THE UN IV ER SIT Y OF MI CHI GAN COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Mathematics Technical Report No. 9 SEMINORMALITY AND UPPER SEMICONTINUITY IN OPTIMAL CONTROL Lamberto Cesari ORA Proj tt i02416 submitted for UNITED STATES AIR FORCE AIR FORCE OFFICE OF SCIENTIFIC RESEARCH GRANT NO. AFOSR-69-1662 ARLINGTON, VIRGINIA administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR December 1969

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SEMINORMALITY AND UPPER SEMICONTINUITY IN'OPTIMAL CONTROL* Lamberto Cesari In the present paper we discuss properties of upper semicontinuity of variable convex closed sets in Euclidean spaces, taking into consideration the modification of Kuratowski's concept of upper semicontinuity [4] which we denoted in [la,b] as property (Q). We have used this property in the proof of lower closure theorems in Lagrange and Mayer problems of optimal control. These theorems reduce to well known lower semicontinuity statements for usual free problems of the calculus of variations, Lower closure theorems are used to prove existence theorems for Lagrange and Mayer problems of optimal control [la,b]. The same property (Q) mentioned above was used again in recent studies by J. R. LaPalm [5], A. Lasota and C. Olech [6], C. Olech [8], L. Cesari, T. Nishiura, and J. R. LaPalm [2], and in recent papers by Cesari [lc,d,e,f] concerning existence theorems for Lagrange problems with multiple integrals and partial differential equations. In these papers we requested property (Q) for variable sets Q(x) of the form Q(x) = [(z,z) Iz > f (x,u), z = f(x,u), u e U(x)] C En+l where f and f = (flooff n) are given continuous functions, In the present paper we give criteria (Sections 2, 3, 6, and 8) for property (Q) of the sets Q(x) in addition to those already stated in [la,b]. In particular we show (Sections 3 and 8) that a slight particularization of *Research partially supported by AFOSR research project 62-1662 at The University of Michigan. 1

property (Q) for these sets Q(x) can be expressed in a form which is similar to Tonelli's seminormality condition [9] for free problems of the calculus of variations. Thus, property (Q) of the sets Q(x) is shown here to represent a generalization —-for Lagrange problems-of the well known seminormality condition for free problems. In Sections 4, 5, and 7 we state a number of properties of convex real-valued functions on a convex subset of E related to the n concept of seminormality, and we use these results in Section 8. In Section 6 we prove another criterion for property (Q) of the sets Q(x) when f is linear in u and f is convex and seminormal in u. 0o 1. PROPERTIES (U) AND (Q) OF VARIABLE SETS Let A be a given subset of the x-space E, for every x E A let U(x) be a given subset of the u-space h, and let M be the set of all (x,u) with x E A, u E U(x). Thus, M is the graph of U(x) in the space E x E. For every x E A and b > 0 let N (x) denote the set of all x E A at a distance < 3 from x. For every x E A and S > 0 let U(x;E) denote the union of all U(x) with x E N(x), or U(x;~) = [u e E Ju E U(x), x E N(x)]. We say that the sets U(x) have property (U) at a point x e A if U(x) = nf cl u(x;). (1.1) We say that the sets U(x) have property (Q) at x e A if U(x) = n cl co U(x;). (1.2) Here cl and co denote the closure and the convx hull respectively of the sets under consideration. We say that the sets U(x) have property (U) [(Q)] in A 2

if this property holds at every point x E A. Property (U) is Kuratovski's concept of uppersemicontinuity of sets [43, and was used, for instance, by G. Choquet [3] and E. Michael [7]. Note that in (1.1) and (1.2) the sign c holds trivially, and thus the actual requirements can be written in the form U(x), n cl U(x;F), or u(x) D n& cl co u(x;8), respectively. The following statements are easily proved: (l.i) If U(x) has property (U) at x, then U(x) is closed. (1.ii) If U(x) has property (Q) at x, then U(x) is closed and convex. (1 iii) If A is closed, then U(x) has property (U) in A if and only if M is closed. A number of other statements concerning properties (U) and (Q) have been stated in [la,b], and will not be repeated here. If f(x,u) = (fl,-.'ff), (x,u) E M, is a given vector function, f:M + E, we shall denote by Q(x) C E the set Q(x) = f(x,U(x)), or Q(x) = [z = (zl,..., n Z )xz = f(x,u), u E U(x)]. 20 THE SETS Q(x) AND A FIRST CRITERION FOR PROPERTY (Q) In Lagrange problems of optimal control and the calculus of variations, besides the vector function f(x,u) = (fl,..a,f ), also a scalar function f (x,u) is given, f = M + El. 15

If f(x,u) = (f,f) = (f,fl1,...,f), then we may denote by Q(x) c E+l the set f(x,U(x)), or Q(x) = [(z,z) z = f (x,u), z = f(x,u), u E U(x)]. Also, we shall denote by Q(x) the set Q(x) = [(z0,Z) Iz > f_(Xu), z = f(x,u), u C U(x)] We may say that Q(x) is the "figurative," and that Q(x) is the set of points "above the figurative." Note that for every x C A the set Q(x) is the projection on the z-space E of the set Q(x) c E +1' Thus, if Q(x) is convex, then certainly Q(x) is also convex. We shall say that a function g(x,u) is "of slower growth than f (x,u) as Jul + m uniformly in some subset A of A, provided given e > 0 there is some u = u(,A ) > 0 such that x E A, u E U(x), uj > u implies Ig(x,u) | < ~ f (x,u). 0 (2.i) (a criterion for property (Q) under a growth condition) Given A closed, M closed, f (x,u) and f(x,u) = (fl,) o-,f ) continuous on M, assume that 1 and f are of slower growth than f as Jul + +o uniformly on some neighborhood A of a point x c A. If the set Q(x) is convex, then the sets Q(x) satisfy property (Q) at x. A proof of (2.i) has been given in [lb]. 3. A SECOND CRITERION FOR PROPERTY (Q) Note that if the sets Q(x) satisfy property (Q) at a point x c A then Q(x) nf cl co Q(x;6)

