable function, and x'(t) = f(t, x(t), u(t)) a.e. in [t t> t 2]. The argument is therefore exactly the same as for (11.4.i).

Under conditions (Lz), (L~), (L~) for (11.4.1-2), conditions (L2), (L3), (L4) hold for (11.1.3-4), and we have seen in Sections 10.8 and 11.1, that (L3), (L4) are actually particular cases of (L2), that is, in any case, there are a locally


integrable function tjJ(t) ~ 0 and a constant c ~ 0 such that

fo(t,x,u) ~ -tjJ(t) - clf(t,x,u)l, or T(t,x,z) ~ -tjJ(t) - clzl·

Thus, there is at most a set of measure zero of points T where T(T,.t, z) may take the value - 00 for some (x, z). Otherwise, the extended function T(t, x, z) takes only finite value or + 00. As in the proof of (l1.4.i), the problem (11.4.3-4) satisfies condition (C*) of Section 11.1 and all requirements of (l1.l.ii). The proof now continues as for (l1.4.i). D

Remark 4. If the convexity condition in (11.4.ii) is not satisfied, there still exists an optimal generalized solution, provided the sets R(t, x) = co (2(t, x) have property (K) with respect to x. Cf. the analogous Remark 4 of Section l1.1A.

In applications often the following simple corollary of (l1.4.ii) suffices:

1l.4.iii (AN EXISTENCE THEOREM BASED ON WEAK CONVERGENCE OF THE

DERIVATIVES). Let A be compact, B closed, M closed, fo and f continuous on M, with either fo(t, x, u) --+ + 00, or If(t, x, u)I--+ + 00 as lul--+ + 00, or both, locally uniformly for (t, x) E A. Assume that for almost all T the sets Q(Y, x), x E A(T), contain a fixed ball Bo = [z E Rnllzl ::;; y], and that the sets Q(Y, x) are convex. Let g be lower semicontinuous on B. Let Q = {(x, u)} be a nonempty row-closed class of admissible pairs, and assume that the class {x'} of the derivatives of the corresponding trajectories is equiabsolutely integrable. Then the functional I[x, u] in (11.4.1) has an absolute minimum in Q.

It is enough that we limit ourselves to the nonempty part QM of Q of all elements (x, u) E Q with I[ x, u] ::;; M. From (8.S.vii) the sets Q(t, x), x E A(T), have property (K) with respect to x on the closed set A(T). Theorem (l1.4.iii) is now a corollary of (11.4.ii) and (L4).

B. Existence Theorems for Problems of Optimal Control with a Comparison Functional

We consider now the problem (11.4.1-2) when a comparison functional, say

(11.4.5) C[ x, u] = jt2 H(t, x(t), u(t)) dt, Jr, is assigned, and we consider classes Q of admissible pairs x(t), u(t), t1 ::;; t ::;; t2 , for which both I[ x, u] and C[ x, u] are finite and C[ x, u] ::;; M for a given constant M. For any (t, x) E A we shall denote by QH(t, x) the set - 0 I 0 QH(t, x) = [(3, z ,z) 3 ~ H(t, x, u), z ~ fo(t, x, u), z = f(t, x, u)], U E U(t, x). Also, let] denote the (n+ l)-vector function J(t, x, u)=(fo,f) = (fo,f1' ... ,j~).

As usual, we shall consider the problem of the minimum of I[ x, u] in the class Q of all admissible pairs x, u satisfying C[ x, u] ::;; M. Alternatively, we may consider the same problem in a smaller class Q of such admissible pairs,


and in this case we shall require a mild closure property on the class Q. We say that Q is row-closed with respect to H,fo, f provided the following variant of our definition at the beginning of this section holds: (a) Qx is r w-closed (Section 11.1), and (b) XE Qx, (x,u) admissible, C[x,u]::;; M, implies that (x, u) E Q. Actually, it would be enough to assume instead of (b) the following less demanding requirement: (b') x E Qx, (x, u) admissible, C[ x, u] ::;; M, implies that either (x, u) E Q, or that there is some other u such that (x, u) is admissible, C[x, u] ::;; M, lEx, u] ::;; lEx, u], and (x, u) E Q. Also, we shall denote by Qll{t,x) the set of all (3,zO,z) with 3~H(t,x,u), zO=fo(t,x,u), z = f(t, x, u), u E U(t, x).

1l.4.iv (AN EXISTENCE THEoREM WITH A DOMINANT COMPARISON FUNCTIONAL

AND ISOPERIMETRIC PROBLEMS). Let fo, H satisfy condition (C') or (C'*). Let A be bounded, B closed, and glower semicontinuous on B. Assume that H satisfies one of the growth conditions (g1'), (g2'), (g3') with respect to J = (fo,f), and that fo satisfies one of the conditions (Li) with respect to f. Assume that (k) for almost all t E Ao the sets QH("f,X) are convex and have property (K) with respect to x on the closed set A(I). Let Q be a nonempty row-closed class of admissible pairs.

(a) Let D be a constant such that the subclass QD of all (x, u) E Q with C[ x, u] ::;; D is not empty. Then lEx, u] has an absolute minimum in QD.

(b) Let N be a constant such that the subclass Q~ of all (x, u) E Q with lEx, u] ::;; N is not empty, then C[ x, u] has an absolute minimum in Q~.

(c) Here we require for g to be continuous on B, we assume that (k) holds for the sets Q~(t, x), and we do not require that fo satisfies a condition (Li) with respect to f. Let N be a constant such that the subclass K~ of all (x, u) with l[ x, u] = N is not empty. Then C[ x, u] has an absolute minimum in K~.

Proof. For a proof of (11.4.iv) we introduce the additional state variable Xo with dxo/dt = fo(t,x,u), XO(t 1) = 0, so that l[x,u] = g(e[x]) + XO(t2). Let x = (XO, x), J = (fo,f). We only sketch a proof of the various parts.

