Dynamic properties of optimal solutions of variational problems

Non&-r-,

DYNAMIC

Ikory,

Meah&

& Appticd,

Vol. 21, No. 8, pp. 895931,1996 Cqyright 0 19% Elscvier Science Ltd Printed in Great Britain. All rights resewed 0362-546X/% J15.00 + 0.00

PROPERTIES OF OPTIMAL SOLUTIONS VARIATIONAL PROBLEMS

OF

A. J. ZASLAVSKI Department of Mathematics, Technion-Israel

Institute of Technology, Haifa 32OW, Israel

(Received 10 September 1994, received in revised form 23 January 1995; meiwd forpublication February 1995)

21

Key words andphrases: Intinite horizon, weakly optimal function, good function, turnpike property.

1. INTRODUCTION

The study of variational and optimal control problems defined on infinte intervals has recently been a rapidly growing area of research. These problems arise in engineering (see Anderson and Moore 111,Artstein and Leizarowitz [2]), in models of economic growth (see Rockafellar [3], Carlson 1411,in infinite discrete models of solid-state physics related to dislocations in one-dimensional crystals which are under discussion in Aubry and Le Daeron [S], Zaslavski [6] and in theory of thermodynamical equilibrium for materials (see Leizarowitz and Mizel [7], Coleman et al. [8D. In this paper we analyse the existence and the structure of optimal solutions of autonomous variational problems. Given an x0 E R” we study the infinite horizon problem of minimizing the expression joTf(x(t), x’(t))dt as T grows to intlnity where a function x : [O,w) 4 R” is locally absolutely continuous (a.c.) and satisfies the initial condition x(0) = x0, and f = f(x, U> is an integrand. The following notion known as the overtding optirndity criterion was introduced in the economics literature by Gale [9] and von Weizsacker [lo] and has been used in control theory by Artstein and Leizarowitz [2], Brock and Haurie ill], Carlson [4], Carlson et al. [12], Carlson et al. [13]. An a.c. function x : [O,w) + R” is called (f)-ouertaking optimal if for any a.c. function y : [O, m) + R”-satisfying y(O) =x(O) lim sup T-*-

r[f(x(r), /

x’(t))

-f(y(t), y’(t))ldt 5 0.

0

In this paper we employ the following weakened version of this criterion. An a.c. function x : [O,a) 4 R” is called (f)-we&y optimal if for any a.c. function y : [0, 00) 4 R”-satisfying y(O) =x(O) T

lim inf T-m

[f(x(t), x’(t)) -f(y(t), y’(t))1 dt I 0.

/ 0

895

896

A.J. ZASLAVSKI

Another type of optimality criterion for infinite horizon problems was introduced by Aubry and Le Daeron [51 in their study of the discrete Frenkel-Kontorova model related to dislocations in one-dimensional crystals. More recently this notion was used by Moser [14,15], Bangert [16], Leizarowitz and Mizel[7] and Zaslavski [6]. Let I be either [0, + m) or (-w, +m). An a.c. function x : Z -S R” is called an (f)-minimal solution if for each numbers Tl E Z, T2 > Tl and each a.c. function y : [T,, T2] + R” which satisfies y(q) =x(q), i = 1,2 the following relation holds Tz~f(x(f),x’(t)) I =1

-f(y(t),y’(t))ldt

< 0.

Clearly every (f)-weakly optimal function is an (f)minimal solution. We can establish the existence of (f)-minimal solutions using the following scheme. Let x0 E R”, N 2 1 be an integer and let xN : [0, N] -+ R” be an optimal solution of the following variational problem N

/0

&r(t),

x’(t))dt

--) min,

x(0) =x0.

When the integrand f satisfies certain assumptions we can show that there exist a subsequence
+ x(t) uniformly i-tar

in [0, k], xIN,~T~x

weakly in L’([O, kl; R”)

and x is an (f)minimal solution. Various existence results of overtaking optimal and weakly optimal functions are nicely collected in Carlson et al. [13]. The most typical infinite horizon optimization problem for which the existence of overtaking optimal function has been established is an autonomous variational problem with a convex integrand j,Tf(x(t), x’(t))dt studied by Rockafellar [3], Brock and Haurie [ll] and Leizarowitz [17]. When the existence of optimal solutions over an infinite horizon was established for a nonconvex problem a convexity assumption was usually replaced by other assumptions which were restrictive and did not hold in general. Our frrst goal is to show that the existence of weakly optimal solutions is a general phenomenon which holds for a large class of variational problems. An optimal solution u : [0, T] + R” of the variational problem T

/0

f(t(t),

z’(t))dt

--) min,

z(O) =x, z(T) =y,

09

z : [0, T] --*R” is an a.c. function always depends on the integrand f and on x,y, T. We say that the integrand f has the turnpike property if for large enough T the dependence on x, y, T is not essential. Namely, for any E > 0 there exist constants L,, L, > 0 which depend only on 1x1,lyl, e and a measurable set E c [O, T] such that the Lebesgue measure of E does not exceed L, and for each TE [0, T - L,]\E the set {u(t): t E [T, T+ L2j) is equal to a set H(f) up to E in the Hausdorff metric where H(f) c R” is a compact set depending only on the integrand f. Moreover, the set E is a finite union of intervals and their number does not exceed a constant which depends only on 1x1, lyl and E.

897

Dynamic properties of optimal solutions of variational problems

More formally we say that an integrand f=f(x, U) E C(R2n) has the turnpike property if there exists a compact set H(f) c R” such that: for each bounded set KC R” and each E > 0 there exist a number 1 > 0 and integers L, Q 2 1 such that for each T 1 L + lQ, each x, y E K and an optimal solution u : 10, T] -+ R” for the variational problem (P) there exist sequences of numbers (bj]i4_ i, CCj~= i C [O, T] for which

qsQ, dist(Hi(f),

{I

Osci-bid,

: t E [T, r+ L])) I E

j=l

,..., q,

for each TE [O,T-L]\

(J [bj,cj]. j=l

(Here dist(*, -1 is a Hausdorff metric). The turnpike property is well known in mathematical economics. It was studied by many authors for optimal trajectories of a von Neumann-Gale model determined by a superliner set-valued mapping (see Makarov and Rubinov [181 and the survey [19]). In the control theory the turnpike property was established by Artstein and Leixarowitz [2] for a tracking periodic problem. In both cases we have an optimal control problem with a convex cost function and a convex set of trajectories. Our second goal is to show that the turnpike property is a general phenomenon which holds for a large class of variational problems. We consider the complete metric space of integrands ?l described below and establish the existence of a set of .Fc % which is a countable intersection of open everywhere dense sets in ‘?I and such that for each f ES the following properties hold: for each z E R” there exists an (f )-weakly optimal function Z : [O, 03) + R” satisfying Z(O) = z; the integrand f has the turnpike property. Moreover we show that the turnpike property holds for approximate solutions of variational problems with a generic integrand f and that the turnpike phenomenon is stable under small perturbations of a generic integrand f. Denote by I.1 the Euclidean norm in R” and denote by ‘3 the set of continuous functions f : R” X R” + R’ which satisfy the following assumptions: Assumptzim. 6) for each x E R” the function f(x, .) : R” + R’ is convex; (ii) the function f is bounded on any bounded subset of R” x R”; (iii) fb, 4 2 sup{Jl(lxD,@4DlulI - a for each (x, u) E R” x R” where a > 0 is a constant and + : [O,m) -+ [O,m) is an increasing function such that +(r) + +w as t --) 01; (iv) for each IV, E > 0 there exist I, 6 > 0 such that

If(x,,u,> -f(X,,U,)lI

~sup~f~x~,u~~,f~x~,u~~l

for each ui, u2, xi, x2 E R” which satisfy

IXJ lM,lUjl2z I% = 1,2), supux, -x2l,Iu1- 241 IS. It is an elementary exercise to show that an integrand f = f(x, u) E C’(R2”> belongs to ‘?I if f satisfies assumptions (3, (iii) with a constant a > 0 and a function + : [0, a~) + [0, 00) and there exists an increasing function

sup{ldf/Mx,

+a : [O,QJ) + [0, a> such that for each x, u E R”

UN,laf/au(x, UN I &(lxl)(l + ~hlM).

898

AJ. ZASLAVSKI

For the set VI we consider the uniformity EW,

which is determined

e, A.) = {(f, g) E QI x 9I : IfC x,u)

-g(x,u)ls

by the following base

l (u,x~R”,Inl,lul~N),

~lf~x,u~l+1~~lg~x,u~~l+1)-‘~~A-‘,h1~x,u~R”,I~I~N~} where N>O, r>O, A> 1 [20]. Clearly, the uniform space W. is Hausdorff and has a countable base. Therefore % is me&able [20]. It was established in [21, proposition 2.21 that the uniform space % is complete. We consider functionals of the form ZW,,T*,x)

=

/?

T2 f(x(t),

wheref~VI,OsT,
(1.1)

isana.c.function. weset

-inf{Zf(Tl,T,,x):x:[Tl,T,]

is an a.c. function satisjing

x’(t)) dt

+R”

x(T,) =y, x(T,) =z},

af(T,,T,,y)=inf(tl~(T,,T,,y,u):tc~R”}.

(1.2)

(1.3)

It is easy to see that -~
of (f)-good

for each T E (0, ml. functions and established

the following

THEOREM 1.1 [21, theorem 1.11. For each f E 2l and each z E R” there exists an (f )-good function Zf : [O,m) + R” sati@ing Zf(O) = z and such that: 1. For each f E ‘8, each z E R” and each a.c. function y : [O, 03) + R” one of the following properties holds: (i) Zr(O, T, y) - Zf(0, T, Zf) --* + 00 as T + CL; (ii) sup(lZf(0, T, y) - Zf(0, T, Zf)l : T E (O,m)} < 00,sup(ly(t)l : t E [0, a)) < m. 2. For each f E !!I and each number M > 0 there exist a neighborhood SYof f in ‘?I and a number Q > 0 such that sup(lZg(t)l: t E [0, Q))} I Q for each g E g and each z E R” satisfying It] 5 M. 3. For each f E 91 and each number M > 0 there exist a neighborhood V of f in ?I and a number Q > 0 such that for each g E %, each z E R” satisfying lzl I M, each Tl 2 0, T, > Tl and each a.c. function y : [T,, T,] + R” satisfying Iy(Tl)I I iU the relation Zg(Tl, T,, Zg) I Zg(Tl, T,, y) + Q holds. 4. For each f E %, each z E R”, each Tl 2 0, T2 > Tl the following relation holds Uf(Tl, T,, Zf(T,), Zf(T,N = Zf(TI, T2, Zr).

Dynamic properties of optimal solutions of variational problems

Let f~ ‘3. For any a.c. function

n : [0, m) --) R” we set

J(x) = lim infT’I’(0,

T, x).

T-r-

Of special interest is the minimal

(1.4)

long-run average cost growth rate

p(f) = i&J(n) : x : [Op) + R” is an a.c. function}. Clearly -a < p(f) < +w and for every (f)-good function x : [O,m) --) R” p(f) =I(x). In Section 4 we will establish the following result. ~OPOSTION

899

(1 S) (1.6)

1.1. For any a.c. function x : [O,m) --) R” either I/(0, T, x) - T,u(f) + +m

LtST+=J

or sup{llr(O, T, xl - Tp(f)l: T E (O,CQ)} < to. Moreover (1.7) holds if and only if n is an (f)-good function. 2. MAIN

(1.7)

RESULTS

We denote d(x, B) = inf{lx - y(: y E B) for x E R”, B c R”. Denote by dist(A, B) the Hausdorff metric for two sets A c R”, B c R” and denote by Card(A) the cardinal&y of a set A. For every bounded a.c. function x : [O,=J) -+ R” define n(n) = {y E R”: there exists a sequence (t,}~~o c (0,~) for which ti+w,x(fi)+yasi+w

1.