0 This means that, if a point (z,z) belongs to the set nf cl co Q(x;S), then (z0,z) E Q(x) = [(Zz)z0 > f (xu), z = f(x,u), u E U(x)]; hence z eQ() = [z = f(x,u), u E U(x)] In other words, the following property (a) is a necessary condition for the sets Q(x) to have property (Q) at x: (o) If (z,z) E n5 cl co Q(x;E), then z e Q(x). For free problems of the calculus of variations n = m, f = u, or f. u, i = l,...,n, and U(x) = E. For these problems then the sets under conn sideration reduce to Q(x) = E and Q(x) = [(zu) z = f(xu), u E E E C + Q(x) = [(zu) z >f (x,u), u e En] C E+1 Thus, property (a) is trivially satisfied for free problems. We shall now introduce the following "condition' (X), at a point x E A: (X) for every z E Q(x), there is at least one point u e (U(x) with z = f(x,u) and the following property: given E > 0 there are numbers 6 > 0, and r, b = (bl,.l,b ) real, such that (X') f (x,u) > r + j.bjfj(x,u) for all x E N (x) and u e U(x); (x") f (XU) < r +Zbif.(1)+ + For free problems (that is, m = n, f = u, U = E ) the present property (X) reduces to the following one concerning the function f only: 5 0~~~

(Xf) For every u = (u,...,u ) E Em and E > 0 there are numbers 6 > 0 and r, b = (bl,...,bm) real such that (XI) fo(x,u) > r + Z.b.uj for all x E N (x) and all u = (ul,...,um) Em; (xf) f (x,u) < r + E.b.u3 +. f 0 J As we shall see in no. 5 below, this condition (Xf) is the well known weak seminormality condition of the function f at (x,u) for all u E E. (3.i) (a criterion for property (Q) under conditions (a) and (X)) If conditions (a) and (X) hold at the point x E A, then Q(x) is closed and convex, and the sets Q(x) satisfy property (Q) at the point x. Proof. To prove that the sets Q(x) satisfy property (Q) at x (and hence Q(x) is closed and convex), we have only to prove that, if z = (z,z) E n cl co:Q(x;6), then z = (z,z) E Q(x). From condition (a) all we know is that z E Q(x). Hence, there is some u E U(x) such that z = f(x,u), (hence (x,u) C M), and statements (X'), (tX") hold. For every 5 > 0 we have z = (z,z) c cl co Q(x;6), and thus for every -0O want from z = (z,z). Thus, there is a sequence of numbers 6k > 0 and of points zk = (zkz k) C co Q(x;6k) such that Sk 0, zk -* as k+ a. In other words, for every integer k there is a system of points xk c N5 (x), y = 1,..., ~~~~~kk 6

1 = z 0=Z z z k= z7k k k' k k k k = z zoy > f (xYuY)u Z = f(xku), (13.1) k - o kk k kk where y = l,...,v, where Z ranges over all y = l,..,v, and Zk - X, Zk Z o -o zk Z Zk as k -+ oo, 7 = l,...,v. k k By condition (X') there is a neighborhood N (x) of x in A, and numbers r, b = (bl...y b ) real, such that V f (x,u) = f (x,u) - r - bf(x,u) > 0 for all x E N6(x) and u E U(x); (3.2) (x,u) (xu) r bf(x,) - r) <. (3.3) For k sufficiently large, so that Ixy - x| < 6, y = l,,.,v, we have now, from (3.1) and (3.2), k k k k o(xk') > k k[ k k) = Z 2k[r + bz rb~ b. Y Zk = r + b-zk k k k k k As k + oo we obtain z > r + b z; hence, from (.13), -0 z > r + brz r + b f(x,) f(x,u) - f (x,u) > f (x,U) - -o Here ~ > 0 is arbitrary; hence z > f (x,u), while z = f(x,u). This shows that z = (z,z) E Q(x). We have proved that the setsQ(x) satisfy property (Q) at the point x E A. Statement (3.i) is thereby proved.

4. SOME PROPERTIES OF CONVEX FUNCTIONS If U is a given subset of E and F(u), u E U, a real-valued function, then F(u) is said to be convex in u provided U is convex, and ul, u2 E U, 0 < C < 1, implies F(Cau + (1 - c)u2) < aF(ul) + (1 - a) F(u2). The following statements are well known: (4.i) If U is a convex subset of E and F(u), u E U, a given real-valued function, then F(u) is convex if and only if uj E U, j. > O, j = l,...,v, v finite, +. + * + A = 1, u = ju.,u implies F(u ) <.=1 jF(uj). v o j=l o -=1 j (4.ii) If U is a convex subset of E, and F(u), u E U, a given realvalued function, then F(u) is convex if and only if the set Q = [(z,u) z > F(u), u E U] C E is convex. n+l A linear scalar function z(u) = r + blul +... + b u = r + b-u, u E E, n n r, bl,., b real, is said to be a supporting plane of F(u), u E U, at a point u c U. provided F(u) = z(u) and F(u) > z(u) for all u E U. (4.iii) If U is a convex subset of E, and F(u), u c U, a given realvalued convex function, then F(u) has a supporting plane at every interior point u of U. Proof. We know already that the set Q = [(z,u) Jz > F(u), u c U] c E n+1 is convex, and hence there exists some supporting hyperplane V at (z,u), -z = l;. [(zu)Ip~z - c o], O = (l n F(u). If V = [(z,u)|p z + p-u - c _> ], p~, p = (,.,.,p ), c real, then 8

p z + p.u - c = 0 and p z + p.u - c > 0 for all u E U and z > F(u). Let us 0 0 prove that p + O. Indeed, if p = O, then we have p.u - c = 0, p.u - c > 0 for all u E U. If ul e E is a point where p-ul - c > 0, and ~ real, then for u = U(E) = ~U1 + (1 - s)u we have p.u(E) - c > 0 for all 6 > 0, and pu(s) - c < O for all ~ < O, with u(~) + U as ~ + 0. Since u E int U, then both u(s), u(-~) both belong to U for s > 0 sufficiently small, and pu(-~) - c < 0, a contradiction. We have proved that p $ O. Actually, we must have p > 0 since p z + ppu - c > 0 for all z > F(u). Finally, if we take z(u) = (-p-u + c)/, then z(u) = F(u) and F(u) > z (u) for all u E U. Given a set U we denote as usual by int U the subset of its interior points. If U has no interior points, that is, int U = ~, statement (4,iii) has the following inplication. First, let us denote by R the hyperspace of E of minimum dimension r containing U. Then UC R c E, 0 < r < n. If U is reduced to a single point then R = U and r = 0. Otherwise, 1 < r < n, and we denote by Rint U the certainly nonempty set of points of U which are interior to U with respect to R. Thus, int U C Rint U c Uc R c E. Statement (4,iii) has now the following corollary: (4.iv) Under the same hypotheses as in (4.iii), F(u) has a supporting plane at every point u c Rint U. The following statement also is relevant: (4r.v) Under the same hypotheses as in (4oiii), F(u) is continuous at every point u e Rint U. 9