For part (a) let i = inf[I[ x, u] I (x, u) E QD]. For a minimizing sequence Xk, Uk we have e[ Xk] E B, g(e[ Xk]) + X~(t2k) -+ i, and for '1k(t) = H(t, xk(t), Uk(t)), ~k(t) = dXk/dt = J(t, Xk(t), Uk(t)), ~ = (e~, ... ,W, we have

('1it), ~k(t)) E QH(t, xk(t)), t E [tlk' t2k] (a. e.),

Srt2k Srt2k ° ° '1k(t) dt ::;; D, ek(t) dt = Xk(t2k), tlk tlk

k = 1,2, ....

Because of the growth properties of H with respect to J = (fo,f) we derive that the functions ~k(t) are equiabsolutely integrable, and because A is bounded there are an AC (n + 1)-vector function x(t), t1 ::;; t::;; t 2 , an Lintegrable (n + 1)-vector function ~(t) = (~O, ~), and a subsequence, say still [k], such that Xk -+ x in the p-metric, ~k -+ ~ weakly in L 1. Finally, we can apply the same process as for (10.7.i) where now the needed bounds below for '1k(t) are provided by the growth properties of H as usual. Then there is an L-integrable scalar function '1(t) such that ('1(t), !(t)) E QH(t, x(t)),


tE[t1,t2] (a.e.), and J:~Hdt~D. Hence, e[xk]~e[x],x~(t2k)~XO(t2)' J:~ ~O(t)dt = XO(t 2), and g(e[x]) + X°(t2) ~ i because of the lower semicontinuity of g. By the implicit function theorem we derive now the existence of a measurable function u(t) with u(t) E U(t, x(t)), such that

l1(t) 2:: H(t, x(t), u(t)), ~O(t) 2:: fo(t, x(t), u(t)),

~(t) = f(t, x(t), u(t)), t E [t 1, t2] (a.e.).

Thus, f(t, x(t), u(t)) is L-integrable as being equal to ~(t), H(t, x(t), u(t)) is L-integrable as being between l1(t) above and the L-integrable functions below which are provided by the growth conditions, and fo(t, x(t), u(t)) is L-integrable as being between ~O(t) above and the L-integrable functions below which are provided by the properties (LJ Finally, C[ x, u] = J:~ H dt ~ D, and

I[x,u] = g(e[x]) + ft 2 fo(t, x(t),u(t)) dt ~ g(e[x]) + X°(t2) ~ i. Jtl Now (x,u)EoOn, hence I[x,u] 2:: i, and by comparison I[x,u] = i. Part (a) is proved.

For part (b), let j = inf[ C[ x, u] I (x, u) E oO~], and let [Xk' Uk] be a minimizing sequence, that is, g(e[ xk]) + X°(t2k) ~ N, C[ Xb Uk] ~ j. The proof is now the same as before, and for the limit element x and ~ = dx/dt we have g(e[x]) + XO(t2) ~ N, and finally I[x,u] ~ N, C[x,u] ~j. Thus (x,u) E oO~, C[ x, u] 2:: j, and finally C[ x, u] = j. Part (b) is proved.

For part (c), we have g(e[xk]) + Xk(t2k) = N, and now Xk ~ x, e[xk] ~ e[ x] imply that g(e[ x]) + X°(t2) = N since 9 is continuous on B. Because of the definition of the sets Q:J.(t, x), by the implicit function theorem we derive now the existence of a measurable function u(t) with u(t) E U(t, x(t)) such that

l1(t) 2:: H(t, x(t), u(t)), ~0(t) = fo(t, x(t), u(t)),

~(t) = f(t,x(t),u(t)), t E [t 1,t2] (a.e.).

Thus, both fo(t, x(t), u(t)) and f(t, x(t), u(t)) are L-integrable because ~(t) has this property, and H(t, x(t), u(t)) is L-integrable because of the growth properties. Finally, C[x, u] = J:~ H dt ~ j, and I[ x, u] = g(e[ x]) + J:~ fo dt = g(e[x]) + X°(t2) = N. Part (c) is proved and so is Theorem (11.4.iv). D

For fo = 0, the problem (11.4.1-2) with comparison functional C[ x, u] required to satisfy C[ x, u] ~ M reduces to a Mayer problem with a comparison functional C[ x, u], and we have the following simpler statement, where the sets Q(t, x) now replace the sets Q(t, x).

1l.4.v (AN EXISTENCE THEOREM FUR MAYER PROBLEMS WITH A COMPARISON

FUNCTIONAL). Let A, H, f satisfy condition (C') or (C'*). Let A be bounded, B closed, and glower semicontinuous on B. Assume that H satisfies one of the growth conditions (gl'), (g2'), (g3') with respect to f. Assume that for almost all Y E A o the sets QH(t, x) are convex and have property (K) with respect to


x on the closed set A(T). Let D be a nonempty class of admissible pairs x, u for which in addition C[x, u] ~ M. Let D be row-closed with respect to H, f. Then the functional l[ x, u] = g( e[ x]) has an absolute minimum in D.

C. Existence Theorems for Problems of Optimal Control with Differential System Linear in u

We consider now functionals of the same type (11.4.1) with systems of differential equations which are linear in u. In other words we consider control problems of the form

(11.4.6)

lEx, u] = g(t1,X(t1), t2,X(t2)) + f l2 fo(t, x(t),u(t)) dt, JII

x'(t) = Ao(t, x(t)) + B(t, x(t) )u(t),

x AC, u measurable, fo(·, x(·), u(·)) E L 1,

(t,x(t)) E A = [to, T] x Rn, (tt.X(t1),t2,X(t2)) E B,

x(t) = (xl, ... , xn), u(t) = (ul, ... , um), u(t) E U = Rm,

where Ao is an n x 1 matrix and B(t, x) is an n x m matrix. For such problem, any pair of functions x(t), u(t), t1 ::; t::; t2, X AC, u

measurable, satisfying the above relations, is said to be admissible. If D = {x, u} is any class of such pairs, we denote as usual by Dx the class of all trajectories x, or Dx = [x I (x, u) ED], and also we denote by Du the class of all control functions, or Du = [u I (x, u) E D].

Obviously, all statements of Section 11.4 holds also in the present situation and we do not repeat them.