(2.1)

We will establish the following results. THEOREM 2.1. There exists a set 9c ?I which is a countable intersection of open everywhere dense subsets of 9I and such that each ~EF has the k&wing property: (B)fNu,) = iI for each (f&good ~UIKGOIIS Ui : [0, 0~) + R”, i = 1,2. Theorem 2.1 describes the Iimit behavior of (f)-good functions for a generic f~ %!I. The following result establishes the existence of an (f&weakIy optimal function for each f~ ?I which has property (B) and each initial state x E R”. THEOREM 2.2. Assume that f~ M and there exists a compact set H(f) c R” such that O(u) = H(f) for each (f)-good function v : [0, a~) + R”. Then for each x E R” there exists an (f&weakly optimal function X : [O, w) + RR saw X(O) =x. It fohows from theorems 2.1 and 2.2 that for a generic f~ QI and every x E R” there exists an
THEOREM

iI

900

A.J. ZASLAVSKI

there exist an integer L r 1 and a neighborhood each (g)-good function v : [0, 00)-+ R"

9 of f in 9x such that for each g E % and

dist(Mf), (v(t) : t E [T, T + L]}) s TV for all large T. Theorems 2.1 and 2.3 show that for a generic f E %?Ithere exists a compact set H(f) c R" such that for any (f)-good function the turnpike property holds with the set H(f) being the attractor. ‘DEOREM 2.4. Assume that f~ ‘% and there exists a compact set H(f) CR" such that Q(v) = H(f) for each (f)-good function v : [O,@J)+ R”. Let M,, M,, Q> 0. Then there exist a neighborhood %Yof f in 8, numbers 1, S > 0 and integers L, Qt r 1 such that for each g E %, each TI E [0, m), T, E [ TI + L + lQ, , a) and each a.c. function v : [T,, T,] + R” which satislks lv(zysM,,

i-1,2,

P(T,,T,,v)

s UB(T~,T*,v(T~),v(T*))

+M,

the relation Iv(t)] s S holds for all t E [T,, T2] and there exist sequences of numbers Cbi]c i, , QsQ.,

dist(H(f), (v(t):tE

[T,T+LI))

s fz

for each TE [T,,T,-L]\

6 [bi,Ci]* i-l

Theorems 2.1 and 2.4 imply that for a generic f~ ?I there exists a compact set H(f) CR” such that for any t&rite horizon problem the turnpike property holds with the set H(f) being the attractor. Let k 2 1 be an integer. Denote by 91k the space of all integrands f~ ‘8 n Ck(Rzn). For p=h ,..., p2”ba ,... 7k)“’ and f~ Ck(R2”) we set

IpI= Epi9

Ivy= a’p’f/dyp.. . ayp.

i-l

For the set ?Ik we consider the unifcumity EW,

which is determined

by the following base

e, A) = ((f, g) E 8, x 2I t:lDPf(n,u)-DPg(x,u)ls~((u,n~R”,lxl,lulsN,

p4L..,

k12? lpl s k), lf( x, u) -g(x, dl s l b, x E R”, 1x1,lul s N),


E[A-‘,A](~,~ER”,IxI~N))

where N>O, E>O, A> 1 [20]. Clearly, the uniform space QIk is Hausdorff and has a countable base. Therefore ‘8, is met&able [2O]. It is easy to verity that the uniform space Sk ls complete. We will establish the following result. THEOREM 2.5. Let k 2 1 be an integer. There

exists a set 9, c 8, which is a countable intersection of open everywhere dense subsets of VIZ, and such that for each f~& and each (f)-good functions vi : [O,m) -B R”, i = 1,2 n(v,)

= NV,).

Dynamic properties of optimal solutions of variational problems 3. AUXILIARY

901

RESULTS

In [21] we established the following results. PROPOSITION 3.1 a21], theorem 1.2). For each f~ %?Ithere exists a neighborhood V of f in % and a number M > 0 such that for each g E %! and each (ghgood function x : [O,ml--, R” lim sup]x(t)l 0. Then there exist a neighborhood V of f in 9x and S > 0 such that for each g E %, each Ti E [O,m) and T, E [Tl + c, ~1 the following properties hoId: (9 for each x,y E R” satisfying ]xl,]y] 5 Ml and each a.c. function u :[T,, T,] + R” satisijing v(T,) =x, v(T,) =y, Zg(T,, T2, u) I Ug(T,, T,, x, y) + M2 the following relation holds (3.1) lu(t)lsS(r~[T,,T,lk (ii) for each x E R” satisfying Ix] s Ml and each a.c. function u : [T,, T,] + R” satis@ng u(T,) =x, Zg(Tl, T2, v) s og(T,, T,, x) + M2 relation (3.1) holds. PROPOSITION3.3 ([21], proposition 2.7). Let f~ 8,O c c1 < c2 < 00,M, Q> 0. Then there exists S > 0 such that for each T,, T, ;r 0 satisfying T, - Tl E [cl, c2] and each y,, y,, zl, z2E R” satisfying Iyil, IZil s M (i = 1,2), sup(IyI -y,l, ItI -z2D s 6 the relation IUf(TI,T2,yI, zl) Uf(TI, T2, y,, ZJ I E holds. PROPOSRION 3.4 ([21], proposition 2.5). Assume that f E !?I, Mi > 0, 0 s TI < T2, Xi : [TI, T21 --) R”, i = 1,2... is a sequence of ax. functions such that Zf(T,, T,, Xi) 5 Ml, i = 1,2.... Then there exist a subsequence{Xi,~-1 and an a.c.function x : [T,, T2] 4 R” such that Zf(TI, T2, x)
PROPOSITION 3.5 a21], corollary 2.1). For each f~ VI, each numbers T,, T, satisfying 0 s Tl < T2 and each zi, z2E R” there is an a.c. function x : [T,, T2] -B R” such that x(T,) = Zi> i = 1,2, If@,, T,, x) = UfU’,, T2, zl, z2). PROPOSITION 3.6 ([21], corollary

2.2). For each f E B, each numbers T,, T2 satisfying0 s Tl < T2 and each z E R” there existsan a.c. function x : [T,, T21 + R” such that x(T,) = Z, Zr(TI, T2, X) = d(T,,

T,, z).

PROPOSITION3.7 ([21], proposition

2.6). Let f~ %!I,0 < cr < c2 < 03,c3 > 0. Then there existsa neighborhood $Vof f in ‘?I such that the set {Ug(T~,T2,zl,z2):g~~,T~~[O,~),T2E[T,+c,,T,+c21,z1,z2~R”,

lzi]~c3 (i=1,2)] isbounded. 3.8 ([21], proposition 2.8). Let f~ ‘?I, 0 < cl < c2 < w, D, E> 0. Then there is a neighborhood V of f in % such that for each g E V, each T,, T2 2 0 satisfying T2 - T, E [cl, ~21

PROPOSITION

902

AJ. ZASIAVSKI

and each ax. function x : [T,, T2] + R” satisfying inf{Zf(Tr, T2, x), ZW,, T2, XI} I D the relation IZf(TI, T2, n) - Zg(TI, T,, x)] I l holds. 3.9 (1211, proposition 2.9). Let f~ 8,O < cl < c2 < 00,c3, E > 0. Then there exists a neighborhood V of f in 91 such that for each g E V, each T,, T2 2 0 satisfjing T, - TI E [cr, c2] and each t, y E R” satisfying lyb It] s c3 the relation lUf(T,, T2, y, t) - Ulg(TI, T2, y, z)l s E holds.

PROPOSITION

3.10 a211), proposition 2.4). Let Mr,e > 0, 0 < r,, < or. Then there is S > 0 such that for each f~ 8, each T,, T, r 0 satisfying T2 - TI E [T,,, ~~1,each a.c. function x : IT,, T,] + R” satisfying Zf(T,,T2,x) s MI and each t,,t, E [T,,T,] which satisfy [tr - r2] I 6 the relation 1x0,) -x(Qi 5 l holds. PROPOSITION

PROPOSITION

that

3.11 ([21]), proposition

2.1). Let f E 8, M, E > 0. Then there exist r, S > 0 such ~4) for each ur, u2, x,, x2 E R” which satisfy

If(x,, ~4~)-f(x,, uJ s l inf(f( x1,ul), f(x2, IXjl I M, lUil2

lul- U*l,

r(i = 172),

lx, --+I I 6.

3.12 ([21]), proposition 2.3). Let MI > 0, 0 < r,, < rr. Then there exists M2 > 0 such that for each f~ %, each T , T2 z 0 satisfying T, - TI E [rO, or] and each a.c. function x : [T,, T21 + R” which satisfies Z1(T,, T2, x) s MI the relation Ix(t) s M2 holds for all t E [T,,T,l. PROPOSITION

3.13 (Leizarowitz [22}). Let Kc R” be a compact set, u : K X K --$ R’ be a continuous function and define