Proof. We may well assume that U is not a single point, that 1 < r < n, and Rint U f 4. Let u be any point u E R int U, and let z = c + p.u the supporting plane at u, so that F(u) = c + p.u. Assume, if possible that for some a > 0 and sequence of points uk c Rint U, uk - u as k -* o we have F(uk) - F(u) < - a for all k. Then, F(Uk) > c + p-Uk, and hence -a > F(uk) - F(u) - P(uk - u). As k - oo, we have -a > O, a contradiction. Assume now, if possible, that for some a > 0 and sequence of points uk c Rint U, Uk + u as k + o, we have F(Uk) - F(u) > a for all k. Then the points = 2u - uk = u - (uk - u) are also points of Rint U for k sufficiently large, and the relations hold u = 2 uk + 2l u, F(u) > 2' F(uk) + 2'F(u') Then F( u) < 2F(u) - F(uk) < F(u) - a k - with uk + u as k +, and, as we have seen, we are led to a contradiction. This proves that F is continuous at every point u E Rint U. (4.vi) Under the same hypotheses as in (4.iii), F(u) is bounded below on every bounded part K of U. Proof. Indeed, if K contains more than one point, then K contains some point z e Rint U, and if z(u) = c + p.u is a supporting plane at u, then F(u) > c + p.u for all u c K C U, and c + p-u has a finite lower bound on K. 10

(4.vii) Under the same hypotheses as in (4.iii), F(u) is upper semicontinuous at every point u E U- Rint U along any-segment s issued from u and contained in U. Proof. Let s be the segment s = uu, s c U. Assume if possible, that there is a sequence of points uk E s C U, Uk + u as k -+, with F(Uk) - F(u) > a for all k. Then, all points interior to the segment s are certainly points of Rint U, say u = (1 - c)u + Mo, 0 < a < 1, and since F(u) < (1 - a) F(u) + otF(u ), we see that F is bounded above on s. Since hk = uk - u + 0 as k -+ o, there is a sequence of numbers 5k > 1 with k -' 00 khk O ~ as k + o. Hence, the points u' = u + k(uk - u), k = 1,2,..., are on the straight line from u containing s, and u' + u as k + o Thus u k E s, uk C k k k Rint U for all k sufficiently large, and the relations hold Uk =r-1uI + - l)u, k kk k k k - k k k F(u') > kF(uk) - (k - 1) F(u) > F(u) + Bk k-k k k k Hence, F(u') -+ +oo as k + +c, a contradiction since F is bounded above on s~ k We have proved that F is upper semicontinuous at u along s, A function F(u), u E U, convex on a convex set U may not be continuous at the points of U - Rint U, as the following example shows. Take U = [uIO < u < 1], and F(u) = 0 for 0 < u < 1, F(u) = 1 for u = 0 and u = 1o (4.viii) If U is a convex subset of E, if F(u), u c U, is a given real11

valued convex function on U, and the set Q = [(z,u) z 0 F(u), u E U] is (convex and) closed, then the function F(u) is lower semicontinuous at every point u C U - Rint! U, and even continuous on each segment s issued from u and contained in U. Proof. Assume, if possible, that there is a number a > 0 and points u, Uk, k = 1,2,..., with u C U - Rint U, uk c s c U, F(uk) < F(u) -c for all k. Take z = F(u), and note that all points (z -, uk) are in Q. Then, as k + a, we see that (z - a,u) is in the closed set Q, a contradiction, since (z,u) E Q if and only if z > z= F(u). The last part of the statement is a consequence of (4.vii), A function F(u), u E U, convex on a convex set U may not be continuous at the points of U - Rint U, even if the set Q is closed, as the following example shows. Take U = [(u,v) Il < u < 1, 0 < v < 1 - (1 - (1 - u)2)1/2, F(u,v) = v if 0 < u < 1, 0 < v < u, F(u,v) = (2u)-(u2 + v2) if 0 < u < 1, u < v 1 (1 (1 1 - u)2)1/2. Obviously, U is convex, F is convex in (u,v), but F is not continuous at (0,0) since F(O,O) = O, F(u,(l - (1 - u)2)l/2) = 1 for all 0 < u < 1. Given a convex set Uc E and a scalar function F(u), u c U, we say that F(u) is convex at the point u c U provided F(u) <. X F(u.) for any con-j=l j vj vex combination u =.. u of points u. E U. j=l j j j (4.ix) If U is a convex subset of E, and F(u), u c U, a given realvalued function, then F(u) is convex at an interior point u of U if and only if F(u) has a supporting plane at u, 12

A proof of this statement can be found in L. Turner [10]. We repeat here the proof for the convenience of the reader. Proof. Suppose F convex at the point u c int U. Then the smallest convex set co Q containing Q = [(z,u) z > F(u), u E U] c E +1 is the set of all n zl points (z,u) = v1 j(zj,uj) with (zj,uj) E Q, X. > 0, X +... + = 1, finite. Now (z,u) 4 co Q if z < F(u) since, for every convex combination (z,u) =. 1 Xj(z.,d) with u = u, u =.u we have z = jAjz >. jF(uj) j=l j 33 j j 3,=z>Fj > F(u). Hence z > z, and therefore (F(u),u) is a boundary point of co Q. Then, there is a hyperplane V = [(z,u)p oz + p.u - c = 0] c E+1 such that p F(u) + p u - c = 0 and p z + p.u - c > 0 for all (z,u) E co Q. For every convex combination z = j.u.j and numbers zj > F(uj), we have (zjuj) e co Q, and p Zj + p.u, - c > 0. Therefore, p [Z U.zj] + pou - c > 0, p F(u) + p.u - c = 0, and p [Z \jzj - F(u)] > 0. Since this is true for ar_ 0 i3 i bitrary large zj and A: > O0, we conclude that p > 0. But p = 0 implies pou - c > 0 for all u c U, which is impossible, as in the previous proof. Thus p > 0 and this hyperplane V can be written now in the form z = b-u + r, with b -P/Po, r = -c/p, and z > b.u + r for all (z,u) E co Q, F(u) = b-u + r. Thus z(u) = b.u + r is a supporting plane for F(u) at u = u. Conversely, if F(u) has a supporting plane z(u) = b-u + r at u c U, then for every convex combination u = ZA juj of points uj c U we have Z.jF(uj) > Zj.jz(uj) = Zjx [b-.u + r] = b-u + r = F(u), and F(u) is convex at u, The following statement (4.v) concerns the case where U = E, F(u) is convex in u in E, and (4ov) gives a characterization of those F which are linear on no straight line of E F n