We shall consider first the case in which fo(t, x, u) is convex in u for all (t, x), and satisfies one of the conditions (Li) of Section 10.8A, i = 1, 2, 3,4.

Alternatively, we shall consider the case in which fo(t, x, u) satisfies one of the growth conditions (yl), (y2), (y3) with respect to u of Section 11.1.

In either case it is convenient to assume a topology for the control functions u, say Lp for some p ;?: 1. Thus we shall consider classes D = {x, u} of admissible pairs x, u with x AC and u E Lp.

In this situation we shall say that Q is r~w-complete provided: If (Xk' Uk) E

D, k = 1,2, ... , is a sequence of pairs in D, and X k --+ x in the p-metric, x AC, xl. --+ x' weakly in L 1, Uk --+ u weakly in L p , u E Lp , and the pair (x, u) is admissible, then (x, u) E D. The class of all admissible pairs is of course r~w -complete.

1l.4.vi (AN EXISTENCE THEOREM WITH A TOPOLOGY ON THE CoNTROL

FUNCTIONS). Let A be a subset of [to, T] x Rn, U = Rm, B closed, and 9 lower semicontinuous and bounded below in B. Let fo(t, x, u), (t, x, u) E A x Rm,

be a function satisfying either (C') or (C'*), satisfying one of the conditions (L i), and convex in u for every (t,x). Let Ao(t, x), B(t,x) be matrices of the types n x nand n x m respectively, and satisfying either (CL) or (CL *) of


Section 1O.8B. Assume that for some functions </>, t/I ~ 0, </> E L 1, t/I E Lq ,

p,q> 1, lip + 1/q = 1 (or p = 1, q = (0), and constants c, C we have

IAo(t,x)1 ~ </>(t) + clxl, IB(t,x)1 ~ t/I(t) + Cjxl, (t,x) E A.

Assume that the class Q of admissible pairs x, u is nonempty and row -closed, that each trajectory x E Qx has some point (t*,x(t*)) on a given compact set P, and that Qu is known to be relatively sequentially weakly compact in Lp. Then the functional I[x, u] in (11.4.6) has an absolute minimum in Q.

Proof. Let i = inf[ I[ x, u], (x, u) E Q], - 00 ~ i < + 00. Note that g is bounded below by hypothesis, but at present we have no bound below for the integral in (11.4.6). Let xk, Uk' k = 1, 2, ... , be a minimizing sequence, that is, (Xk' Uk) E Q, I[ xk, Uk] -+ i. Thus, the elements Uk belong to Qu, hence, there are a function u(t), t1 ~ t ~ t2, U E L p , and a subsequence, say still [k], such that Uk -+ U weakly in Lp. Then, Ilukllp is bounded, say Ilukllp ~ J1.. By the conditions (Li) we conclude now that the integral J:~~ fo(t, xk(t), uk(t)) dt is bounded below. Then I[ xk, Uk] is also bounded below, and i is finite. We have

x~(t) = Ao(t, xk(t)) + B(t, xk(t) )uk(t), t E [tIk' t2k] (a.e.), k = 1, 2, ....

By (1O.8.iv) there are functions x(t), u(t), t1 ~ t ~ t2, X AC, u E Lp , and a subsequence, say still [k], such that X k -+ x in the p-metric, Uk -+ U weakly in L p , x~ -+ x' weakly in L 1 , and

x'(t) = Ao(t, x(t)) + B(t, x(t) )u(t),

By (1O.8.i) we have now I[ x, u] ~ lim inf I[ xk, Uk], or I[ x] ~ i. By the closure property of Q we derive that (x, u) E Q, hence I[ x, u] ~ i, and finally I[x, u] = i.

D

1l.4.vii. The same as (11.4.vi) where now we assume that fo(t, x, u) satisfies one of the growth conditions (y1), (y2), (y3) with respect to u, and is convex in u for every (t, x). Assume that Ao(t, x) and B(t, x) are as in (11.4.vi) with p = 1. For the class Q of admissible pairs x, u, u ELI' we assume only the Q is nonempty and r~w-closed, and that each trajectory x E Qx has some point (t*, x(t*)) on a given compact set P. Then the functional I[ x, u] in (11.4.6) has an absolute minimum in Q.

Proof. The proof is analogous to the one for (11.4.vi), but here we derive from Section 10.4 that the sequence [Uk] is equiabsolutely integrable. Then there is a subsequence, say still [k], such that tlk -+ t1, t2k -+ t2, to ~ tl ~ t2 ~ T, and by Dunford-Pettis there is an Lcintegrable function u(t), tl ~ t ~ t2, and a further subsequence, say still [k], such that Uk -+ U weakly in L 1• Directly by the growth conditions we know that there is an L 1-integrable function t/I in [to, T] such that fo(t, x, u) ~ -t/l(t); hence I[x,u] is bounded below and i is finite. By (10.8.iv), case p = 1, we know that there are an AC function x(t), tl ~ t ~ t2, and a further subsequence, say still [k], such that


xk -+ X in the p-metric, Uk -+ U weakly in L l, x~ -+ x' weakly in L to and x'(t) =

A(t, x(t» + B(t, x(t))u(t), t E [t1' tz] (a.e.). By (10.8.i) we have now I[ x, u] S lim inf I[ Xk, Uk], or I[ X, u] s i. By the closure property r~w for p = 1 we conclude that (x, u) E Q, hence I[ x, u 1 2: i, and finally I[ x, u] = i. 0

As an important particular case we consider now quadratic integrals with linear differential equations:

I[ x, u] = r 1 (t2 [x*(t)P(t)x(t) + u*(t)R(t)u(t)] dt, Jtl (11.4.7) x'(t) = F(t)x(t) + G(t)u(t), t E [tl> tz], u(t) E U = Rm,

A = [tQ' T] x Rn, (ttoX(tl), tz,x(tz) E B,

where B is a closed set, and P, R, F, G are n x n, m x m, n x n, n x m matrices respectively, p* = P, R* = R, all with say continuous entries in [to, TJ. We assume that Q is the class of all admissible pairs x, u with x AC and u E L z, and that Q is nonempty.