PROPOSITION

N-l ~(U)=inf

~~~,-’

C

i

U(t~,tj+~):(ti}im_OCK

(3.1)

)

i=O

i

N-l T”(X)=iXlf

%jr$ i

C

[U(Z~~Z~+~)~~(U)l~(Lj]~~~cK,Z~~X

,

i=O

(3.2)

1

P(x,y) = u(x,y) - p(u) + 7?(y) - P”(X)

(3.3)

for x,y E K. Then W” : K + R’, 8” : K x K --, R’ are continuous functions, 8” is nonnegative and E(x) = {y E K : 8 “(x, y) = 0) is nonempty for every x E K. 4. PROOF

OF PROPOSITION

1.1

Let f~ 8, z E R” and let an a.c. function Zf : [0, a) --) R” be as guaranteed in theorem 1.1. By (1.4), (1.6), theorem 1.1 and assumption (iii)

p(f) = lim iI&v-‘Zf(0,

N,Zf)

N+-

where N is an integer. Together with theorem 1.1 this implies that N-l

p(f)

= li~+iffN-l

C W(i, i + l,Zf(i),Zrti i=O

+ 1)).

(4.1)

Dynamic

By theorem

properties

1.1 and proposition

of

optimal

solutions

903

problems

3.1 there exists a number Q > sup{lZfW:

such that for each (f&good

of variational

(4.2)

t E V-b>1

function x : [O, m) + R” lim suplx(t)l 1-P-J

(4.3)

< Q.

Set BP = {x E R” : Ix/ I Qj, u(x,y)

= Ur(o,l,x,y)

(x,y++

(4.4)

= Uf(O,l,x,y)

(x,y~R~,i=O,l.).

(4.5)

Clearly + l,x,y)

Uf(i,i

By proposition

3.3 the function v : BP

x

B, --) R’ is continuous.

We will show that

N-l

(4.6) sup I C [v(Zf(i),Zf(i+l))-&Ill:N=l,2...
(Osi
O”(Xl,Xi+l)=O

1)).

(4.7)

- p(v).

(4.8)

3.13 that

N-l

c

b(xl,x;+l)

- P(V)1 = TV(Z) - T”(XN-J

+ U(XN-l,XN)

i=O

Proposition

3.13 and (4.2) imply that N-l C

+ 1)) - p(v)1 2 a”(z)

[u(Zf(i),Zf(i

(4.9)

- 7r”GV(N)).

i=O

On the other hand by (4.4), (4.9, (4.7), assertion 4 of theorem 1.1, pqosition N-l

C

v(Zf(i),Zf(i

+ 1)) =

i=O

C Ur( i,i+

l,Zf(i),Zf(i

+ 1)) =Zf(O,N,Zf)

i-0

N-l

I

C

N-l

Uf(i,i+l,Xt,Xi+l)=

i=O

Combining

3.5

N-l

C

v(xl>Xi+l)a

(4.10)

i=O

(4.8-4.10) we obtain that for any integer N 2 2 N-l

I C Iu(Zf(i),

Zf(i + 11) - p(o)JJ I 2~up(Jv”(y)l:

Y E BP}

i=O

+ 2sup{lv(x,

y)l: x, y E Bp}.

(4.11)

It follows from (4.1), (4.111, (4.4), (4.5) that p(f) Together with (4.11) this implies (4.6).

= p(u).

(4.12)

904

A.J. zA!LAvsKI

We will show that s~p{ll~(O,T,Z?)

- Tp(f)l:T~

(O,=J))
(4.13)

Let T > 0. There exists an integer N 2 0 such that N I T < N + 1. By assumption (iii) If(T,N+

l,Z,)

On the other hand by (4.21, (4.41, (4.51, theorem

2 -a.

(4.14)

1.1 and assumption (iii)

I,(T,N+1,Zr)=Ir(N,N+1,Zr)-Ir(N,T,Zr)~;v(Zf(N),Zf(N+1))+a. It follows from this relations, theorem IZf(0, T,Zf)

- T/df)l

1.1, (4.41, (4.9, (4.14) that

s II’CO, N + 1, Zr) - (N + l)~(f)l +Ilf(T,

N+ l,Z,)

I I $ [ zdzw,

- (N+

1- T)p(f)l

Zf(i + 1)) - P(f)ll

i=O

+I1L~f~I+a+sup~lv~x,y~l:x,y~B~}. Together with (4.6) this implies (4.13). Proposition 1.1.

1.1 now follows from (4.13) and theorem

S.WEAKENEDVERSIONOFTHEOREM2.3 PROPOSITION 5.1. Let g E 3, y : [O,=J) + R” be a (g&good exists T,, > 0 such that for each T r T,,, F > T

P(T,

T, y) s WT,

function and let CE> 0. Then there

T, y(T), y@)) + E.

proof: Let us assume the converse. Then there exist sequences {T&i, that ~ug(T,,Ti,y(~),y(T,))+c Set F’= 0. By proposition (tE[q, T+r],i=O,l...),

<@-i

c (0,~) such

(i=L2...).

3.5 there is an a.c. function x : [0, CO)+ R” such that x(t) = y(t)

Zg(Tj,~,x)=Ug(l;,~,x(~),x(~)),

i=1,2...

It is easy to verify that Zg(O, T, y) -HO, z, x) + + 03 as i + 00. Therefore y is not a (g&good function. The obtained contradiction proves the proposition. n PROPOSRION

5.2. L.et fE a, S,, S, > 0 and let x : [O,m) + R” be an a.c. function

Ix(t)l I S, for all t E [O,m) and Zf(O,i,x) Then x is an (f&good

function.

5 Uf(O,i,x(0),x(i))

+Sj,

i = 1,2...

such that

Dynamic properties of optimal solutions of variational problems

905

Proof: Fix z E R" satisfying ItI I S, and let an a.c. function Zr: [O,m) --) R" be as guaranteed in theorem 1.1. By theorem 1.1 and proposition 3.1 there exists a number Q > supWr(t)l : t E [0, m)) + S, such that lim sup, ~ m ly(t)l < Q for each (f)-good function y : [0, m)

+R".

SetB,=(yER”:lylIQ),u(yl,yZ)=Uf(O,l,yl,yz)

(YIPY, q?)*

By proposition 3.3 the function u : BP x B, + R' is continuous. It was shown in Section 4 that CL(f) = I.&J). Fix an integer N zz 2. By proposition 3.13 there exists a sequence (yJ&, c B, such that y, =x(O), 8”(yi, y,, i) = 0 (0 4 i < N - l), y, =x(N). It follows from these relations, the conditions of the proposition and propositions 3.13, 3.5 that N-l

- /.&)I = ~"(x(0)) - n”(y,+,)

c My,,yj+J

+ dyN-+(N))

- EL(U),

i-0

Ir(O, N, n) - Np(f) I Uf(O, N,x(O), x(N)) + S, - Np(u) N-l s

IS, It follows from these relations n proposition is proved.

I”(Yi9Yi+l)

iFo

+ 2sup{l~“(h)l:

and proposition

-

Au)1

hE

+

sl

Be) + 2sup{ldh,,h,)l:

1.1 that x is an (f)-good

h,,h, E BP}. function.

The

The following result is a weakened version of theorem 2.3. THEOREM 5.1. Assume that f~ ‘8 and there exists a compact set H(f) c R" such that Cl(u) = H(f) for each (f)-good function u : [0, m) + R". Let E be a positive number. Then there exist an integer L r 1 such that for each (f&good function u : [0, m) --t R" dist(H(f),

{u(t):

t E [T,T+L]})

I E

for all large

T.

Proof: Let us assume the converse. Then for every integer N 2 1 there exists an (f)-good function xN : [0, 00) + R" such that lim sup dist(H(f),

{x,(t)

:t E

[T,T + N])) 2 E.

(5.1)

T-tm

By proposition

3.1 there exists M,, > 0 such that for each (f&good lim suplx(t)l

CM,.

function

x : [O, 00) --* R" (5 2)

t+m

By the definition

of MO, propositions

1.1 and 5.1 we may assume that

Ix,Wl
(tE[O,m),N=1,2...),

(5.3)

If(T,,T,,x,k Ur(T~,T,,x,(T,),x,(T,))+N-' (5.4) for each numbers Tl 10, T, > Ti and each integer N 2 1. It is easy to verify that for every (f)-good function x : [0, m) + R" and every number y > 0 the relation d(x(t), H(f)) I y holds

906

A.J. ZASLAVSKI

for ah large t. Therefore we may assume that &x,(t),

H(f))

(tE[O,w),N=1,2...).

s4-L

(5.5)

Let N 2 4 be an integer. By (5.1) there exists 7” E [0, m) such that disttHtf),

{x,(t):t

E [TN,TN +N]})

Together with (5.5) this implies that there exists h, E H(f)

r (7.8-l)~.

(5.6)

for which

dth,,{x,tt):tE[T,,T,+N]))22-1~.

(5.7)

There exists an integer jN 2 0 such that jN I TN
< MJ

(t E [O, N - 2lA

I Ur(T,,T,,v,(T,),v,(T,))

(5.9) + N-’

(5 .lO)

for each numbers Tl E 10, N - 2), T2 E (T,, N - 21, (5.11)

d(h,,{v,(t):t~[O,N-21))22-'~. Fix an integer j 2 1. It follows from (5.9), (5.10) and proposition

3.3 that

sup{Zj(O, j, vN) : N is an integer, N > 4 + j} ~2sup(Uf(O, j, v,(O) ,v,(j)):Nisaninteger,Nz4+j)+l<

+=J.

Together with proposition 3.4 this implies the existence of a subsequence {v,,~,, function v, : [O, w) + R” such that for any integer j 11 vj&)

+ u*(t)

Vhk --) v:

as k+~uniformlyin[O,j], as k + 03wetiy

in L’(R”;

and an a.c.

(5.12) (0, j))

and Zr(O, j, v, ) I liF+5fZf(0,

j, vNk).

(5.13)

(5.9), (5.12) imply that Iv,(t)1 I M,-, for all t E [O,m). By (5.12), (5.13) (5.10) and proposition for any integer j r 1

3.3

ZftO,j,v,)=Uf(O,j,u.tO),v,tj)). Together with proposition 5.2 this implies that v, is an (f&good function. We may assume without loss of generality that there exists (5.14)

h, = lilimhNkEH(f). Since v. is an
proves the theorem.

n

i c (0, m)

Dynamic properties of optimal solutions of variational problems 6. CONTINUITY

OF THE

FUNCTION

907

T,, X, Y)

@(T,,

THECIREM 6.1 Assume that f~ 9I. Then the mapping (T,, T,, x, y) + Uf(T1, Tr, x, Y) is continuousfor T,E[O,~),T~E(T~,~),X,~ER”. proof: LBWU semicontinuity. Let X, y E R”, Tl E [0, W), Tz E CT,, m), {Xirw ,oCyirz 1 CR”, ITlir=,, {Tzirml C[O,~),T~,> Tli (i=1,2*.*.), Xi~X,yi~y,T,ijT~T*i~T2 asi+w. (6.1) thay proposition

3.5 for each integer i 2 1 there exists an a.c. function Zi : [Tli, TziI + R” such Zi(Tli)

By proposition

=Xi> Zi(Tzi) =yi,

Zf(Tli,T~i,Zi)=Uf(Tli,T~i,xi,yi).

(6.2)

3.7 and (6.1) sup{(Uf(T,i,T,i,xi,yi)l:i=

We may assume that there exist limi+Jf(Tii,

1,2...}


(6.3)

Tzi, xi, yj). Fix S E (0,l) and set

T,(S) = sup{O,T, - S}.

(6.4)

There exists a natural number N(6) such that T,+S2T~irT,(6),

for all integers i 2 NCS).

T,--~sT,~IT,+S

(6.5)

Set ~=~~p(lf(h,O)l:h~R”,Ihlssup{(xil+lyil:i=1,2...)).

(6.6)

For every integer i 2 N(S) we define an a.c. function

zgi : [T,(6 ), T2 + Sl + R" as follows

Z,,(t) =Zi(Tli)(t E [TIC6), TliI), Z,(t) =zi(Tzi) (t E lITzipTz+ 6 I).

z,i(t>

(I E IiTli, TziI)r

=Zitt)

(6.7)

By (6.2), (6.3), (6.5), (6.7) the sequence {Zf(T,( a), T2 + S, z~~)~=~(~) is bounded. It follows from proposition 3.4 that there exist a subsequence {z~$=~ {il ziV(S)) and an a.c. function zg : [T,(6), T2 + S] + R" such that ask-+~uniformIyin[T,(6),T,+Sl,

zfji,(t) + Z,tt) Zbi,