(4.x) If F(u), u E E, is convex in u, then there are no points u, ul c E with ul O such that F(u) = 2-'[F(u ) + Xul) + F(uo - Xul)] for all real %, (4.1) if and only if there is a linear function w(u) = r + b.u, u E E, r, b = (bl,...,b) real, such that F(u) > w(u) for all u E E, and f(u) - w(u) +oo. This statement was essentially proved by L. Tonelli [9] under smoothness conditions on F. The proof below, based only on continuity and convexity properties, can be found in L. Turner [10], and is repeated here for the convenience of the reader. Proof. (a) Let us prove the sufficiency. Assume, if possible, that such a linear function w(u) as above exists and that also (4.1) holds for some uo, ul c E, u1 f O. Let z(u) = r + b-u, u e E, r, b = (b1,...,bn) real, be a supporting plane of F(u) at u. Then F(u) > z(u) for all u c E, and F(u + \ul) > r + b.(u + Xul) F(uo - Xul) > r + b.(uo + ful) F(u ) = r + b.u 0 0 By difference then we have 14

F(uo + \u1) - F(uo) > b-(Xul) F(u o u - F(u ) b(-Xu1) and by using (4o1) also 2-1 [F(uo + u1) - F(u - \u1)] > b ( ul) 2-1[F(uo - %uj) - F(uo + )ul)] > b(-ul). Since the sum of these relations is 0 = O, we conclude that = sign holds in both; hence F(uo + Xu1) - F(u ) = 2-1 [F(uo + Xul) - F(u - Xu1)] = b-(Xu1), F(u - Xu1) - F(u ) = 2' [F(uo - u1) - F(u + \u1)] = b(-Xu1) and finally F(u0 + Xul) = F(uo) + b'(Xul) = r + b-(uo + \ul) From F(u) > w(u) we deduce now F(u + Xul) = r + b.(u + u1) > r + b(u + u1), 0 0 - 0 and hence r - r + (b - b)u > x(b - b) ou for all % real. Since the first member is a constant, we must have (b - b)-ul 0, and then 15

F(uo + %ul) - w(uo + ul) = r + b.(u + Xul) - r - b(u + %ul) r - r + (b- b).u, 0 where the last member is a constant. This contradicts that F(u) - w(u) - oo as lul + +.o We have proved the sufficiency of the condition. (b) Let us prove the necessity. First assume that F(u) > O for all u E E, with F(O) = 0. Let T be the set of all real vectors b = (bl,...,bn) for which there is some real number r such that F(u) > r + b-u for all u E E. If bl, b2 e T and rl, r2 are the corresponding numbers, then for 0 < a < 1 F(u) - [crl + (1 - o)r2 - (Obl + (1 - c)b2).u] = a[F(u) - (r1 + b, u)] + (1 - a) )[F(u) - (r2 + b2u)] > O for all u e En. Hence, abl + (1 - a)b2 e T, and T is convex. Moreover, T contains the origin since F(u) > 0 for all u e E. Let us prove that T is not contained in any (n - 1) dimensional subspace of E. If it were, there would be a unit vector e such that e-b = 0 for all n b E T. Since F(Xe) + F(-Xe) > 0 for some A + O, then either F(Xe) > 0 or F(-Xe) > O. Suppose F(Xe) > 0 to be concrete. Let z(u) = F(Xe) + b-(u - Xe) be a supporting plane for F(u) at the point Xe. This supporting plane exists by force of (4.iii). Then F(u) > z(u) for all u, so b E T, ebb = 0 and z(ye) = F(Xe) + bi(ye - Xe) = F(ke) > 0 for all y real. Thus, in the directions + e the function z(u) is constant and positive. But z(O) < F(O) = 0, a contradiction. Thus, T is n-dimensional. We know that a convex set in E contained in no (n - l)-dimensional mani16

fold has an interior point. Therefore, let b, E > O, be such that b E T and lb - bJ < ~ implies b E T. Let r be a constant such that F(u) > w(u) r + b-u for all u E E. Suppose that lim inf [F(u) - w(u)] +oo, where lim inf is taken as Jul + +x. Then, there is a constant a > 0 and a sequence [Uk] such Juki I +o, F(Uk) - W(Uk) < a for all k. Without loss of generality we can assume that uk /Ju k converges to a unit vector u as k -+ o. Then b + ~ u E T, and there is a constant rl such that z(u) = rl + (b + ~u).u < F(u) for all u. Thus F(uk ) ) > r( + (b + u).uk - r - buk r - r + Eu.uk rl - r + eUklu.(Uk/fUk|) ) + as k -+ o, a contradiction. Thus, F(u) - w(u) * +oo as Jul | +o. We have proved the statement for functions F with F(u) > 0 and F(O) = 0. For an arbitrary F(u) let z(u) = F(O) + bl-u be a supporting plane for F(u) at the origin. Let G(u) = F(u) - z(u). Then G(u) > 0 for all u e E and G(O) = O. Thus, G satisfies the hypotheses assumed at the beginning, and there exists w2(u) = r2 + b2'u such that G(u) > w2(u) for all u and G(u) - w2(u) + +m. Let w(u) = z(u) + w2(u). Then F(u) - w(u) = G(u) - w2(u) > O for all u E En, and lim [F(u) - w(u)] = lim [G(u) - w2(u)] = +o, where both limits are taken as Jul + +o. Statement (4.v) is thereby proved. 5. SEMINORMALITY OF CONVEX FUNCTIONS As usual let A be a closed subset of the x-space, and f (x,u) a given 17