If we assume that P is positive semidefinite for all t, and R is positive definite for all t, in the sense that there is some constant A > ° such that u* R(t)u 2: Alulz for all t E [to, T] and all u E Rm, then

fo(t,x,u) = x*Px + u*Ru 2: Alulz.

In other words, fo satisfies growth condition (yl) with respect to u, fo is convex in u, and (11.4.vii) applies. The integral I[x,u] in (11.4.7) has an absolute minimum in Q.

If we only know that R(t) is semidefinite positive but Q = {x, u} is the class of all admissible pairs with J:~ lulZ dt for some D > 0, then the class Q

is relatively weakly compact in L 2 , (11.4.vi) applies, and I[ x, u] has an absolute minimum in Q. For instance, for R(t) == 0, still lEx, u] has an absolute minimum in Q.

If U is any compact and convex set of Rm, none of the specific conditions above is needed, and I[ x, u] has an absolute minimum in Q because of the theorems of Chapter 9.

Thus, the following problems with n = 1, m = 1, x(l) undetermined,

I = SOl (XZ + uZ)dt, x' = u, U E R, x(O) = Xo,

I = S: X Z dt, x' = u, lui s 1, x(O) = Xo,

I = S: X Z dt, x' = u, U E R, S: u2 dt s 1, x{O) = Xo,

have an absolute minimum. Instead, the problem

(11.4.8) I = S: XZ dt, x' = U, U E R, x(O) = Xo,

has obviously no absolute minimum if Xo -# 0, while for Xo = ° it has the optimal solution x(t) == 0.


Remark 5. We consider here in some detail problem (11.4.7) with t1 , t2 fixed, x(t 1) = Xo, R(t) only positive semidefinite (possibly identically zero) and no weak Lz-compactness on Qu • The following operational approach is of interest. Let V = [t1' t2J.

If <P(t) denote any fundamental n x n matrix of the linear differential system dy/dt = F(t)y, we take cjJ(t, T) = <P(t)tP- 1(T). Then the differential equation x' = F(t)x + G(t)u(t) with x(t1 ) = X o has the unique solution

(11.4.9) x(t) = cjJ(t, t1)xo + r' cjJ(t, T)G(T)U(T)dT = SXo + Tu, J"

where Sand T are linear operators S:Rn ---+ (L2(v))n, T:(L2(v))m ---+ (L2(V))". Then

T*v = G*(t) r cjJ*(T, t)v(T)dT, T*:(L2(V))" ---+ (L2(V)t,

is the dual operator of T in the sense that, if (x, y) = S:~ x* y dt denotes the usual inner product in (L2(v))n and (Lz{V)t, then (Tu, y) = (u, T*y) for all u E (L 2(v)t and y E (L 2(V) In. Indeed,

(Tu,y) = r (s.: cjJ(t,T)G(T)U(T)dT)* y(t)dt

= r (1" U*(T)G*(T)<P*(t, T)y(t) dt) dT

= r'2 U*(T)G*(T) dT r'2 cjJ*(t, T)y(t) dt = (u, T*y). Jtl Jr: By substituting (11.4.9) in the expression of I[ x, u] we have the new functional

J[u] = 2- 1(u,Au) + (u, w) + jo,

A = T*PT+ R, w = T*PSxo,

We have obtained a functional depending on u alone, u E (L2(v)t, and no other parameter or constraint. It is easy to see that A: (L2(V) t ---+ (L2(V) t is selfadjoint.

ll.4.viii. The quadratic functional J[u] has an absolute minimum in (L2(v))m if and only if (a) A is positive semidefinite, and (b) w is in the range of A, that is, w = Au for some u.

Proof. Since A is linear, it is easy to see that for all u, h E (L2(v))m we have

J[u + h] - J[u] = (Au + w, h) + (Ah,h).

If J has an absolute minimum at u, then J[u + h] - J[u] ~ ° for all h. Hence, by the usual argument we must have (Au + w, h) = ° for all h, and then Au + w = 0, or w = - Au = A( - u), and also (Ah, h) ~ ° for all h. That is, w is in the range of A, and A is positive semidefinite. The sufficiency is obvious. D

If we know that the linear map A is bounded and coercive, that is, IIAulb ~ Cllulb (Au, u) ~ cllull~ for some constants c, C > ° and all u E (L 2(V)t, then A is onto (and has an inverse A -1 which is also bounded and coercive) and both (a) and (b) are certainly satisfied.

It is interesting to apply (11.4.viii) to the problem (11.4.8). For this case n = m = 1, P = 1,R = O,F = O,G = 1, Tu = S~ u(T)dT, T*x = Sl x(T)dT, w = T*xo, A = T*T,and (Au, u) = (T*Tu, u) = (Tu, Tu) ~ 0, that is, A is certainly positive semidefinite. Now


w = Au means here

that is

Then by differentiation, S~ u(-r:)d-r: = xo, u(s) = 0, Xo = 0.

11.5 Elimination of the Hypothesis that A Is Bounded in Theorems (11.4.i-v)

The hypothesis that A is bounded in the theorems (11.4.i-v) can be easily removed as we have done in Section 11.2 and, before, in Chapter 9. As already stated, all we have to do is to guarantee that a minimizing sequence can be contained in some compact subset Ao of A.

1. If the set A is not bounded, but A is contained in a fixed slab [to::;; t ::;; To, X ERn] of Rn + 1, then Theorem (11.4.i) is still valid if we know for instance that (h'd 9 is bounded below on B, say 9 ~ - M 1, and (C~) Q is a given nonempty row-closed class of AC trajectories x, each of which has at least one point (t*, x(t*)) on a given compact subset P of A, and where t* may depend on the trajectory. The proof is the same as in Section 11.3.

2. If A is neither bounded nor contained in any slab as above, then Theorem (11.4.i) is still valid if we know for instance that (h~) and (C'l) hold, and in addition that (ez) there are constants fl > 0, Ro ~ 0, such that fort, x, u) ~ fl > ° for all (t, x, u) E M with t ~ Ro. The proof is the same as in Section 11.3.