j Zb

‘as k+wweakIyin

L'(R";(T,,T,)),

Z~(T,(~),T*+~,Z,)II~~~Z~(T,(S),T~+

S,Zsik)*

(6.8)

By (6.6-6.8), (6.2) for any integer k 2 1 Zf(Tl(6),Tz

+ 6,Z~i~)=Zf(T~(~),T~il,~~i~)+Zf(T~~i,T~i~tzsi~)

+Zf(T~i,,T~+ S,Zgik)I (Tlik- Tl(S))y+ Uf(Tlit,Tzi~,xi,,Yi,) +(Tz + S- Tzi,)rS

Uf(T~it,Tzit,Xi*,yi,>

+4ya*

Together with (6.8) this implies that Z~(T,(~),T,+S,Z,)I

lim Uf(T,i,T*,,Xi,yi)+4rs. j-tm

(6.9)

908

A.J. ZASLAVSKI

By this relation, (6.4) and assumption (iii) Zj(Tl,T,,z,)~

limUf(T,i,T,i,xi,yi)+4ys+2aS. i-m To complete the proof of the lower semicontinuity it is sufficient to show that z,(q)

=x,

z,(T,)

=y.

(6.10)

By (6.8) z,;p1)

+z,(T,),

It follows from (6.1), (6.3), (6.9) and proposition Zu,(Tl)

ask+m.

zaj,(Tz) +Z,(Tz)

-Zsik(Tli,)

3.10 that

* 0, Zai,(T2) -z8i,(T2ik)

Together with (6.11), (6.7), (6.2), (6.1) this relation proved.

+ 0 a~ k + 00.

implies (6.10). The lower semicontinuity

is

Let x, y E R”, Tl E [O, ~1, T, E (T,,m),

Upper semicontinuity.

Tli By proposition

(6.11)

(i=

1,2....),

(6.12)

asi-+m. Xi +x, yi+y, Tli* Tl, T2; + Tz 3.5 there exists an a.c. function z : [T,, T2] + R” such that zU’J

=x,z(T,)

=y,

Zf(Tl,T,,z)

(6.13)

= UfUl,T2,x,y).

Fix 6 E (0,l) and define y, T,(6) by (6.6),(6.4). There is an integer N(6) 2 1 satisfying (6.5). Define an a.c. function zs : [T,(6), T2 + S] + R” as follows z,(r) =x

(f E D’,W,T,l),

z,(t) =y

(t E [T*,T*

+

z,(t) =z(t)

0 E D’,, TJ),

SD.

(6.14)

For an integer i 2 NC S) we set bi= [yi+z,(T,;)-Xi-Z,(T,i>](T,,-T~i)-’, z,;(r) =z&)

+ Q; + b;t

ui =x; -z,(TJ

(rE [TlW,T2

+ SI).

- biTli, (6.15)

Clearly Z,i(Tli)

=xi, Z,i(Tzi) =J’i,

(i is an integer, i 2N(6)),

ai + 0, bi * 0 as i --) a.

(6.16) (6.17)

We will show that Zf(Tl(6),T,+

6,~~;) +Zf(Tl(6),T,

It is easy to see that IZf(T,(6),T,

+ 6,~~)

asi+m.

(6.18)

+ 6,z,)J < w Set

s,=sup(lz,(r)l:rE[T~(S),Tz+61).

(6.19)

Fix E > 0. There exists a number A E (0,l) such that A(]Zf(T,(S), (recall a in assumption (iii).

T2 + 6, zs)l + a(Tz - Tl + 2s)) < 8-b

(6.20)

Dynamic properties of optimal solutions of variational problems

By proposition

3.11 there exist I,, > 0, 6, E (0,l) such that If(hl,u,)

for each q, z+, h,, h, E

-f(h,,u,)l

I Ainf{f(hl,u,),

f&,+)1

(6.21)

R" which satisfy

lhil I So + 4, IUil 2 I?, (i = 1,2), 1111- U*I, Ih, -h,lS Since the function

909

f

s,*

(6.22)

is continuous there exists 8, E (0, S,) such that

If&,uJ -f(h,,u,)l for each h,, h,, q, u1 E R" which satisfy

I W’, - Tl + 2)1-1e

lhil, Iu~I 5 ro + So + 6, i = 1,2, sup{(ht

-hzI,

1~1 -uzl1

(6.23)

I 6,.

(6.24)

By (6.17) there is an integer Nr > N(6) such that for any integer i 2 ZVl lbil~ 2-‘S,, IUi + bitI I 2-‘6,

(t E [zp),T,

(6.25)

+ 81).

Assume that an integer i 2 ZVr.We wih estimate zp(T,(6),T,+6,Z,i)-Zf(T~(G),Tz+6,Z*). Set Ei={tE

[T,(S),T*+S]:l~~(t)l2r,+l},

E, = [T,(S),T,

+ 6l\E,.

(6.26)

We have (6.27)

~~r(~l(S),T~+S,Z*i)-zr(T~(S)~T*+S,z~)I=~l+aZ

where

u,=

/ Ei

If(z~(t),z~(t))--f(z~i(t),z~i(t))ldt,

i = 1,2.

(6.28)

We will estimate or, a, separately. Let t E E,. It follows from (6.261, (6.191, (6.251, (6.15) and the definition of I,,, S, (see (6.211, (6.22)) that

If(Zs(t),~~(t))-f(Z,i(t>,Zbi(t))II

Af(Z,(t),z~(t))*

By this relation, (6.20) and assumption (iii> a,
f(zs(t>,zb(t))dtIA(Z’(T,(6),T2+S,zg)+u(T2-T,+2~))j8-1~. / El

(6.29)

Let t E E,. It follows from (6.26), (6.19), (6.25), (6.15) and the definition (6.24)) that

Iftz,(t)y z;(t)) -f(z,i(t),

of 6, (see (6.23),

zbi(t))l I [NT* - Tl+ 2)I-1 l *

Therefore a, I 8-le. Combining

(6.30)

(6.27), (6.29) (6.30) we obtain that

IZf(T,(6),T,+6,Z,)-Zf(T,(6),T*+ for each integer i r iV1. This implies (6.18)

S,Zai)IlE

910

A.J. ZASIAVSKI

By (6.16), (6.4), (6.5), assumption (iii), (6.18) for any integer i r N( 6) U’(Tli,T*i,Xi,yi)

5Zf(Tli,T2i,Pgi)

~zf(T*(S),T,

+ ‘,‘,i)

+4as

asi+m.

jz’(T,(6),T,+6,z,)+4aS Together with (6.14), (6.4), (6.6), (6.13) this implies that limsup i+m

~~(~,i,T,i,xi,yi)~Zf(~,(S),T,+S,t,)+4a~~Zf(’l;,T,,z)+4a~+2Sy s4a6+2Sy+Uf(Tl,T,,x,y).

This completes the proof of the upper semicontinuity. 7. DIWRETE

TIME

CONTROL

The theorem is proved.

a

SYSTEMS

Consider a continuous function ZI: R” x R” + R’ satisfying as 1x1+ lyl + m. dx,y) + +w We have the following result (Leizarowitz [22, theorem 8.1]). 7.1. Given a compact set C CR” there is a ball B c R” such that for every sequence {z&a c R” not included in B with z,, E C there exists a sequence {s&‘,,, C B with sO= z. such that ~OPosITION

5 u(~k,~k+l) k-0

Let XGR”.

< f

u(zk,zk+l)

for all large N.

k=O

De%e N-l

p(u)=inf

l$iiffN-’

.

(7.1)

= inf lF+kf kgo [u(zk, zk+r) - p(D)]: {zk)I=o CR”, zO =x i e”(x,y) = o(x,y) - ptu> + m”(y) - 7r”(x). CcR” wedefhre uC:CXC+R’ by

(7.2)

i Clearly CL(U) E R’ and is independent

c u(zk,zk+l):

{zk)~,ocR”,

zo=x i

k=O

of x. For x, y E R” we

set

N-l

r”(x)

Foracompactset Propositions ~ROPOS~N.

uC(x, y) = dx, y) (x, y E 0. 3.13, 7.1 imply the following result. 7.2. 1. Given a compact set C c R” there is a ball B 3 C such that

p (uB) = /L(u),

T”(X) = 7/(x),

B”(x,y)

= e%,y)

2 V’(X) (~a), e”(X,Y) 2 e”(X,Y) for every x E C there etists y E B satis@ing W’b, y) = t9%, 2. ~0, 8” are continuous fwnctions, 8” is nonnegative. au”(x)

(x, y E 0,

(x~C,y-1; y) = 0.

(7.3)

Dynamic properties of optimal solutions of variational problems PROPOSITON

911

7.3. a”(X) + +m aS 1X1--) +O”.

proof: There is a number c,, > 0 such that infIu(z,

y) : z, y E Rn,

lzl + lyl r co) 2 41pb)l + 4.

Let XER”,

{z~};-~ CR”,

zo ‘XT

) - p(u)1 s T”(X) + 1. Set E=(kE{1,2,...I:Iz~l~c,}.

Clearly, E#@.Set

r”(x)

+ 12 Iiminf A’+-

m=inf(i:iEE}.

We have

m-l

N

C [u(zk,zk+i)

k-0

-P(U)]

2 C

-

[u(z~,z~+~)

P(U)]

k-0

+ dz,)

2 u(x, z,) - p(u) + inf(r”(y)

: y E R”,lyl

s co} + +m

as Ix) + +w. This completes the proof of the proposition. PROPOSITION 7.4. Let {x$-O

c R”. Then one of the properties below holds:

t Mxk,xk+‘) - cl(u)1+ +a,

asN-++m;

k=O

the sequences (z&-o

and (~~~,,[LJ(x~, xk+,) - ~(u)&,~

are bounded.

Proof: Assume that the first property does not hold. Then for every integer k 2 1

k-l 2 ~“(xJ

+

c

e”(xq,xq+l).

q=o

Hence EJ, o 8 “(xq, x4+ i) < + a~. Together with proposition 8. PROOF

OF THEOREM

7.3 this completes the proof.

n

2.2

Let f~ ?I. It is easy to see that Uf(t, t + T, x, y) = Uf(0, T, x, y) for each x, y E R”, T, r E (0, co). For every T > 0 define a function u{ : R” X R” + R’ as follows u&x,y)=Uf(O,T,x,y)(x,y~R”).

(8.1)

Theorem 6.1 and proposition 3.12 impIy that for any T > 0 the function and u{(x, y) --) +m as 1x1+ lyl + 01. We define p$ = T-b(u), where u = u$ (see 7.1-7.3)).

m{ = d‘,

04 = 8”

u$ is continuous (8.2)

912

A.J. ZASLAVSKI

PROPOSITION

8.1. pf = p(f)

for every T > 0 (recall p(f)

in 0.5)).

Proof Let T > 0. By theorem 1.1 there exists an (fhgood function x : [0, 03) + R”. Proposition 1.1 implies that sup(lZr(O, T, x) - T&f)1 : T E (0, ml) < ~0. By this relation, (7.1), (8.2) N-l

kf I 7-l 1iminfW’ N-m

C u,f(x(ir),

x((i + 117))

i=O

N-l

57-l

1iminfN’ N-+-

IT-*

limj$N-‘Zf(O,

C tY(h,(i

+ 117, x(k),

x((i + 117))

i=O

N7, xl I z@).

(8.3)

It follows from proposition 7.2 that there exists a sequence {yi)~SO c R” such that e,f(yi,yi+l)=O, i=O, l,... By propositions 7.3 and 7.4 N-l

sup

(8.4)

(I By proposition x(h)

=yi,

3.5 there exists an a.c. function x : [0, m) + R” such that Zf(iT,(i+

1)7,X)=Uf(i7,(i+1)7,yi,yi+l)‘U7f(yi,yi+l)

(i=O,l,...).

It follows from these relations, (1.41, (1.5) and (8.4) that N-l

/L(f)<

l$n+it$T-lZf(O,T,x)I

liF&f(rN)-i

C Uf(yi,yi+i)=/J$. i=O

Together with (8.4) this completes the proof of the proposition.

n

Let m(ds) be Lebesgue measure. The following result was established in Leizarowitz and Mizel [7, lemma 4.31. LEMMA 8.1. tit (d,};,, be an increasing sequence of positive numbers such that d, + + m as k + +m. Consider numbers T > 0 with the property that for every p r 1, Ff{d,-iT:k>p,irO,d,riT}=O.

(8.5)

Then there is a set D c LO,00) with m( [O, 03) \ D) = 0 such that every TED every integer p 2 1. By a simple modification of the proof of proposition establish the following result.

satisfies (8.5) for

4.4 in Leizarowitz and Mizel[7]

we can

THEOREM 8.1. There exist a continuous function rf : R” + R’, a continuous nonnegative function (T, x, y) --, @(x, y) E R’ defined for T > 0, x, y E R”, and a set D C LO, + 00) with

dynamic properties of optimd solutions of variational problems

913

m( [0, + 00) \ D) = 0 such that d(x)

= T{(X)

foreveryxER”

d(x)

I T{(X)

for every x E R” and every T > 0;