scalar function continuous on A x E n The function f (x,u) is said to be weakly seminormal in u at the point (x,u) E A x E provided, given ~ > O, there are numbers 5 > 0, and r, b = (bl,...,bn)) real, such that (XI) f (x,u) > r + b.u for all x E N (x), u E E; (X") f (x,u) < r + b.u + f 0o The function f (x,u) is said to be weakly seminormal in u at the point x E A if it has the just mentioned property at (x,u) E A x E for every u E E. The function f (x,u) is said to be seminormal in u at the point (x,u) e A x Em provided, given ~ > O, there are numbers o > 0, v > 0, and r, b = (bl,...,b ) real, such that (SN') f (x,u) > r + b'u + vju - uI for all x E N (x), u E; (SN") f (x,u) < r + b-u + ~ The function f (x,u) is said to be seminormal in u at the point x E A if it has the just mentioned property at (x,u) c A x E for every u c E. These concepts of seminormality are essentially due to L. Tonelli [9]. Requirement (SN") is often stated in the stronger form (SN"*) f (x,u) < r + b-u + ~ for all x c N (x), u c E, lu - u| < 5. As we shall see, statement (5.i) below holds for both form (SN") and (SN"*)o (5.i) If f (x,u) is continuous in A x E, then f is seminormal in u at x if and only if f (xu) is convex in u, and for no u, ul Em, ul = O, it 18

occurs that f (x,u) = 2-l[f (x,u + ul1) + fo(x,u - \ul)] for all X > 0. This statement was proved by L. Tonelli [9] under smoothness conditions on F. The proof below, based only on continuity and convexity properties, can be found in L. Turner [10], and is repeated here for the convenience of the reader. Proof. (a) Suppose f (x,u) seminormal in u at the point x c A. Then for every u E E there are constants r, b = (bl,...,bm) real and v > 0 such that m (SN') and (SN") hold. Let:(u) denote:(u) = r + b.u. Then, if Z.\.u. is any convex combination of points u. E E, with u = ZA u., then o'(x,;) < l(u) + E = Ejj z(uj) + E < j fo(x,uj) + E where E is arbitrary. Thus f (x,u) is convex in u at the point u = u. If there were points u, ul E E with ul / 0 such that 2- [f (x,u + ul) + fr(x,u - Xul)] = fo(x) (5.1) for all real X, then by force of (SN') f (xu) = 2 [fo(x,u + Xul) + fo(x,u - Xul)] > 2-1'[(u + Xul) +:(u - Xul)] + 2vJXl Jlu = l(u) + 2vllJ JuJll This is impossible since X can be arbitrarily large. We have proved the necessity of the condition. (b) Let us assume that f (x,u) is convex in u and that for no points u, ul c Er, ul $ 0, the relation f (x,u) = 2- [fo(x,u + Xu1) + fo(x,u - Aul)] 19

holds for all X > 0, and let us prove that f is seminormal in u at x c A. Let u be any point of Em, and let v(u) = rl + bl u be the supporting plane of f (x,u) at u = u. Let w(u) = r + b-u be the function satisfying the requirements of (4.v) for f (x,u) thought of as a function of u alone. Then, for 0 < a < 1, and all u we have f (x,u) - [aw(u) + (1 - a) v(u)] = a[f(X,u) - w(u)] + (1 - a)[f(x,u) - v(u)]>O. Let a be so small that a |w(u) - v(u)j < s/4, and let z(u) = W(U) (1 - o) v(u) - ~/4 0 0 Then fo(x,u) - z(u) = C [f (x,u) - w(u)] + (1 - C) [f o(x,u) - v(u)] - c/4 > E/4 for all u c E; (5.2) lim[f (x,u) - z(u)] = +C as lul + +o; (5.3) fo(x,u) - Z(U) = V() - () a (u) - W() ] + 7/4 < ~/2 (5.4) From (5.3) we conclude that, for some m > 0 we have Inf [f (x,u) - z(u)] > 2E u-u I =m Now define q(x) = Inf[f (x,u) - z(u)], where Inf is taken for Iu - u | = m Then q(x) is a continuous function of x for x E A, and T(x) > 2~. Then (5.2) and (5.3) above, and the continuity of 3(x), imply that there is t > 0 such that 20

f(x,u) - z(u) > ~/8 for Ix - xI < 6, Ju - ul = m; (5.5) r(x) > 9E/8 for Ix- x < S; (5.6) f (x,u) < z(u) + ~ for Ix - xI < 6, |u - ul < 6. (5.7) Relation (5.7) is requirement (SN") (actually, the stronger statement (SN"*)). If v = E/8m, then (5.5) implies fO(x,u) - z(u) - VIu - uI > (E/8) - vlu - u| > S/8 - E/8 = 0 for Ix - xl<, lu - ul < m. For u - ul > m, let a = m/lu - ulI, so 0 < a < 1, and let us define ul = a(u - u) + u. Then l U - u = ()(/lu - )(u - ) = m, ul = cu + (1 - ca)u, and thus, for Ix - xl < 5, f (xu1) < c f (x,u) + (1 - c) f (x,u), O'' -- 0 0 fo(x,u) - z(ul) < a[fo(x,u) - z(u)] + (1 - c)[f (X,U) - z(u)] f (x,u) - z(u) > f (x,u) - z(u) + (l/a)[fo(x,U 1) Z(U,)]- [f (x,u) z(u)] > 0 + (l/o) ([rl( ) - ~] > (l/cl)(9Se/8 - 8) = ~/8a Since c lu - ul = ~/8, we have f (x,u) - Z(u) - VIU - u I > (~/8a) (/8a) = 0, or f (x,u) > z(u) + vlu - u I for all u and Ix - xl < 8. This is requirement 0 O (SN'). Statement (5.i) is thereby proved. 6. A THIRD CRITERION FOR PROPERTY (Q) We give here a simple criterion for property (Q) of the sets Q(x) of no. 2 for the case in which f is linear in U, 21