Concerning Theorem (11.4.ii), if we assume that condition (gl '), or (g2'), or (g3') hold, then the same considerations hold as for Theorem (11.4.i). If we do not want to invoke conditions (g'), then the following holds. If A is not compact but closed and contained in a slab [to::;; t::;; T, x ERn], then (11.4.ii) is still valid, provided we know that (h~), (Cd hold, and for instance (C~) there are constants c > 0, R ~ 0, and a locally integrable function t/J(t) ~ 0, t E R, such that fort, x, u) ~ - t/J(t) + clf(t, x, u)1 for all (t, x, u) E M with Ixl ~ R.

If A not compact, nor contained in any slab as above, then (l1.4.ii) is still valid, provided we know, for instance, that (h'l), (C~), (C~), (C~) hold.

For Theorem (11.4.iii) the same considerations hold as for (11.4.ii). For Theorem (11.4.iv) it is enough that we transfer to H the assumptions we have

made for fo in (l1.4.i). For (11.4.v) the same considerations hold as for (11.1.iii), that is, we may require the additional assumption (h z) of Section 11.3 for A contained in a slab, or (h3) for A unbounded and contained in no slab.

Alternatively, instead of condition (C~) we may assume that

C'. There is a constant C ~ ° such that xlfl + ... + x"J" ::;; qlxl 2 + 1) for all (t, x, u) E

M.

A particular case of (C') is of course that If(t, x, u)1 ::;; qlxl + 1) for all (t, x, u) E M.

11.6 Examples 397

Instead of (C') we may consider the more general condition (CN ) There is a scalar function V(t,x) of class C1 in A and a positive constant C

such that gradx V(t, x) . f(t, x, u) + 8V/8t :5; CV(t, x)

for all (t, x, u) E M, and, moreover, for every a, b, IX real, the set [x I V(t, x) :5; IX for some (t, x) E A, a :5; t:5; b] is compact.

The argument is the same as in Sections 9.4, 9.5. The reader may also extend to the present situation the remaining remarks in Sections 9.4, 9.5.

11.6 Examples

1. l[x,u] = J:f(1 + tu + u2)dt, with system x' = u, U E U = R, t1 = 0, n = m = 1, x(O) = 1,0:5;t2 :5; 1, (t2,X(t2)) on the locus r:t=(1+x2)-1, -oo<x<+oo (a Lagrange problem of the calculus of variations written as a problem of control). Take for A the slab [0 :5; t :5; 1, x E R], a closed set; then M = A x R is also closed, and so is B = {O} x {1} x r. The sets Q(t, x) are all closed and convex in R2. Also fo ~ (Iul- 1)2, and thus the growth condition (g1') of Section 11.4 holds with ifJ(C) = (C - 1)2. Here A is a closed slab, and condition (C'l) holds, since (t1,xd = (0,1) is fixed.

2. l[x,u] = g(1 + t)u2dt with system x' = U, UE U = R, n = 1, m = 1, t1 = 0, t2 = 1, x(O) = 1, x(1) = O. Here we can take A = [0,1] x R closed, M = A x R closed, B= {O} x {1} x {1} x {O} compact. The sets Q(t,x) are all convex and closed. Here fo ~ u2, so growth condition (g1') holds with ifJ(C) = C2, and condition (C~) holds, since B is compact. The functional l[ x, u] has an absolute minimum under the constraints.

3. l[x,u] = JA t·u2dt, 0:5; IX < 1, with system x' = U, UE U = R, t1 = 0, t2 = 1, x(O) = 1, x(1) = 0 (a Lagrange problem ofthe calculus of variations written as a problem of control). Here we take A = [0 :5; t :5; 1, x E R] closed, n = 1, m = 1, M = A x R also closed, B = {O} x {1} x {1} x {O} compact. Growth condition (g2') of Section 11.4 holds. To prove this, note that, for any e > 0, the function ",,(t) = e- 1C· is L-integrable in [0,1]' Now, either lul:5; e- 1t-·, and then If I = IUI:5; e- 1C·:5; "',(t) + efo; or lui> e-lt-·, and then 1 < et"lul and If I = luJ:5; et·u2 :5; "')t) + efo. In any case If I :5; I/I.{t) + efo, and condition (g2') holds. The sets Q(t) are here Q(t) = [(ZO, z)lzO ~ t·z2, - 00 < z < + 00], 0 :5; t :5; 1, and they are all closed and convex. Because of t· being positive everywhere in (0, 1], we see that for all Y, 0 < Y :5; 1, the sets Q(Y, x) have growth property (ifJ) of Section 10.5 with respect to x only; hence, the same sets have property (K), as well as property (Q), with respect to x only (see (lO.5.iii) with g = 0, or (G') of Section 10.5). Here, A is a closed slab, and condition (C~) is satisfied, since the initial point (0,1) is fixed. The functional has an absolute minimum under the constraints. (See also Exercise 5 in Section 2.20C).

4. lex, u] = J\f [(1 + Xl)U2 + 1] dt with system x' = (1 + x)u, u E U = R, m = n = 1, t1 = 0, t2 ~ 0, x(O) = 0, (t2,X2) on the locus r:tx = 1, t > O. Here we can take A = [0:5; t < + 00, X E R] closed, M = A x R closed, B = {O} x {O} x r also closed. A is not contained in any slab, but condition (Cz) is satisfied, since fo ~ 1. Condition (C~) is satisfied, since (t1,X1) = (0,0) is fixed. Finally, (1 + X)2 :5; 2 + 2X2, fo = (1 + X2)U1 + 1 ~ 2- 1IfI 2, and the growth condition (g1') of Section 11.4 holds with ifJ(C) = 2- 1,2.