Uf(0, T, x, y) = T,u(f) + w’(x)

andevery TED;

- rf(y)

+ @(x, y)

for each x, y E R” and each T > 0; forevery T>OandeveryxER”

thereisyER”satisfying

@(x,y)=O.

THEOREM 8.2. For every x E R”

d(x)

= inf( p$f[Zf(O,

-# m

T,u) - p(fWl

: u : [OP)

-+ R” is an a.c. function satisfying u(O) = x) . proof: Let x E R” and u : [0, m) + R” be an ax. function satisfying u(0) =x. By theorem 8.1 and proposition 8.1 it is sufficient to show that &(x1 By proposition

s l$n+i~f[Zf(O,T,u)

- p(f)T].

1.1 we may assume that u is an (f&good

function and

sup{lZ~(O, T, u) - p.(f)TI: T E (O,m)} < ~0, Proposition 3.1 implies that sup{lv(r)l: T E (0,~)) < 00. There exists a sequence of positive numbers {d&i such that dk+ 1 - dk r 1 (k = 1,2,. . .)

= l~~iimnf[Zf(O,T,u)-CL(~)Tl.

lim [Zr(O,d,,u)--ddk&l]

k-rm

(8.6)

By lemma 8.1 there is a set D c (0,~) with m([O, 03) \ D) = 0 such that every T E D satisfies (8.5) for every integer p 2 1. Let TED. It follows from the definition of D and (8.5) that there exists a sequence of positive integers Mj --) TVas j -+ m (j = 1,2,. . . ) and a subsequence {dkjys 1 such that dkj-l Assumption

(j=1,2


,... >,

dkj-MjT+Oasj+m.

(iii), (8.7) imply that for j = 1,2. . . .

Zf(O,TMj,u)-TMjkCL(f)-

[Zf(O,d,j,u)-dkjLL(f)]

s -Zf(TMj,dkj,u)

+I w(f)l(d,, - TMj) s (dkj - TMj)(I k(f)1 + a) + 0 Together with proposition d(x)

(8.7)

I T{(X)

as j+m.

8.1 and theorem 8.1 this implies that

I l$-$f

5 [U%T,

(i + l)T, u(iT),

u((i + UT))

- T/~.(f)1

i=O

I ‘~~~f[Zf(O,TMi,U)-TMjp(f)]

I lipl,i,nf[Z’(O,d,j,U)-d,j~(f)].

The theorem now follows from this relation and (8.6).

914

AJ.

ZAsLAvsKl

8.3. For every x E R” there exists an (f)-good u(0) =x and the relation

THEOREM

Z’U,,T,,u)

= (T, - TJy(f)

function

+ Trf(u(T,))

u : [0, m) + R” such that

- TfhJ(T2))

holds for each Tr E 10, m), T2 E (T,,m). Pro$ Let n E R”. Choose a number T > 0 such that n{(y) proposition 7.2 there exists a sequence (x$‘~~ c R” such that x0 =x,

e~txj,xj+,)

= m’(y)

for all y E R”. By

(i=O,l...).

=0

(8.8)

It follows from propositions 7.3 and 7.4 that the sequence I.$=, is bounded. By proposition 3.5 there is an a.c. function u : [0, W) --) R” such that for every integer i 10 u(iT)

==xi,

Zf(iT, (i + l)T, u> = Uf(iT,

(i + l)T,u(iT),

u((i + 1)T)).

Since the sequence (x$=~ is bounded it follows from (8.9) and propositions sup{lu(t)l:r

E[O,W))

72, u) 2 (72 - 7J/df)

+ Wf(U(T,))

It follows from (8.8) and (8.9) and the definition Zf(iT, (i + l)T,u)

= Tp(f)

3.7 and 3.12 that (8.10)

< 03.

Propositions 1.1, 8.1 and (8.8), (8.9) imply that u is an (f)-good each T] E IO, 4, ?2 E tq,4 Z’h,,

(8.9)

function. By theorem 8.1 for (8.11)

- ~f(U(4.

of T that for any integer i 2 0

+ nf(u(iT))

+ l)T)).

- rf(u((i

H

(8.12)

The theorem now follows from (8.11) and (8.10). 8.4. Assume that there exists a compact set H(f) c R” such that n(x) = H(f) for every (f)-good function x : [O, a) + R”. Let u : [0, QJ)+ R” be an a.c. function such that for each Tl E to, ~1, Tt E CT,,4

THEOREM

Zf(TI,T2,u) Then u is an (f)-weakly tj-)+wasj+~suchthat

= (T2 - TJptf)

optimal

function.

+ rrf(u(T,>) Moreover,

there exists a sequence of numbers

limsup[zf(o,fj,ubzfto,~j,w>~ J.--P

- 7rfb(T2)).

so

00

for each a.c. function w : 10, w) + R” satisfying u(O) = w(0). proof. There is x, EH(~) such that mf(x, ) r mf(z) for all z E H(f). It follows from theorem 8.1, propositions 1.1 and 7.3 that u is an (f)-good function. To prove the theorem it is sufficient to note that there exists a sequence of numbers {tj& 1 such that fj + +a, u(t,)-*x, as j--+00. a Theorem

2.2 now foknvs from theorems 8.3. and 8.4.

91.5

Dynamic properties of optimai sohhons of variational problems 9. PRELIMINARY

LEMMAS

FOR

THEOREM

2.1

The following result was established in Aubin and Ekeland [23, Chapter 2, Section 31. PROFOWION 9.1. Let fi be a closed subset of Rq. Then there exists a bounded nonnegative function + E C’(RQ) such that Jk = {x E Rq : &x) = 0) and for each integer m 2 1 and b&ch sequence of nonnegative integers pi, pz, . . .p,,, the function &‘l$/Jx~. . .%x2 : Rq + R’ is bounded where 1pl = Cy.! 1pi.

Let fe %?I.By proposition

3.1 there exists a number M > 0 such that limsup(u(t)l


(9.1)

u is an tf)-goodfunction}.

(9.2)

t4’p

for each (f)-good

function u : [O, m) + R”. Define 0(f)

= {LItv):

LEMMA 9.1. There exists H* E 0(f)

such that for every D E 0((f)

\ (H*)

D\H*z0. Roo$ Let D,, D, E 0 (f). We will say that D, I D, if and only if D, CD,. We will show that there exists a minimal element of the ordered set 0((f). Consider a nonempty set E co(f) such that for each D,, D2 E E one of the relations below holds: D, zzDzi D, ZGD,. By Come’s lemma it is sufficient to show that there is 6 ~0 (f> such that D c D for any D E E. Set B= l!2ED~ Clearly D # 0. We will show that there exists an (f)-good n(u) CD. For every integer p r 1 there exists DP E E such that

function u : [0, m) + R” such that

dist( D, DP) sp-’ and there exists an (f)-good

function up : [0, m) + R” such that fNu,)

By proposition

(9.3)

= DP.

(9.4)

5.1 we may assume without loss of generality that Zf(T,,T*,v,)

5; Uf(T,,T*,u,(T,),u,(T,))

+ 1

(9.5)

. . . .).

(9.6)

for each p ;i: 1 and each T1 E [0, m), T2 E (T,, co), d( u,(t), D,) sp-’

(tE[O,m),p=1,2

It follows from (9.5), (9.6) theorem 6.1 and proposition 3.4 that there exist a subsequence {uPj~=i and an a.c. function u : [0, m) -+ R” such that for every integer Nr 1 u,, -+ u(t) as j + m uniformly

in [O, N] and

Zr(0, N, u) I Uf(0, N, u(O), u(N))

+ 1.

(9.7)

916

A.J. ZASIAVSKI

It follows from (9.3) and (9.6) that zsposition .

u(t) E b for ah t E [O,m). (9.8) 5.2, (9.7) and (9.8) u is an (f)-good function. This completes the proof of the n

LEMMA 9.2. Let u: [0, co) +R” be an (f)-good function. Then there exists an (f)-good function u : [0, a) + G(u) such that for each T1 E LO,m), T2 E (Tt,m) Zf(T1,T*,u)

= ur(T~,T,,vtT,),u(T,)).

PRX$ We may assume that By proposition

for all t E [O,m). d(u(t),n(u)) 4 1 5.1 we may assume without loss of generality that

(9.9) (9.10)

Zf(T,,T,,u) 5 ur(T~,T,,u(T~),utT*)) + 1 for each I’i E [0, w), T, E (Tl,w). For every integer p 2 1 we set u,(t)

= 240 +p)

By (9.9), (9.10), theorem 6.1 and proposition 3.4 there exist a subsequence (r+$ function u : [0, a) + R” such that for every integer N 2 1 u,(t) ‘I

(9.11)

tt E [OP)).

-+ u(t) as j -+ 00uniformly zfto, Iv, u) I l~+~I’(O,

i and an a.c.

in [0, Nl and (9.12)

N, up,).

Together with (9.11) this implies that (9.13)

u(t) E Q(u) (t E KP)). It follows from (9.11), (9.12), proposition 5.1 and theorem 6.1 that zqo, N, u) = ufto, N,u(O), u(N)) Together with (9.13) and proposition lemma is proved. H

for any integer N 2 1.

5.2 this implies

that u is an (f)-good

By lemma 9.1 there exists H* ~0 (f) such that for every D E0((f) \ (H*} relation holds D \ H* # 0. By lemma 9.2 there exists an (f)-good function wr : [O, m) + H* such that Zf(Tl,T2,w,)

function.

The

the following

= ~~(T,,T,,w,CT,>,w,CT,))

(9.14)

for each T, E [0, 001,T, E CT,, 00). Clearly iI

LEMMA 9.3. Suppose that 4 : R n + [0, 01) is a continuous

H*c{x~R”:4’(n)=O} For r E (0, l] we set gx,

Lo =f(x,u)

(9.15)

= H*.

cH* + r@(x)

bounded function such that

lJ{x~R”:IxlrM+3}. (x,u E R”).

(9.16) (9.17)

917

Dynamic properties of optimal solutions of variational problems

Then f, E ‘$I for all I E (0, l] and for any neighborhood r, E (0,l) such that f, E V for every r E (0, r,,).

% of f in % there e&s

a number

LEMMA 9.4. Suppose that C$: R” --) [0, a~) is a continuous bounded function which satisfies (9.X), r E (0, 11 and a function f, : R” x A* + R’ is defined by (9.17). Suppose also that IJ : [O, 00) + R” is an (f,)-good function. Then a( U) = Z-Z*. Proof Let us assume the converse. Then n(u) #H*. It is easy to see that I = p( and wr is an (f,)-good proposition 1.1 this implies that u is an (f)-good function and sup {IZ~40,

T, u) - T&)1)

function.

< m, sup {IZ’CO, T, 0) - T/L(f)ll

T>O

(9.18) Together with (9.19)

< 03.

T>O

It follows from (9.18) and the definition of H* that there exists a sequence {qr- 1 C (0,~) such that Ti + QJ,u(Tj) *z E Q(o) \ H* asi+m. (9.20) By the definition

of M (see (9.1)) and proposition

5.1 we may assume that

Iu(t)l
(9.21)

for each Tr E[O,m), T2 E (T,,m). We may assume without loss of generality that i= 1,2.... (9.22) dtutl;,), H*) 2 y, T+1 - ‘I 2 10, with a constant y> 0. By propositions 3.10, 3.7 and (9.21) there is SE (0,l) such that ldt,) - u(t,)l I 8-r y for each t,, t, E [0, w) satisfying ItI - t,l5; 6. It follows from the definition of 6 and (9.21), (9.22) that for every integer i 2 1, every tE[Ij:,q+Sl dutt),