(6.i) (a criterion for property (Q) for f linear in u). If A is closed, U = E, M = A x E, if fo(x,u) is continuous on M, convex in u, and seminormal in u at a point x c A, if f(x,u) = B(x)u + C(x) where the matrices B,C have entries continuous on A, then the sets Q(x) staisfy property (Q) at x, Proof. By seminormality we know that there is a neighborhood N (x) of 0 x in A and real numbers r, b = (bl,...,b) such that f (x,u) - r - bou > 0 for n - all x E N (x), u E E. By replacing f by f - r - b'u if necessary, we see m o 0 0 that it is not restrictive to assume f > 0 for all x E Nt (x) and u e E o0- m 0 Thus, f (x,u) > 0 for all x E N (x), u E E, and the sets Q(x) are de0 fined by [(z,z) z0 > f (x,u), z = f(x,u), u E E ]. We have to prove that z = (z,z) E 0l Cl co Q(x;o) implies z e Q(x). Let z be a given point z = (z,z) E nl cl co Q(x;&) and let us prove that z E Q(x). For every b > 0 we have _ — 0 - - 0 z = (z,z) E cl co Q(x;E) and thus, for every 8> 0 there are points z = (z,z) -0 E CO Q(x;E) at a distance as small as we want from z = (z,z). Thus, there is - a sequence of numbers 8k > 0 and of points zk = (Zkzk) c co Q(x;Sk) such that +k 0+ Zk + z as k -+. In other words, for every integer k there is a system of points xk e Nb (x), = 1,...,v, say v = n + 2, corresponding points Zk =(Z07z7) E E(x), points uk E E and numbers 7 0 < 1, y = 1 v1 k k k k m kk k such that 1 = zZxz k' k k Azk' Zk kzk Zk kk (6.4) zi 7 7 k7 f(xu z) B(xk)uy + C(x) z -f Zo k Xk')k k k 22

Y where y = l,...,v, k = 1,2,..., where Z ranges over all r = l,...,v, xk Y o -0 c N (x), and x, Zk, Z Z z as k - oo, 7 = By seminormality of f at the point x there are numbers E', 0 < V' < v > 0, r real, so that f (x,u) > r + vlul for all x E N (x). If k is sufficiently large so that Ek < 5' K<, and hence Ix - x| < k < 5', and because = r + blul is a convex function of u we have 0 = 7 Y 07 Y Y k z = xzjk > k fo(xkuk) > L Xk[r + vuk] > r + v Xuk ~Y Y 0~0 -0 Thus Z Ak u ki < v-l[zk - r] where z k z as k + o. This proves that k k k k /Z iU, k = 1,2,.., is a bounded sequence of points of E. By a suitable k k m extraction, there is a subsequence, say still [k], such that uk = k Uk k k k u E E as k -. m 0 -o oy 7 From the third relation of (6.4), where zk + z, Zk_ > 0, 0< k < _ we deduce that each of the v sequences [kk zk k = 1 2,.o ] 7y = lO wv is bounded. From the fifth relation (6.4) we then deduce k k > fz(xxu) > X(r + vuj), k k- o k' k-k k and hence kk<uk7 < v'[kzky + Irl]. Thus each of the v sequences [Xkuk, k k k kk k = 1,2,..O,], 7 = 1,2,,..,v, is bounded. If we denote by A7 the expression k A C[(B(x)x)) + C(x)) (( + ))] k k- Lk k2k k k1Uk 25

or (B(x B(x))(Xu ) + x(C(xk) - C(x)) k k k k k k and because of the continuity of B and C, and of x x, < <1, we conk kclude that A- 0 as k -, y = l v Given ~ > 0, by the seminormality of f (x,u) at x we can determine new numbers 5" > 0, and r, b = (bl,...,b ) real so that n f (x,u) > r + b.u for all x c N,,(x), u e E, and f (x,u) < r + b.u + o _ m o Now we have 0 7 7 Y Y Y z > > f (x,uk) > Z X[r + b.u ] = r + bou Zk =k k k0 okk k k k r + b.u + b.(u - u) < f(xu) + b( - u) - k ok0 Zk = Z z = Z k[B(xk)uY + C(xk)] = E k[B( (x)] k k k k kk k k k + Ak = B(x)uk + C(x) + Ak At the limit as k + co we obtain "O z > f (x,u) -, z = B(x)u + C(x) -- 0 and because ~ > 0 is arbitrary, also z > f (xu), z = f(x,u); hence z = (z,z) E Q(x). Statement (6oi) is thereby proved. 7. THE FUNCTION T Given A, U(x), M, f, f = (xfl f ), Q(x) as usual (A,M closed,

f, f continuous on M), let us remind here that the sets Q(x) are the projections of the sets Q(x) c En+1 on the z-space E. Hence, for every (z,z) C Q(x) we have z e Q(x), and z = f(x,u), z > f (x,u) for some u E U(x). Conversely, if z c Q(x), hence z = f(x,u) for some u c U(x), all points (z, z) 0 with z > f (x,u) certainly are in Q(x). -0 O For any fixed x C A let us consider the scalar function defined on Q(x): T(z;x) = Inf[f (x,u) z = f(x,u), u E U(x)] 0 = nf[z 1(z,z) E Q(x)], z Q(x) Then, for x E A, we have -a < T(z;x) < +oo for all z e Q(x). We shall consider T(z;x) as a function of z in Q(x). Note that the convexity of Q(x) C E +1 implies the convexity of Q(x) C E, but Q(x) may be not closed even if Q(x) is closed. Also, we shall denote by R the linear manifold in E containing Q(x) of minimum dimension r, thus, Q(x) C R c E, 0 < r < n. As usual, we shall denote by int Q(x) the set of all z E E which are interior to Q(x) (with respect to E ), and by Rint Q(x) the set of all points z which are interior to Q(x) with respect to R; thus int Q(x) c Rint Q(x) c Q(x) c R = E F n (7.i) If Q(x) is convex, then either T(z;x) = -o0 for all z C Rint Q(x), or T(z;x) > -o for all z c Q(x). In the latter case, T(z,x) is finite everywhere and a convex function of z in Q(x), T(z;x) is bounded below on every bounded subset of Q(x), and T(z;x) is continuous in the convex set Rint Q(x) (open with respect to R)o Finally, if Q(x) is convex and closed, and T(z,x) > 25