The functional has an absolute minimum under the constraints. 5. l[x,u] = g(1 + u2)'dt, e > t, with system x' = u, U E U = R, tl = 0, 0:5; t2 :5; 1,

x(O) = 0, (t2,X2) on the segment r = [0:5; t:5; 1, x = 1], A = [0:5; t:5; 1,0:5; x:5; 1] (a


problem of the calculus of variations written as a problem of control). Here A is compact, M = A x R is closed, B = {O} x {O} x r is compact, the fixed set Q is closed and convex. The growth condition (gl') is satisfied with cp(() = e'. The functional has an absolute minimum under the constraints. (This is not the case for [; = t. See Example 2 in Section 11.7 below).

6. I[ x, y, u] = Sb (tzxZyZ + XZ + yZ)lul dt with system x' = (x + y)u, y' = u, U E U =

R, x(O) = 1, y(l) = 0, in the class Q of all admissible systems x(t), y(t), u(t), 0:::;; t:::;; 1 with Sb(x'Z + y,z)dt:::;; 4. Here n = 2, m = 1, A = [0,1] X RZ is closed, M = A x R is also closed, x(I), y(O) are undetermined, B closed. Also Hly'l dt :::;; m (x'Z + y'z) dt)l/Z :::;; 2, so that the point (x(O), y(O» is certainly in the compact set P = [(x, y) I x = 1, IYI :::;; 2], and condition (C~) holds. Here Ifl-+ + 00 as u -+ + 00; hence, the sets Q(t, x, y) certainly have property (K) by (8.5.vii). Note that for no t do the convex sets Q(t, x, y) = [(tZxZyZ + XZ + y2)lul, ZI = (x + y)u, ZZ = u, U E R] satisfy property (Q) with respect to (x, y). However, the functional has an absolute minimum under the constraints by (11.4.ii) and conditions (Cl') and (K).

7. I[x, u] = Sb fo(t, x, u)dt with differential system x' = f(t, x, u), x(O) = Xl, x(l) = Xz, u E U = R, with fo(t, x, u) = ao(t, x)lul + bo(t, x), f(t, x, u) = a(t, x)u + b(t, x), ao(t, x) z 0, a(t, x) =I- 0, (t, x) E A = [0,1] x R, a, b, ao, bo continuous functions on A, Q the class of all admissible pairs x(t), u(t), 0 :::;; t :::;; 1, with S6 X'2 dt :::;; C for C > (xz - Xl)z. Here m = n = 1, A is closed, M = A x R is closed, B = {O} X {Xl} x {I} x {xz} is compact, and condition (C1) is satisfied. Note that fo(t, x, u) z bo{t, x) and bo is bounded below in each compact part of A. Thus, a condition (L~) holds in each compact part of A. Also, Ifl-+ + 00 as lul-+ + 00 uniformly in each compact part of A, and thus the sets Q(t, x) have property (K) in (t, x). Note that for no t do the same convex sets Q(t, x) satisfy property (Q) with respect to x. However, the functional I[ x, u] has an absolute minimum under the constraints by (11.4.ii).

8. I[ x, y, u, v] = S6 fo(t, x, y, u, v) dt with differential system x' = fl, y' = fz, (u, v) E

U = RZ, x(O) = y(O) = 1, Xz, Y2 undetermined, tz = 1, in the class Q of all admissible systems x(t), y(t), u(t), v(t), 0 :::;; t :::;; 1, with H (x'Z + y'z) dt :::;; 4. Here fo(t, x, y, u, v) = ao(t, x, y)lul + bo(t, x, y)lvl, f(t, x, y, u, v) = (fl, fz), /; = Gi(t, x, y)u + bi(t, x, y)v, i = 1,2, and there are positive constants c, Co, d, D such that ao z Co, bo z Co, D(uZ + VZ)l/Z z Iflz d(uZ + VZ)I/Z. Then Ifl-+ 00 as (u z + VZ)I/Z -+ 00, and by (8.5.vii) the sets Q(t, x, y) have property (K). Also, for (u z + VZ)I/Z Z R at least one of lui and Ivl is z R2 ~ liZ and we havefo = aolul + bolvlz C02~1/zR z rl/zcoD~ llfl. Conditions (Cd, (C3) hold. Note that for no t do the sets Q(t, x, y) satisfy property (Q) with respect to (x, y) (unless ai' b1, az, b2 are constant with respect to (x, y)). However, the functional I[x, y, u, v] has an absolute minimum under the constraints by (11.4.ii).

11.7 Counterexamples

1. I[x, u] = S6 tuZ dt with system x' = u, n = m = 1, x(O) = 1, x(l) = 0, u E U = R, A = [0 :::;; t :::;; 1, x E R]. Here A is compact, M = A x R is closed, B = {O} x {1} x {1} x {O} is compact, the sets Q(t,x) = [(zO,u)lzO z tuZ, uER] are all closed and convex, and the growth condition (gl') holds at every 0 < t :::;; 1, but does not hold at t = O. Here I z 0; hence i z O. On the other hand, for the admissible pairs Xk(t), Uk(t), 0 :::;; t :::;; 1, k = 2, 3, ... , defined by Xk(t) = 1, uk(t) = 0 for 0:::;; t < k~l, xk(t) = -(log t)/(log k), Uk(t) = - t~ l/log k for k~ 1 :::;; t :::;; 1, we have I[xk' Uk] = (log k)~ 1. Hence I[xk, Uk] -+ 0

Bibliographical Notes 399

as k·--> 00, and thus i ~ O. Finally, i = 0, and I cannot have the value zero, since that would imply tuZ = 0 a.e. in [0, 1], u = 0 a.e. in [0, 1], x a constant. The functional I[ x, u] above has no absolute minimum under the constraints. (A control version of Section 1.5, no. 4).

2. I[x,u] = S\f (1 + UZ)l/Z dt, with system x' = U, UE U = R, t1 = 0,0 ~tz ~ 1, x(O) = 0, (tz,x z) on the segment r = [0 ~ t ~ 1, x = 1], A = [0 ~ t ~ 1, 0 ~ x ~ 1] (the problem ofa path C in nonparametricform, C:x = x(t),O ~ t ~ tz, x AC in [O,t z], of minimum length joining (0, 0) to n. Obviously, the problem has no absolute minimum. The growth condition (gl') is nowhere satisfied. (A control version of a remark in Section 1.5, no. 3).