H* 12 2-l y.

Together with (9.21), (9.16) this implies that there exists a constant yr > 0 such that for every integer i 2 1 and every t E [q, & + 6 ] 44uO))

2 71.

It follows from this property and (9.19) and (9.17) that zf~to,T,,u)

- T&(f)

This is contradictory

rZftO,T,,u)

- TN/.&(f) +v&N-

to (9.19). The obtained contradiction

1) + +w proves the lemma.

as N+w. n

Lemmas 9.3 and 9.4 imply the following result. LEMMA 9.5. There exists an everywhere dense subset E c ‘8 such that for each f~ E and

each (f)-good

functions ui : [O, w) + R”, i = 1, 2 i-G,)

= ntu,).

918

A.J. ZASLAVSKI

Proposition

9.1 and lemmas 9.3 and 9.4 imply the folIowing result.

9.6. Let k 2 1 be an integer. Then there exists an everywhere dense subset Ek such that for each f~ Ek and each (f&rod functions ui : [O, 00) + R”, i = 1,2

LEMMA

n(q) 10. PRELIMINARY

LEMMAS

(f)-good

a,

= mu,). FOR

THEOREMS

2.3 AND

2.4

Assume that f~ B and H(f) c R” is a compact set such that n(u) -Z-Z(f) (fl-good function u : [0, CQ)+ R”. LEMMA

c

for every

10.1. Let go E (0, l), K,, MO > 0 and let I be a positive integer such that for each function x : [0, Q))+ R” dist(H(f),

{x(t) : t E [T, T + 11)) I 8-l.~~

(10.1)

for all large T (the existence of 1 follows from theorem 5.1). Then there exists an integer N 2 10 such that for each a.c. function x: [O, NZ] --, R” which satisfies (x(0)1, Ix(M)] SK,, IftO, NZ, x) I V’(O, NZ, x(O), x(M)) + MO there exists an integer i, E [0, N - 81 such that dist(H(f), {n(t): t E [T, T + I])) s co for all T E [i,l, Go + 7)1]. proof: Assume that the lemma is wrong. Then for every integer N 2 10 there exists an a.c. function xN : [0, NZ] + R” satisfying

IqJO)l, Ix,woI

SK,,

IftO, NZ, xN) I UrtO, NZ, x,(O), x,WZ)) such that for any integer i E [0, N - 81 there is T(i) E [iZ, (i + 7)Z] for which dist(H(f),{x(t) By proposition

: t E [T(i),

T(i) + Z]}) > co.

+ MO

(10.2) (10.3)

3.2 sup(lxN(t)l:

t E [O, NZI, N is an integer, N 2 lo] < 00.

(10.4)

It follows from (10.2), (10.4) and theorem 6.1 that for any integer j 2 1 sup{Zf(O, jZ, x,) : N is an integer, N 2 sup(j, 10)) < 00. By (10.5) and proposition 3.4 there exists a subsequence x : 10, m) + R” such that for each integer q r 1

(10.5)

{xNJzl

and an a.c. function

I liminfZf(O,qZ,x,,$.

(10.6)

x,(t)+x(t)asi+~uniformlyin[O,qf],x~i+x’asi+~weakIy in L’CR”;

(O,ql)),

Zf(O,qf,x)

i-am

(10.2), (10.6) and theorem 6.1 imply that IftO, qZ, x) I U(O, ql, x(O), x(ql))

+ MO for any integer q 2 1.

(10.7)

It follows from proposition 5.2, (10.7), (10.4), (10.6) that x: [0, m) + R” is an (f)-good function. By the definition of I there exists an integer Q 2 1 such that (10.1) holds for all T E [QZ, w). By (10.6) there exists an integer p r 1 such that Np 2 2Q + 20, IxNp(t) -x(t)1

I 8-‘co

(t E [0, (2Q + 2O)Zl.

(10.8)

919

Dynamic properties of optimal solutions of variational problems

It follows from (10.1) which holds for every T E [Q1, w) and (10.8) that

This is contradictory the lemma. n

to the definition

of xN p (see (10.3)). The obtained contradiction

proves

LEMMA 10.2. Let q, E (0, l), K,, M,, > 0 and let 1 be a positive integer such that each (f)-good function x : [0, w) + R” satisfies (10.1) for all large T. Then there exists an integer N 2 10 and a neighborhood W of f in %!I such that for each g E %‘, each S E [0, w) and each a.c. function x : [ S, S + NZ] --) R” satisfying IXW,

Zg(S,S+NI,x)~Ug(S,S+NZ,x(S),x(S+NI))+M,

Ix(S + WI 5 K,,

(10.9)

there exists an integer i, E [0, N - 81 such that for all T E [S + i,l, S + (i. + 7)Z] dist(H(f),{x(t):tE

[T,T+I]])

Proof By lemma 10.1 there exists an integer x : [O, Nl] + R” which satisfies

Ix(O)I, lxwz)I I K,,

Zr(O, Nl,

(10.10)

I Ed.

N r 10 such that for each a.c. function

x) I Ur(O, N&x(O), x(M)) + MO+ 8

there exists an integer i, E [0, N - 81 such that (10.10) holds for all T E [i,l,(i, ft = sup{lUf(O, M, y, z)l: y, z E R”,

(10.11)

+ 7111). Set

lyl, lrl I 2K, + l].

(10.12)

By Theorem 6.1 .!?< 03. By proposition 3.9 there exists a neighborhood %I of f in ‘?I such that for each g E szl,, each S E [O, w), and each y, z E R” satisfying ly], IzI I 2K, + 1 the following relation holds (10.13) IUf(S,S+NI,y,z)-Ug(S,S+~,y,z)Ij8-’. By proposition 3.8 there is a neighborhood % of f in % such htat Y/C %Y1and for each g E %, each S E 10, a), and each a.c. function x : [S, S + NZ] + R” satisfying inf@(S,

S + M, x), P(S, S + Nl, x)] I 4 + M, + 3

(10.14)

the following relation holds (10.15) IZ’(S,S+M,x)-Zg(S,S+NZ,x)ls8-l. Assume that g E %, S E [0, w) and ax. function x : [S, S + NZ] --, R” satisfies (10.9). By (10.9) and the definition of W1

IUr(S, S + M, x(S),x(S + NO) - WCS, S + NZ, x(S), x(S -I- NO)1 I 8-l.

(10.16)

(10.16) (10.12) and (10.9) imply that I~(S,S+M,X)~U~(S,S+NI,X(S),X(S+NI))+M,+~-’~~++~+~-‘. It follows from (10.17) and the definition Combining

(10.17)

of YL that

IZg(S,S +M,x) -Zf(S,S (10.17) and (10.18) we obtain that

+NZ,x)l

I 8-l.

Zf(S, S + Nl, x) I UfLS,S + NZ, x(S), x(S + NO) + M,, + 4-l.

(10.18)

920

A.J. zAsuvsI
By the definition of N there exists an integer i, E [0, N - 81 such that (10.10) holds for all T E [S + i,f, S + (i,, + 7)1]. The lemma is proved. n LEMMA 10.3. Let .S> 0. Then there exists 6 > 0 such that for each xl, x2 E R” which satisfy d(xi, H(f)) s 6, i = 1, 2 there exists an a.c. function u : 10, TI --) R” for which

Tr 1,

u(O) =x1,

IftO, T, u) - vf(xJ proof: By theorems z2 E R” satisfying

+ 7d(x2) - Tp(f)

6.1, 8.1 there exists S, E (O,inf& dtyi, H(f))

I l,

(10.19)

u(T) =x2, I; E.

t 10.20)

E)) such that for each y,, y,, q,

dtZi9 Htf))

(10.21)

i = 1,2

5 l,

(10.22)

i=1,2

IYi -zil I 6(j, the following relation hold IUf(0, 1, y,, y2) - Uf(0, 1921, 22)/S 8-b,

I~ftyi)

-

Tf(Zi)

5

i = 1,2.

8-l&,

By theorem 5.1 there exists an integer G r 1 such that for each (f&good R” dist(H(f),{u(t): for all large T. By theorem w(0) = 0 and the relation Zf(TI,T2,w)

t

function v : [0, a> + (10.24)

E [T, T + G]}) I 4-‘S,

8.3 there exists an (f)-good = CT2 - T&(f)

+ rf(w(TI))

function

(10.23)

w : [0, a~) --, R” such that

- 7rf(w(T2))

10.25)

holds for each TI E 10, ~1, T, E (T,, 03). It follows from the definition of G that there exists To > 0 such that dist(H(f),

{w(t) : t E [T, T + GI)) I 4-‘S,

10.26)

for any T 2 To. Set S= 8-‘6,. Let x1, x1 E R”, d(x,, H(f))

(10.27)

5 S, i = 1, 2. There exist hi, h2 E H(f)

IXi-hilS6,

such that

i = 1,2.

(10.28)

It follows from (10.26) which holds for any T 2 To that there exist TI E ET,, To + Gl and T2 E [To + lOG, To + llG] such that

Iw(7J) - hiI <4-*s,,

i = 1,2.

(10.29)

There exists an a.c. function u : [0, T2 - TI] + R” such that 40) =x1,

u(t) = wtt + T,)

tc E [l, T, - Tl - II),

Zf(0, 1, u) = Uf(O, 1, u(O), u(l)), Zf(T2 - TI - l,T, - T,,u) = Uf(T, u(T2 - TIN.

- TI - l,T,

- T,,u(T,

VU, - T,) =x2,

- T, - 11,

(10.30)

’

921

Dyuamic properties of optimal solutions of variational problems

by the definition

of 6, (see (10.21-10.23))

(Uf(O,l,u(O),La)) luf(T,

and (10.301, (10.2% (10.27) ad (10.29)

- U’(T,,T,

- Tl - l,T, -Uf(T,

+ l,w(T,),w(T,

- TI,v(T,--

- l,T,,w(T,

+ 1111s 8%

TI - lh.0,

- TIN

- l),w(T,))I

I8-‘8.

- rrf(q)

+ mfb,)

(10.31)

It follows from (10.30) that Zf(0, T2 - T,, u) - (T2 - TI)p(f)

= [ Zf(O,l,v)

- p(f)

- wf(x,) + &J(l))]

+ [ Zf(l, T2 - Tl - 1, U) - CT2 - TI - 2)p.(f)

- mf(dl)) (10.32)

- T, - l))]

+ ~rf(v(T,

+ [Zf(T2 - Tl - l,T,

- wf(v(T,

- T,,u) - p(f)

+ afMT,

- T,))

- TI - l))].

(10.30) and (10.25) imply that Zf(l,T,

- Tl - 1,~) - p(f)(T,

- TI - 2) - ~r~(u(l))

+ dMTz

- T, - 1)) = 0. (10.33)

By the definition of 6, (see (10.21-10.23)) and (10.28), (10.27), (10.29) and (10.31) l$w(Q - 7rf(xJ s 8- lo, i = 1, 2. Together with (10.30), (10.31) and (10.25) this implies

lUf(O, 1,uKo,dl)) 1, w(T,), w(T, + l))l+ brf(w(T,)) - rf(x,)l

lZf(0, 1, v) - 7rfCxJ + 7rf(v(l))

-Uf(T,,

Tl +

+IUf(T,,T,

- p(f)1 s

+ l,w(T,),w(T,

+ wf(w(T, + 1)) - p(f)\

+ 1)) - mfMTIN I 4-b,

IZf(T2 - Tl - 1, T2 - TI,u)

- rf(v(T,

- TI - 1)) + mf(v(Tz - T,)) - p(f)1

I IUf(T, - Tl - 1, T, - T,, uCT2 - TI - l),v(T,

-Uf(T,

- mfMT,

w(T,)) - rf(w(Tz Combining

- l),w(T,))l

- l,T,,w(T,

+I~rf(w(T~))

- TIN

- T,NI + IUf(T, - l,T,,w(T,

- 1)) + rrf(w(T&

- &)I

- l),

I 4-b.

these. relations and (10.32) and (10.33) we obtain that 0 rZf(O,T,

- T,,v) - OFF

This completes the proof of the lemma. LEMMA 10.4. Let E E (0,l) function u : [0, m) + R”

+ rf(x,)