-to for all z c Q(x), then T(z;x) is lower semicontinuous at every point z E Q(x) - Rint Q(x). Proof. If Q(x) is a single point, then r = O, Rint Q(x) = l, and nothing has to be proved. Assume that Q(x) is not a single point. Then 1 < r < n, and Rint Q(x) +$. Let z be any point z C Rint Q(x). Assume that at some point zl E Q(x) we have T(zl,x) = -ao, and let us prove that T(z;x) = -X. For any integer k there are points (kzl) E Q(x) with k < -k, k = 1,2,... Take X = zl - z, and choose t > 0 so small that z2 = z - kF E Rint- Q(x). Take any point (z2, z2) c Q(x), and note that all points (az0 + (1 - )zk 2 + (1 - )z), 0 < a < 1, belong to Q(x). In particular, for a = (1 + ))-1 we have aZ2 +(1- c)z1 = ( (z- X) + (1 - a)z z - (1 - 0)(z- z1) - z + (1- a - c) = z - - 0 0 T(z;x) < Ocz2 + (1 - ) k< (1 + -z (1 - (1 + )-l)k k Z where the last term approaches -oo as k + o, hence T(z;x) = -to, Since x is any point of Rint Q(x), we have proved the first part of (7.i). The remaining parts of (7.i) are now a consequence of the definitions and of statements (4.v), (4.vi), and (4,viii). The first of the two cases mentioned in (7.i) may actually occur, even in 26

situations where the sets Q have property (Q) at x. Indeed, take m = n = 1, f = u, f = 0, U = El. Then Q z = 0, [(z,z) z E E1, z = 0], and 0 T = -0. As another case, take n = 1, m = 2, u,v control variables, f = u, 0 f = sin v, U = [(u,v) E E2]. Then Q = [zJ - 1 < z < 1], Q = [(z,z) Iz E E, -1 < z < 1], and T(z) = -o for all -1 < z < 1. In both cases, Q and Q are fixed, closed, convex sets, and certainly have property (Q). As a third example take n = 1, m = 2, u,v control variables, f = (1 - sin2v)u, f = sin v, 0 U = [(u,v) E E2]. Then Q = [zl-l < z < 1], and Q = [(z,z)I z E1 if -1 < z < 1, and z = 0 if z = + 1]. Finally, T(z) = -o for -1 < z < 1, T(z) = 0 for z = + 1. The following example proves that T(z;x) may not be lower semicontinuous on Q(x) if the set Q(x) is not closed. As usual, we shall denote by [g(P)]h the function of P which has the value g(P) if g(P) > h, and the value h if g(P) < h. Now take n = 1, m = 2, (u,v) control variables, fo = [(1 - sin2v)u]-l, f = sin v, U = [(u,v) E E2]. Then Q = [zI -1 < z < 1], and Q = [(z,z) z > -1 if -1 < z < 1, z > 0 if z = + 1]. Finally, T(z) = -1 for -1 < z < 1, T(z) = 0 for z = + 1. The following example shows that, even if the set Q(x) is closed and convex, the function T(z;x) may not be continuous at the points z c Q(x) - Rint Q(x). Let Q be the convex set [(1,r)l0 < 5 K 1, 0 < (1 - (1 - )2)1/2], and let T(, q) be defined by taking T = q for 0 K< K 1, 0 _< ~ K, T = (2~)-1(2 + n2) for 0 < ~ < 1, _< n _< (1 - (1 - )2)1/2. As we have seen in Section 4, T(,tj) is convex and bounded in Q, and continuous in Q except at the point (~ = 0, O = 0). Now let us define the set Q. To this purpose, 27

let U be the union of the two closed disjoint sets U1 = [(u,v,w) O < u < 1, -1 <v<u-l, w > O] and U2 = [(u,v,w) 10 < u < 1, 0 < v < (1 1 - u)2)1/2, w > O], Let q(w,u) be the function defined for all (u,w), 0 < u < 1, w > O, by taking q(w,u) = (w + 1)-1 - u for 0 < u < (w + 1)-1, (w,u) = 0 for (w + 1)-1 < u < 1. Finally, let us define the functions f (u,v,w), fl(u,v,w), f2(u,v,w) continuous on U = U1 U U2, by taking fl = u, f2 = v + 1, f = v on U1, and = f1 = v + (u - )(1 + v)-1/2[(l + V)1/2 - (1 - V)1/2] = f2 = v f = 2(= + n(w,u))-'[21(w,u) + + 2] 0 on U2. Then, if Q denotes the set Q = [(,,)z >, = f1, = = f2 (u,v,w) E U = Uz U U2], and T(S,~) = Inf[z0 (z0,,r) e Q], then T is exactly the convex function defined above on Q, and Q is convex and closed. The following example shows that at points z E Q(x) - Rint Q(x) the supporting plane of Q(x) may be vertical, even if Q(x) is convex and closed, Q(x) is convex and compact, and T(z;x) continuous on Q(x). Indeed, take Q = [(u,v)Ju2 + v 2< 1], T =-(1 - U2 v2)1/2, U f1 =u, f = v, f =T, 0 0 Q = [(z,u,v)Jz0 > T. (u,v) E U]. 8. A CHARACTERIZATION OF PROPERTY (Q) FOR THE SETS Q(t,x) For fixed x E A and 8 > O, let us consider the set Q*(x;5) = co Q(x;5) = co{ NU() Q(x)) c En+1 (81l) 28

and its projection on the z-space E n Q*(x;b) = co Q(x;b) =co U() Q(x) ) (xN ) x n Both sets Q*(x;5) and Q*(x,6) are convex, and E n Q(x;3) = Q(x), En D Q*(x;5) = Q(x) (8.2) As before, we shall consider the scalar function defined on Q*(x;-): T*(z;x,5) = Inf z0 (z,z) e Q*(x,5)), z E Q*(x;E) Thus, we have again -o < T*(z;x,5) < 0 for all z E Q*(x;5). Also, for every z E Q(x), we have T*(z;x,8) < T(z;x). We have now the following characterization of property (Q): (8.i) If T(z;x) > -oo in Q(x), then the sets Q(x) have property (Q) at x if and only if properties (o) and (X) hold at the point x. Proof. We have already proved in (3oi) that the union of (a) and (X) implies (Q). We need only prove that, if T(z;x) > - oo in Q(x), and Q(x) has property (Q) at x, then both (a) and (X) hold at x, We know already that (a) is a necessary condition for property (Q) and thus (a) holds. Also, Q(x) is closed and convex. Since T(z;x) > - oo by hypothesis, we know from (7.i) that T(z;x) is a lower semicontinuous convex function of z in the convex set Q(x). We have already noticed that -oo < T*(z;x,E) < T(z,x) C + oo for all z C Q(x) and 6 > 0. -o Now take any point z C Q(x), and let z = T(z,x). Then by (7i) the 29