3. I[ x, u] = gn Ix - 2tl dt with differential system x' = Ix - 2tlu, u E U = [u I u ~ 1], x(O) = 0, x(2n) = 4n, in the class Q of all admissible pairs x(t), u(t), 0 ~ t ~ 2n, with gn x'z dt ~ lOn. Here n = m = 1, and we can take A = [0,2n] x R closed, M = A x R closed, B = {O} x {O} x {2n} x {4n} compact. Also fo ~ 0; thus condition (L~) is satisfied with l/!(t) = O. Let i ~ 0 be the infimum of I[ x, u] in Q. Note that for the elements Xk(t) = 2t - k- 1 sin kt, Uk(t) = k(2 - cos kt)lsin ktl- I, 0 ~ t ~ 2n, k = 1,2, ... , we have x~(t) = 2 - cos kt = f(t, x(t), Uk(t», Uk(t) ~ 1, measurable and finite almost everywhere in [0, 2n]. Also, gn xiz dt = 9n, and thus (Xk, Uk) E Q for all k. Finally, I[ xk, Uk] = 4k- 1 --> 0 as k --> 00, and hence i = O. The functional does not take the value zero, since this would require x(t) = 2t, x'(t) = 2 = Ix(t) - 2tlu(t), t E [0, 2n] (a. e.), and this is impossible. Condition (C'l) is satisfied, since both initial and terminal points are fixed. The functional has no absolute minimum. Note that here the sets Q and Q are all closed and convex. Indeed, if x = 2t, then Q(t, x) = {O}, Q(t, x) = [(ZO, z)lzO ~ 0, z = 0] = R+; if Ix - 2tl = a > 0, then Q(t,x) = [z = au, U ~ 1], Q(t,x) = [(zO,z)lzO ~ a, z = au, U ~ 1]. The sets Q(t,x) do not have properties (K) and (Q) at x = 2t.

Bibliographical Notes The treatment of the existence theorem of this chapter, and connected lower semicontinuity and lower closure theorems of Sections 10.7-8, reflects a number of remarks which have been made in the last years.

In this connection and for the use of mere uniform convergence of the trajectories the upper semicontinuity properties (K) and (Q) of the relevant sets Q(t, x), Q(t, x) with respect to (t, x), (that is, the same properties for the set valued maps, or multifunctions (t, x) --> Q(t, x) and similar ones), were most natural (L. Cesari [6], 1966) (Section 8.8), and lower closure theorems take the place of lower semicontinuity theorems.

There was then the remark that, in connection with weak convergence of the derivatives, or weak convergence in HI.I, the same properties with respect to x only suffice in the proof of lower closure theorems, that is, the same properties for the set valued maps x --> Q(7, x) and analogous ones for almost every 7 (L. Cesari [13], M. F. Bidaut [1], L. D. Berkovitz [1], independently around 1975) (Section (10.7».

Then there was the remark that the property (K) requirement really suffices, since weak convergence implies a growth property which in turn, by a remark of Cesari [7] (Section 10.5), implies property (Q) for certain auxiliary sets, or maps x --> Q*(7, x), with essentially no further change in the argument of either lower closure or lower semicontinuity theorems (A. D. Ioffe [1], 1977, I. Ekeland and R. Temam [I], L. Cesari and M. B. Suryanarayana [7, 8, 9]). It appears that property (Q) in some weak form,


or equivalent properties, are needed in the proof of the underlying lower closure or lower semicontinuity theorems, whether we name explicitly these properties or not.

Then there was the remark that the terminology is somewhat simplified by a consistent use of the "Lagrangian", or extended function T(t,x,x') (Section 1.12) for which the value + 00 is allowed, and which is related to the sets Q by the simple relation epiz T(t, x, z) = cl Q(t, x) (R. T. Rockafellar [6] in connection with his approach in terms of Convex Analysis and duality) (We shall cover these ideas in Chapter 17). However, in terms of the "Lagrangian" T the proof of the lower semicontinuity does not change much, and the same auxiliary sets Q* are needed with property (Q) in some form.

Actually, property (K) with respect to x of the sets Q(t, x) is equivalent to the lower semicontinuity with respect to (x, z) of the extended function T(t, x, z), and to the closure of the set epi T(t,·,·). Analogously, property (Q) of the sets Q can be equivalently expressed in terms of "seminormality" properties (in the sense of Tonelli and McShane of the function T (L. Cesari [8, 10, 11]) (Section 17.5), as well as in terms of Convex Analysis (G. S. Goodman [1]) (Section 17.6).

In this Chapter we have taken full advantage of all these steps. Indeed we proved the lower closure theorem (1O.7.i) with full use of the auxiliary sets Q*, we immediately derived the corresponding lower semicontinuity theorem (1O.8.i), which is actually equivalent to (1O.7.i), and we proved first the existence theorems (11.1.i) and (11.1.ii) in terms of the Lagrangian T. Then in Section 11.4 we derived the existence theorems for optimal control from the previous ones, that is, by actual use of their corresponding Lagrangian T, but the hypotheses are in terms of original problems of optimal control, or criteria are given.

In general, in applications, it appears that in problems where the Lagrangian is given, the terminology in terms of Lagrangian is more suitable (Section 11.3); in problems of optimal control where certain functions 10, 1 are given containing the control parameters (Section 11.4), the terminology in terms of these functions is more suitable, or the one in terms of sets immediately defined from them, better than in terms of the Lagrangian function T(t,x,x') which seldom can be written explicitly. We tried to show in this book the essential equivalence of the different terminologies.

Theorem (11.1.i) can be thought of as a present day form and far reaching extension of the fundamental 1914 theorem of Tonelli [4,6, and I, vol. 2, p. 282]' We proved it first by relying on the lower semicontinuity theorem (1O.8.i) and therefore on the equivalent lower closure theorem (1O.7.i). We noted that under the growth conditions of (11.1.i) the sets Q already have property (Q) with respect to x, hence for this theorem much simpler version of (1O.7.i) and (10.8.i) suffice (Cf. second proof of (1O.7.i».