- CT2 - Z&(f)

I E.

n

and let L be a positive integer such that for each (f)-good dist(H(f),

{v(t) : t E [T, T + L])) I E

for all large T (the existence of L follows from theorem 5.1).

(10.34)

AJ. ZASLAVSKI

922

Then there exists 6 > 0 such that for each T E [L, 00) and each a.c. function u : [0, Tl + R” which satisfies dMO),

H(f))

d(dT),

5 6,

Zf(0, T, u) - Tp(f)

- ~~(~40))

H(f))

(10.35)

s 6,

+ d(dT))

(10.36)

r; 6

the relation disHHi(

(10.37)

{v(l) : t E [S, S + L]]) I E

holds for every S E [O, T - L]. Proof: By lemma 10.3 for every integer k 2 1 there exists (10.38)

6, E (0,2-V such that for each xi, x2 E R” which satisfy d(Xi, H(f))

i=1,2

s Sk,

(10.39)

there is an a.c. function u : [0, T] + R” for which Tz 1,

40) =x1,

Zf(0, T, u) - &xl)

u(T) =x2,

+ d(xZ)

- Tjdf)

s 2-k. (10.40)

Assume that the lemma is wrong. Then for every integer k 2 1 there exist Tk 2 L, an a.c. function vk : [0, Tk] --f R” and Sk E [0, Tk - L] such that h,(O), zf(o,T,,uk)

H(f))

s Sk,

- T&(f)

dist(H(f),

d(@,),

- dV,(O))

H(f))

(10.41)

< a,,

+ T%k(Tk))

(10.43)

{Uk(t) : t E [Sk, Sk + L]]) > 8.

It follows from (10.41) and the definition of {a,h-, (see (10.38-10.40)) k 2 1 there exists an a.c. function uk : [0, TV]--, R” for which 7k2 1,

(10.42)

< a,,

ukbk) = uk+ l(o),

u,(o) = u,(T,),

If@, Tk, uk) - d&(o))

that for each integer

+ d(t‘khk))

(10.44)

- 7k ,.‘(f) 5 2-k.

By (10.44) there exists an a.c. function u : [O, m)+ R” such that u(t) = u,(t) (t E [O, T, I),

u(t) = I.#

- TJ

(TV [T&T,

+ ql),

k-l C(q+Ti>,

C i=l

9

(K+Ti)+Tk

k-l

U(t)=Uk

t-

C (Ti+7i)+Tk, i=l

i-l

~(T,+~i~

)

k=2,3....

i=l

(10.45)

Dynamic properties of optimal soh~tions of variational probIems

923

Set T,,, 7. = 0. By (10.49, (10&t), (10.42) and (10.38) for any integer k z 2

\

i=l

\

\i=l

i-l

q-1 =

i

[Ifiqi

q=lL

(Ti+Ti)3

q-1

C i=O

\ i-0

(q+Ti)+Ta,U

- 9rf

.

u

I’(

C i i=O

(rrl,+Ti)

(q-1

q-1

is (I;. + 73 + Tq)) -df)Tq+

C (Ti+7i)+Tq,

(Ifi

~(T,+‘~),’

I i=O

i-0

q-1

~(~+~i)+Tq))+*f(U(~o(~+il)))-I.(f)Ts] i=O k

= c [Zf(O,Tq,uq) - T~(u~) +rf(uqUq))- T,p(f) q=l

+rf(o,Tq,Uq)

-

Tf(U,(O))

+

mf(Uq(7q))

-

Tq/L(f)]

<

;

(6,

+

2-4) I4.

q=l

It follows from this relation, proposition 1.1 and (10.45), (10.44) and (10.41) that u is an (f)-good function. By the defkrition of L (10.34) hoIds for all large T. On the other hand by (10.45), (10.43) for every integer k 2 2 (I~:+T~)+S~+L The obtained contradiction

proves the lemma. 11. PROOF

>E.

n

OF THEOREM

2.4

Assume that f E 9I and there exists a compact set H(f) c RR such that n(u) = H(f) for each (f&good function u : [0, a~) + R”. Let MO, Al,, .c > 0. We may assume without loss of generality that Ml > 4 + sqml:

y EMf)),

EC 1.

By theorem 5.1 there exists an integer L 2 1 such that for each (f&good R” dist(H(f),{v(t)

(11 .l) function ZJ: [O, 00) + (11.2)

: r E [T, T + L])) I 4-k

for all large T. By proposition 3.2 there exist a neighborhood go of f in $?I and S > MO + Mi such that for each g E go, each Ti E [0, w), T2 E [Ti + 1, w) and each a.c. function u : [T,, T2] + R” satisfying IUUJ

s Ml,

i = 1,2,

ZgtTl,T2,d

s UgtT,,T,,u(T,),u(T,))

+Mo

(11.3)

924

A.J. ZASLAVSKI

the following relation holds

(t4TJJ).

Iu(t)l IS

(11.4)

By lemma 10.4 there exists a number 6, E (0,8-k)

(11.5)

such that for each T E [L, m) and each a.c. function u : [0, T] + R" satisfying d(U(7), fnf))

~'0, T,

s so,

If(O,T,v)-

Tp(fb

d(dO))+

d(u(T))s

6, (11.6)

the relation dist(H(f),{u(t):tE

[7,T+L]})

14-l&

(11.7)

holds for every T E [0, T - Z,]. Fix a natural number Q. >U)+2(M,+4+2sup{lUf(0,1,x,y)l:x,y~R”, + 2 sup[llr’(x)l:

Ixl,lyI~S)

x E R", 1x1I; S) + 21 ~~tf)l)S;~.

(11.8)

s, E (0, fr’S,)

(11.9)

By theorems 6.1 and 8.1 there exists such that for each yr, yz, zr, z2 E R" satistjing d(y;, H(f))

s 4,

d(Zi, H(f))

54,

Iyi -ql s 84,

i = 1,2

(11.10)

the following relations hold IUr(O,l,y*,yZ)-

u~(o,1,z,,z~)I~(16Q*PS0,

laf(yj) - d(Z,)l s (16Q. 1-l 60, By theorem

5.1 there exists an integer

i = 1,2.

L, 2 1 such that for each (f&good

(11.11) function

u: [O,m)-,R" [T,T+L,l}k8-'6,

dist(H(f),(u(t):te

(11.12)

for all large T. We may assume without loss of generality that

L,28L.

(11.13)

By lemma 10.2 there exists an integer IV2 10 and a neighborhood

(11.14)

%‘1CO

and for each g E %r, each T E [0, m), and each a.c. function satisfies

??ZIof f in VI such that

x : [T, T + NL,] + R" which

Ix(T)I, Ix(T + NLJI s S, I~(T,T+NL,,x)~Ug(T,T+NL,,x(T),x(T+NL,))+Mo+4

(11.15)

there exists an integer i, E [O, N - 81 such that dist(H(f),

{x(t) : t E [T, T + L,])) I 6,

(11.16)

Dynamic properties of optimal solutions of variational problems

for all T E [T + i,L,,

T + (i,, + 7)L,]. Set R",d(x,H(f))

D,,=sup{ld(xh~

By propositions

925

1=5ONL,.

~41,

3.8 and 3.9 there exists a neighborhood

(11.17)

of % of f in ‘?I such that (11.18)

&ICZ,

and the following properties hold (i) for each g E %, each Tr E [0, m), T, E [Tr + 1, T1 + lOON(L, u : [T,, TJ + R" which satisfies

+ I) and each a.c. function

infIZf(T~,T,,u),Zg(T,,T,,u))140,+8+200NL,ICl(f)I

+2sup~lUf(0,1,x,ykx,y~R”,Ixl,IyI~S~

(11.19)

the following relation holds (11.20)

IZf(T,,T,,u)-Zg(T,,T,,u)ls(64Q.)-’6,; (ii) for each g E ?A, each Tr E [0, m), T, E [Tr + 1, Tl + lOON(L, satisfying lyl, Izl < 2s + 8 the following relation holds

+ Z)] and each y, z E R"

I[lf(T~,T2ry,~)-Ug(T~,T2,y,~)I~(64Q.)-161. LA gE’%(, T,E such that buy By the definition

[O,m) T,E[T~+L+ZQ,,

SM,,

i = 1,2,

(11.21)

m) and let u : [T,, T2] + R” be an a.c. function Zg(Tl,T2,u)

I Ug(T~,Tz,u(T,),u(T2))

+M,

(11.22)

of S, %. (11.4) holds. Set

8={h~R~:T~+lONL~1h1T~-lONL~,dist(~(f),{u(t):t~[h,h+L]})>~).

(11.23)

If 8 = 0 then theorem 2.4 is valid. Hence we may assume that I # 0. We set i;=inf{h:hE8) and choose h, E 8 satisfying h, I h + 4- ‘. It follows from the definition (11.14-11.16)) that there are integers i,, i, E (0,. . . N - 8) such that dist(ZZi(f),Mt)

: t E [T,T + L,]])

of gl,

(B) NL, I cq - b4 I 4NL,,

[b,,c, -NL,lfl8+

0,

b4 >c,-,

(11.24)

I S,

for any number T E [h, - (2N - i,)L,, h, - (2N - i, - 7)L,]U[h, + (N + i,)L,, (N + i, + 7)L,]. Set b, = h - 2NL, + i,L,, cl = h, + NL, + i,L,. By induction we define a sequence of intervals [b,, c4] (q 2 1) such that:

N (see

h, t

if 4 2 2,

81 fi [bj,CjIC(Cq,T2];

(11.25)

j=l

Clearly for q = 1 properties (B) and (C) hold. Let k be a natural number. Assume that a sequence of intervals {[ b4, c,]},k_ r was defined

!m

AJ. ZASIAVSKI

and properties (B), (0 hold for q = 1,. . . k. If 81

I; [b,,c,]=B q-1

then the construction of the sequence is compkted and [bk, ck] is its last element. Let us assume the converse. Choose h,~8\ U&,{b,,c,] such that

It follows from the definition (11.24) holds for every

of al, N that there are integers ji, j2 E [0, N - 81 such that

TE[~,-(~N-~,)L,,~,-(~N-~,-~)L,IU[~,+(N+~,)L,,~,+(N+~,+~)L,I. Set ck+ 1 = h, + NL, + j2L1, bk+l = SUp(ck, h, - 2NL, + j,L,}. It is easy to verify that properties (BJ, (0 hold with q = k + 1. Evidently the construction of the sequence will be completed Let [be, ccl be the last element of the sequence. Clearly

in a finite number of steps.

Q

8~ U [b,,c,l.

(11.26)

q=l

By theorem 8.3 there exists an (f>good each TI E [0, co), T2 E (T,, m) I’(T,,T,,w) By the definition

function w : [0, a) + R” such that w(O) = 0 and for

= (T2 - T&(f)

+ d(w(T,))

- d(w(T,)).

(11.27)

of L, (see (11.12)) there exists 70 > 0 such that for alI T 2 T,, dist(H(f),

{w(t) : t E IT,T

Let i E {l,.. .Q]. It follows from properties definition of S, (see (11.5-11.7)) and I that Zf(b,, c;, u) - d(u(b,)) It follows from (11.291, properties (11.19) and (11.20)) that

+ L,]})

(B), (C), (11.13), (11.9) and (11.24) and the

+ dMci))

- (ci - b&(f)

+ mf(u(c,))

- (ci - b,)p(f)

By property (C) and (11.24) and (11.28) there exist t(i,l)E 2L,, t(i, 1) + 3L,] such that 1))

> 6,.

(B), (C), (11.7) and (11.4) and the definition

Zg(bi,ci, v) - &Mb,))

sup{lw(t(i,

(11.28)

I 8-‘6,.

-

U(bi)ly

Iw(t(i,2))

-U(Ci)l

> 6, -W’S,,.

(11.29) of % (see (11.30)

[T,,, m) and t(i,2) E [t(i, 1) + I2Si.

(11.31)

set t(i,3)=t(i,2)-t(i,l).

(11.32)

927

Dynamic properties of optimal soh~tions of variational problems

By proposition

3.5 there exists an a.c. function vi :[t(i, l),t(i,2)1+ uiw = w(r) (t E [rti, 1) + l,t(i,2)

UiCrti, l)) = Utbi), Zf(t(i, l),t(i, Zf(r(i,2)

1) + l,Ui) = V(r(i,

- l,t(i,2),ui)

l),t(i,l)

= Uf(r(i,2)

-

+ l,L$(t(i,

R” such that

111,

1)),

- l,r(i,2),ui(r(i,2)

- d(u,(r(i,l)))