point (z,z) belongs to Q(x), and hence there is some u c U(x) with -o -o z = T(z,x) = f (x,u), z f(x,u). Given ~ > 0, the point P = (z - F,z) is not on the closed set Q(x), and hence has a minimum distance 9 from this set, with 0 < r < E. Since T(z;x) is lower semicontinuous at z, there is some I', O < 9' K<, such that T(z;x) > T(z;x) - T/3 for all z E Q(x) with iz - I < i'. mO Let a be the closed ball in E of center P = (z - E,z) and radius n+l 9'/3. Let a denote the projection of a on the z-space; thus a is the closed O O ball in E of center z and radius i'/3- We shall denote also by a, the closed ball in E of center z and radius 2n'/3. n Now let us consider the convex sets Q*(x;5) = co Q(x,5) defined in (8.1) and their relative functions T*(z;x,5) defined in (8.2). Let us prove that there is some 5 > 0 such that 0 o < T(z;x) - T*(z;x,6) < i/3 (8.3) for all O<8<Fi and zc c1 n Q*(z,-). Indeed in the contrary case, there would be - O numbers Sk > 0 and points zk E al E, k = 1,2,..., with 5k 0 as k + o and T*(zk;x,Jk) _< T(z,x) - 9/3, and hence points (Zkzk) c co Q(X,Sk) with O -0 0 < T(z;x) - 9/3 = z - /3. Hence, for every 5 > O we have (z z ) c k k - k co Q(x,5) for all k sufficiently large, and then also (z - 9/3,Zk) c co Q(x;5). If z' is any point of accumulation of [zk] we have then z' E c1, (z - 9/3,z') e cl - -o, co Q(x,5), and by property (Q) also (z - V/3,z') c Q(x) = n cl co Q(x, ). This implies T(z';x) < z - 9r/3 with z' CE c, IZ' - zj < 29'/3 < l', a contradiction. We have proved that for some - > 0 relation (8.3) holds for all O < 5 < ~ and z E a1 n Q*(x,5). 3o

Let us prove that any two points P = (z,z) c a and P' = (z',z') C Q*(x;S ) have a distance (P,P') >'/3. Indeed, either P' is outside the 0 cylinder (z0 El, z E a1) and then (P',P) > Iz' - zJ > Iz' - z - Iz - zl > 2T'/3 - 1'/3 = n'/3 or P' is inside the cylinder above, and then by (8.3) for 0 < 5 < 5, z > T*(z';x, ) > T(z,x) -'O/3 z - T/3, o -o -o 0 -o o 0 (PtP ~[> z0 - z [z - (z - ~)] + [ z I + -z > - /3- (P,P) > - /3 - /33 = ~/3 Thus, the convex sets a and Q*(x;5) have a distance _> /3, and the same occurs for the convex closed sets a and cl Q*(x,8), (a compact). We conclude that there is some hyperplane n in E 1 separating the two convex sets a and n + cl Q*(x;6). This hyperplane nmust intersect the vertical segment [z - E + O/3 < o -o o z < z,z = z] at some point (z = r, z = z), andn cannot be parallel to the z -axis, otherwise all points of the straight line z = z would be on n, in particular the center P of the ball a, and not all points of a could be separated from Q*(x;5). Thus, n is of the form n: z = r + b.(z- z) = (r - b.z) + bz Q(x) as well as cl Q*(x;b) are above, and thus, (z,z) e el Q*(x;b) implies z > (r - bez) + b.z. In other words, for O < 6 < 5, x E N(x), x e A, 31

u c U(x), we have f (x,u) > (r - b-z) + b.f(x,u). On the other hand, fo(x,u) -o -0 - - = Z = (z - ) + E < r + E = (r - b.z) + b.z + E. We have proved that property (X) holds at the point x E A. Statement (8.i) is thereby proved. 32

REFERENCES 1. L. Cesari, (a) Existence theorems for weak and usual optimal solutions in Lagrange problems with unilateral constraints. I and II. Trans. Amer. Math. Soc. 124, 1966, 369-412, 413-429. (b) Existence theorems for optimal controls of the Mayer type. SIAM. J. Control 6, 1968, 517-552. (c) Existence theorems for multidimensional problems of optimal control. Differential Equations and Dynamical Systems, 115-132, Academic Press, 1967. (d) Existence theorems for multidimensional Lagrange problems. Journal of Optimization Theory and Applications 1, 1967, 87-112. (e) Sobolev spaces and multidimensional Lagrange problems of optimization. Annali Scuola Normale Sup. Pisa 22, 1968, 193-227. (f) Existence theorems for abstract multidimensional control problems. Journal of Optimization Theory and Applications. To appear. 2. L. Cesari, J. R. LaPalm, and T. Nishiura, Remarks on some existence theorems for optimal control. Journal of Optimization Theory and Applications 3, 1969, 296-305. 3. G. Choquet, Convergences, Annales Univ. Grenoble 23, 1947-48, 55-112. 4. C. Kuratowski, Les fonctions semicontinues dans l'espace des ensembles fermes. Fund. Math. 18, 1932, 148-166. 5. J. R. La Palm, Remarks on certain growth conditions in problems of optimal control, Journal of Optimization Theory and Applications. To appear. 6. A. Lasota and C. Olech, On the closedness of the set of trajectories of a control system, Bull. Acad, Polon. Sci., Sero Sci. Math. Astr. Physo 14, 1966, 615-621. 7, E. Michael, Topologies on the spaces of subsets, Trans. Amer. Math. Socd 71, 1951, 152-182. 8. C. Olech, Existence theorems for optimal problems wita vector-valued cost functions. Trans. Amero Math. Soco 136, 1969, 159-180. 9. L. Tonelli, Sugli integrali del calcolo delle variazioni in forma ordinaria, Annali Scuola Norm. Sup. Pisa (2) 3, 1934, 401-450 (= Opere Scelte, Cremonese, Roma 1962, 3, 192-254). 10. L, Turner, The direct method in the calculus of variations. PhDo Thesis, Purdue University, Lafayette, Indiana, 1957.

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