We have completed Section 11.1 with existence theorems for problems with a comparison functional, for problems with an integrand linear in the derivatives, for isoperimetric problems in which one of the integrals is linear, for problems with optimal generalized solutions. Theorem (11.1.vi) can be traced in E. J. McShane [18]; Remark (11.1.vii) and the Theorem (11.l.ix) for n = 1 are in L. Tonelli [I].

The problems of optimal control with unbounded controls are covered in Section 11.4, and their proof is given in terms of the related function T(t, x, z). Most of those theorems had been proved already in terms of orientor fields anyhow, and the proofs are essentially the same. Theorems (11.4.i) and (11.4.ii) correspond essentially to the theorems (11.1.i), (11.1.ii) respectively. The existence theorem (11.4.iv) for problems with a comparison functional and for isoperimetric problems appears to be somehow more elaborated and comprehensive than in other presentations. The existence theorem (11.4.vi) for linear integrals based on the topology of weak convergence in L p , p> 1,

Bibliographical Notes 401

includes a number of previous statements, in particular the result ofM. Vidyasagar [1], who assumed a uniform Lipschitz condition with respect to x.

The ideas underlying the present Chapter have been shown to be relevant in other situations. M. B. Suryanarayana [3, 4, 5] has studied problems of optimization with canonic hyperbolic equations and with linear total partial differential equations. T. S. Angell [1, 2, 4] has studied problems of optimal control with functional differential equations, with hereditary equations, and with nonlinear Volterra equations. H. S. Hou [1-4] has studied problems monitored by abstract nonlinear equations, including parabolic partial differential equations. Both T. S. Angell and H. S. Hou use properties (K) and (Q) in proving the existence of solutions (controllability) as well as in proving the existence of optimal solutions. R. F. Baum [1, 3] has studied stochastic control problems, and problems monitored by partial differential equations in Rn with lower dimensional controls. For extensions to problems in Banach spaces cf. L. Cesari [18]. For an existence theory for Pareto problems, that is, problems with functionals having their values in Rn or in Banach spaces, cf. L. Cesari and M. B. Suryanarayana [4, 5, 6, 7]. For further work on Pareto problems see also C. Olech [2, 5], N. O. Dacunha and E. Polak [1], P. L. Yu [1], P. L. Yu and G. Leitmann [1], L. A. Zadeh [1].

For Remark 5 of Section 11.4 we refer to W. F. Powers, B. D. Cheng and E. R. Edge [1] where a further analysis is made for the characterization of the singular solutions and for the elaboration of rapidly convergent methods for the numerical determination of the solutions.

Along the same lines discussed in this Chapter we mention here the extensive work ofE.1. McShane [5-7, 10, 18], C. Olech [8,9], C. Olech and A. Lasota [1, 2], A. Lasota and F. H. Szaframiec [1], E. O. Roxin [1], M. Q. Jacobs [1].

Many more ideas in existence theorems for one dimensional problems will be discussed in the next Chapters 12, 13, 14, 15, 16 in connection with different topologies and different viewpoints.

S. Cinquini [1-6] discussed problems of the calculus of variations for curves and surfaces depending on derivatives of higher order.

Only mention can be made here ofthe extensive work ofe. B. Morrey [I] on multiple integrals, existence and regularity of the solutions of elliptic partial differential equations, and continuous surfaces of finite area.

We have already mentioned that the concept of generalized solutions was introduced by L. C. Young [1] in 1936 in terms of functional analysis. We refer to L. e. Young [1,1-9], W. H. Fleming [1-4], W. H. Fleming and L. C. Young [1-2], E. J. McShane [12, 13, 14, 18] for work on generalized solutions in one and more variables. In particular W. H. Fleming (loc. cit), in the same frame ofreference, developed a theory for solutions of stochastic partial differential equations.

Finally, only mention can be made here of the fundamental work of 1. L. Lions [I] for multidimensional problems in the frame of differential inequalities, covering quadratic functionals for problems monitored by elliptic, parabolic, and hyperbolic linear partial differential equations.

For parametric problems of the calculus of variations on surfaces S in R 3, under sole continuity assumptions and finiteness of the Lebesgue area, L. Cesari [20, 21] discussed the Weierstrass condition as a necessary and also as a sufficient condition for lower semicontinuity ofthe parametric integrals /[S,/o] ofthe calculus of variations with respect to the topology of the Frechet distance (uniform topology). On the basis of these results, and surface area theory, L. Cesari [22] proved the existence of a parametric surface So for which /[S,fo] has a minimum value among all surfaces S of finite


area and spanning a given simple continuous curve in R3 (if any such surface exists). This is the Plateau problem for general parametric integrals ofthe calculus of variations. For such surfaces (merely continuous and of finite Lebesgue area, with no differentiability assumptions), the concept of the integral J[S,fo] was discussed by L. Cesari [19], in the spirit of surface area theory, as a Weierstrass integral. Later the same integral was discussed by L. Cesari [23, 24] in an abstract form, in connection with any quasiadditive vector valued set function (instead of a mere signed area function), showing that the property of quasiadditivity is preserved by the nonlinear parametric integrand fo and that the Weierstrass integral can be defined both as a Burkill type integral, and as a Lebesgue-Stieltjes integral with respect to the area measure defined by the surface and with the classical lacobians replaced by Radon-Nikodym derivatives ofthe relevant set functions. This work has been continued by many authors (G. W. Warner, 1. C. Breckenridge, A. W. 1. Stoddart, L. H. Turner, T. Nishiura, A. Averna, C. Bardaro, M. Boni, P. Brandi, p. Candeloro, C. Gori, P. Pucci, M. Ragni, A. Salvadori, C. Vinti), and will be presented in III. Abstract lower semicontinuity theorems and a great many other properties of such integrals have been proved. In this connection extensive work has been done by E. Silverman on the lower semicontinuity of integrals on kdimensional manifolds in Rft (cf. III).

optimization—theory and applications || existence theorems: weak convergence and growth conditions

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