=Zf(r(i,l),t(i,l)

- l),u,(r(i,2))).

- l,r(i,2),ui)

I IUf(r(i,l),r(i, -Uf(r(i,

+ 7rf(uj(t(i,2)))

+ l,v,) - d(u,(r(i,l)))

+Zf(r(i,2)

l)), q(r(i,

l), r(i, 1) + l,w(r(i,

-r(i,3)p(f)

- CL(f)

1) + 1))

l)),w(r(i,

1) + 1))I

1))) - &(U,(t(i,

+lUf(r(i,2)

- l,r(i,2),ui(r(i,2)

- l),U,(t(i,2)))

-Uf(r(i,2)

- l,r(i,2),w(r(i,2)

- l),w(rti,2)))1

-

+ 1)) -P(f)

- 1)) + 7rf(ui(f(i,2)))

+Id(wMi,

+Id(w(r(i,2)))

(11.33)

of 6, (see (11.9-11.11))

+ 7d(q(r(i,l)

- lrf(ui(r(i,2)

1) + 1, U,(t(i,

= utCi),

ui(r(i, 1) + 1))~

It follows from (11.33), (11.31), (11.27) and (11.28) and the definition that Zf(r(i,l),r(i,2),q)

Uittti,2))

1)))l

d(Ui(f(i,2)))l

(11.34)

s(4QJ’S,. Property (C) and (11.34), (11.33) and (11.17) imply that Zf(r(i, l), r(i,2), ui) I 3&I p(f)1 + 20, + 1. By induction

(11.35)

we define sequences of numbers <&}g 1, CEi>ipIr as follows &I = b,,

Zi =Si + r(i,3)

(i= l,...Q),

&i+l

=bi+I

+q-ci

(i is an integer, i s Q - 1).

(11.36)

Define an a.c. function u : [T,, i’J + R” as follows u(r) = u(r) (r E IT,,

b,l),

u(r) =u,(r -Si +r(i,l)) u(r) = u(r - zi + ci))

(rE[&,zj],i=l

,... Q),

(rE [ri,&+,],isQ-1).(11.37)

By property (B), (11.23) cQ I T2 - 6NL,,

zp=cp-

ffJ(ci-bi-r(i,3))~

[cO-~QNL,,~~-(N-~)QL,].

i-l

(11.38) It follows from the definition

of U, % (see (11.19-11.21)),

(11.36), (11.37X (11.35) and (11.34)

928

A.J. zAsL4vsKI

that for i = 1,. . . Q zg(I+, Ei, u) - 7rf(u(&.)) I Z’(t(i,

l), t(i,2),

+ mf(ui(t(i,

+ d(UGii))

- (Ci -b&df)

Ui) - 7rf(Ui(t(i,

2))) - t(i, 3)p(f)

1))) + (6412, I-’ 61 (11.39)

s(6L1Q,)-‘6,+(4(2.)-‘6,. There exists an a.c. function zq, : [T,, TJ + R" such that u,(t) = u(t)

(=

u,(t) = w( t - CQ + t(Q, 2))

[w*])9

u,(t) = u(t)

0 E [cp,Tzl),

Zf(h,h + l,u,)

= UWz,h

+ l,u,(h),u,(h

+ 1))

(t E [CQ + 1, CQ- ll),

(h E EQ,CQ - l]).

(11.40)

It follows from (11.40), (11.37), (11.33), (11.28), (11.4) and (11.1) that (11.41)

lup(r)l~s(t~{Zp,~p+l,Cp-l,cp}). By this relation, the definition

of YY(see (11.19-11.21),

(11.40))

Z~~h,h+1,u,~r;sup(lU’~0,1,x,y~l:x,y~R”,Ixl,lyl~S~+~~Q~~-‘~, (h E &pCQ

(11.42)

- 11).

It follows from the definition (11.17) that

of % (see (11.19) and (11.20)) and (11.40), (11.27), (11.28) and

IZg(t,,t,,u,) - af(u,,(t,)) + d(u&)) - (t2 -t,)/.df)l s 64Q, )-%3, (11.43) for each t, E [ze + 1, m) and each t, E (-m, cp - l] which satisfy 1 < t, - t, I lOONL,. It follows from (11.40), (11.37), (11.42) that

+Zg(~~+l,Cp-ll,U~)+Zg(~Q,~Q+l,U~)+

~ [Z’(~i,Ei,u)-Z’(b,,C,,u)] i=l

I it1 [zpi,Ci,u)

-Z’(b,J,,U,]

+zgGQ + l,c,

+2supWf(0,1,x,ykx,y~R”,lnl,ly14 By this relation, (11.39), (11.33), (11.37), (11.36), (11.30)

Zg(Tl,T,,u,)-Zg(T,,T,,u)

- l,u,) +(32Q,)-‘6,.

sZg(Cp+ l,cp-1,~s)

+2supW’(O,1,x,y)l:x,y~R”,Ixl,lyl~S~ Q +i~l[S,(64Q.)-*+(4Q*)-16,+P(f)(t(i,3)-ci+bi) 4,+64-‘s,].

+(32Q,)-‘6,

(11.44)

Dynamic properties of optimal solutions of variational problems

929

Set ti=~,+l+iQ-‘(c,-c,-2)(i=O,...Q). (11.40), (11.27), (11.41), (11.43), (11.38) imply that Q-1 IgCcQ + l, cQ - l, uo) 5 C [ Ig(ti, Ci+l, ug) - ?Tf(UO(ti)) + Tf(U()(ti+l>>] i=O

+ 2 sup{lTf(x)l: + P(f)(CQ

x E R”, 1x1I S} 5 2 sup(l?7f(x)l:

- z, - 2) + QS,WQ*

x E R”, 1x1I S)

)-‘.

By this relation and (11.44), (11.22), (11.40), (11.37), (11.38) -M,~z~(T~,T*,Uo)-z~(T~,T~,u)I2sup{l~f(x)l:xER”,Ix(IS} +2supWf(0,1,x,y)l:x,ydt”,lxl,lylsSJ + Q[ 6,WQ,

1-l + 6,(4&,

I-’

+(32Q.)-‘6, + SOW-’

- ao] + QS,WQ.

)-’ + 2l~lCf)l.

It follows from this relation and (11.8) and (11.9) that Q + 20 I Q, . This completes the proof of the theorem. n 12. PROOF

OF THEOREM

2.3

Assume that f~ ?I and there exists a compact set H(f) c R” such that n(u) = H(f) for each (f)-good function u : [0, @J)--, R”. Let E > 0. By proposition 3.1 there exists a number Mi > 0 and a neighborhood %i of f in !!I such that for each g E %i and each (g&good function u : [0, m) + R” limsuplu(t)l -CM,. (12.1) t-m By theorem 2.4 there exist a neighborhood F?/ of f in ?I, a number I > 0 and integers L,Q,~lsuchthatV~~~aandforeachg~%Y,eachT,~ [0,m),T,~[T~+L+LQ~,m)and each a.c. function v : [T,, T,] + R” which satisfies i= 1,2,

luq;:)l IM,,

Zg(TI,T,,u)

I Ug(T,,T,,u(T,),u(T,))

+ 1

(12.2)

there exist sequences of numbers {b,}E i, {c,}g i c [T,, T2] such that

QsQe,

l,...Q),

(12.3)

{u(t) : t E [T, T + L])) I E

(12.4)

Osci-bisI

dist(H(f),

(i=

for every T E [T,, T, - L] \ U f?=Jb,, ci]. Let g E % and u : 10, m) + R” be a (g)-good function. It is sufficient to show that (12.4) holds for all large T. Let us assume the converse. Then there exists a sequence {tJ;= i c (0, 03) such that ‘i+l

- ti 2 41+ 4,

By the definition

dist(H(f),{v(t)

: c E [ti, ti + L])) > E,

of gl,, M1 (see (12.1)) and proposition Iu(t)l
i=l,2...

(12.5)

5.1 we may assume that (12.6)

(t E [OP)),

I Vg(T,,T,,u(T,),u(T,))

+ 1

(12.7)

930

A.J. ZAsLAvsKl

for each T, E [0, a~), T2 E (T,,m). Fix an integer T>(L+Z>Q,

(12.8)

+tzg,+r+4L+4.

It follows from (12.6), (12.7) and (12.8) and the definition of %, 1, L, Q* that there exist sequences of numbers (&]E r,
j),

p(i) =pt

tj -

ti 5 Cp(i)

to (12.5). The obtained contradiction 13. PROOFS

OF THEOREMS

- bp(i)

5 1.

proves the theorem. 2.1 AND

n

2.5

Proof of theorem 2.1. By Lemma 9.5 there exists an everywhere dense that for each f E E the following property holds: there exists a compact that n(u) = H(f) for every (f )-good function u : [0, W) + R”. Let f E E and p 2 1 be an integer. By theorem 2.3 there exist an SY(f, p) of f in ‘?I and an integer L( f, p) 2 1 such that for each g (g&good function u : [0, 03) --t R”

subset E in ‘?I such set H(f) c R” such open neighborhood E 2&f, p) and each

for ail large T.

dist(~i(f),Iu(t):tE[T,T+L(f,p)l))Ip-’

Set .F= fl~-lUfEENf,p). Let g ET and ui : [0, a~) --) R”(i = 1,2) be a (g)-good functions. Let E E (0,l). There exist an integer p 2 1 and f E E such that p > 8&-l,

It follows from the definition dist(Hi(f),fl<~i>)

i?E Nf,P). of %( f, p), L( f, p) that
i=l,2,

This completes the proof of the theorem.

dist(CI(u,),Nu,>>

I 2p-’

< E.

n

By a simple modification of the above proof (basically by replacing lemma 9.5 by lemma 9.6) we can easily establish theorem 2.5. 14. EXAMPLES

Fix a constant a > 0 and set t)(t) = t (t E [0, ~1). Consider the complete metric space %!Iof integrands f : R” X R” + R’ defined in Introduction. Exumpk 1. Consider an integrand f(x,u) = /xl2 + lu12, x, u E R”. It is easy to see that f E ‘?I. Applying the methods used in the proofs of theorems 2.1-2.4 we can show that MU) = (0) for every (f )-good function u : [0, 00) + R”. Exumpfe 2. Fix a number q > 0 and consider an integrand g(x, U) = qlx121x - e12 + lu12(x, u E R”) where e = (l,l,,,l) E R”. It is easy to see that g E !!I if the constant u is large enough.

Dynamic properties of optimal solutions of variational problems

931

Clearly the functions q(t) = 0, u&) = e (t E [0, 03)) are (g&good and g does not have property (B) (see theorem 2.1). AcknowkdgemeeTbe author thanks A.D. Ioffe, A. Leixarowitx and VJ. Marcus for valuable suggestions.

Mid

for

helpful

d&l&ons

ad

M.

REFERENCES 1. ANDERSON B.D.O. & MOORE J.B., Lincpr @timal Cont&. Prentice-Ha& Englewood Cliffs, NJ (1971). 2. ARTSTEIN Z. & LEJZAROWITZ A, Tracking periodic signals with overtaking criterion, ZEEE Trans. autom. Control AC 30, 1122-1126 (1985). 3. ROCKAFELLAR R.T., Saddle points of Hamiltonian systems in convex problem of Lagrange, 1 Optim Theor. Apple. l2,367-389 (1973). 4. CARISON D.A., The existence of catchina-un optimal solutions for a class of in6nite horizon optimal control problems with time delay, SLQMJ. Control-?.&im.~Ztt, 402-422 (19901. 5. AUBRY S. & LE DAERON P.Y.. The discrete Frenkel-Kontorova model and its extensions I. Exact results for the ground states, Physica IID, 381-422 (1983). 6. ZASLAVSKI AJ., Ground states in Frenkel-Kontorova model, Math. USSR Zzestiya 29(2), 323-354 (19871. 7. LEIZAROWITZ A. & MIZEL VJ., One dimensional iniinite-horizon variational problems arising in continuum mechanics, Arch. r&on. Mech Analysis 1@6,161-194 (1989). 8. COLEMAN B.D., MARCUS M. & MIZEL V.J., On the thermodynamics of periodic phases, Arch rarion. Me&. Analysti 117,321-347 (1992). 9. GALE D., On optimal development in a multisector economy, Reu. &on. Studies 34,1-19 (1967). 10. VON WEIZSACKER C.C., Existence of optimal programs of accumulation for an infinite horizon. Reu. Econ. Studies 32,85-104 (1965). 11. BROCK WA. & HAURIE A., On existence of overtaking optimal trajectories over an inlinite horizon, Math. Optim.

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