JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 78, No. 3, SEPTEMBER 1993

Nonconvex and Relaxed Infinite-Horizon Optimal Control Problems D, A.

CARLSON 1

Communicated by T. S. Angell

Abstract. In this paper, we consider the Lagrange problem of optimal control defined on an unbounded time interval in which the traditional convexity hypotheses are not met. Models of this form have been introduced into the economics literature to investigate the exploitation of a renewable resource and to treat various aspects of continuous-time investment. An additional distinguishing feature in the models considered is that we do not assume a priori that the objective functional (described by an improper integral) is finite, and so we are led to consider the weaker notions of overtaking and weakly overtaking optimality. To treat these models, we introduce a relaxed optimal control problem through the introduction of chattering controls. This leads us naturally to consider the relationship between the original problem and the convexified relaxed problem. In particular, we show that the relaxed problem may be viewed as a limiting case for the original problem. We also present several examples demonstrating the applicability of our results. Key Words. Infinite-horizon optimal control, overtaking optimality, economic growth, chattering states. 1. Introduction Infinite-horizon optimal control problems are used in mathematical economics to model economic growth. The models considered are formulated as Lagrange problems of optimal control in which the cost or objective functional is described by an improper integral. Examples have shown that often this improper integral diverges. Consequently, the usual notion of minimizing the objective functional is meaningless. This difficulty is overcome by considering one of several weaker concepts of optimality

1Professor, Department of Mathematics, University of Toledo, Toledo, Ohio. 465 0022-3239/93/0900-0465507.00/0 © 1993 Plenum Publishing Corporation

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that have been formulated within the economics literature. A detailed introduction and analysis of these notions is found in the monograph of Carlson, Haurie, and Leizarowitz (Ref. 1). In all the results reported in Ref. l, the authors assume at least the usual convexity and seminormality conditions that have been used in establishing the existence of optimal solutions for optimal control problems defined on finite intervals. In this paper, we consider models in which the usual convexity hypotheses do not hold. To do this we formulate the relaxed problem through the introduction of chattering states. For the case of a finite interval, this relaxed problem has been discussed by many authors [see, e.g., McShane (Ref. 2) or Cesari (Ref. 3)]. For the infinite-horizon case considered here, this problem has been discussed very little and apparently has only been considered in the economics literature. We refer the reader to the survey paper of Lewis and Schmalensee (Ref. 4) for a discussion of these results and mention only that they have noted, for the models which they consider, that the existence of optimal solutions has not been rigorously established. The plan of this paper is as folllows. In Section 2, we formulate the nonretaxed model, along with the basic hypotheses assumed throughout the rest of our work. In addition, we give precise definitions of the concepts of optimality that we consider here. In Section 3, we formulate the relaxed problem and present two existence results for these problems. In Section 4, we extend a theorem of Gamkretidze (Ref. 5) that provides conditions under which a relaxed trajectory can be approximated by a sequence of admissible trajectories for the nonretaxed problem in the compact-open topology. We conclude this section by using this approximation result to investigate the relationship between the optimal solutions of the original problem of interest to the optimal solutions of the relaxed problem. In Section 5, we present some simple examples to which the results of Section4 may be applied, and we make some concluding remarks in Section 6.

2. Nonrelaxed Problem The problem that we consider is the Lagrange problem of optimal control defined on [0, + ce). Specifically, we consider the problem of minimizing an integral functional J, described by an improper integral of the form

J(x, u ) =

g(t, x(t), u(t)) dt,

(t)

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over all pairs of functions {x, u}: [0, + ~ ) - ~ R " x R "~ satisfying a control system of the form

2(t)=f(t,x(t),u(t)),

a.e. on [0, +oo),

x(0) = x o,

(t,x(t))sA, u(t)eU(t,x(t)),

(2a) (2b)

on [-0, +oo), a.e. on [0, +oo),

(2c) (2d)

where we assume that A c [0, + oo)x N" is a given closed set, U: A ~ 2 ~m is a set-valued mapping which is nonempty for each (t, x ) e A and has closed graph

M = {(t, x, u):(tcA, u6 U(t, x)}, g: M ~ N is a given real-valued function, and f : M --* R ~ is a given vectorvalued function. We require the function f and g to satisfy at least the regularity requirements given in the following definitions. Definition 2.1. (i) A given function f : M - * ~" is said to satisfy the Carath~odory conditions if f ( . , x,u) is Lebesgue measurable on [0, + ~ ) for each fixed (x, u), (t, x, u) s M, and if f(t, .,.) is continuous for almost all t/> 0. (ii) A function g : M ~ ~ is called a Lebesgue normal integrand if g(t,., .) is lower semicontinuous for all t>~0 and if g is a measurable function with respect to the a-algebra generated by products of Lebesgue subsets of [0, + ~ ) with the Borel measurable subsets of ~n+m The pairs of functions {x, u} over which we minimize are referred to as admissible pairs. As indicated in Section 1, we will not necessarily assume that the functional J is finite. Therefore, for a given admissible pair of functions {x, u}, the map t ~ g ( t , x(t), u(t)) is not Lebesgue integrable on [0, + co). Instead, we merely assume that this map is locally Lebesgue integrable. This leads to the following definition of an admissible pair. Definition 2.2. A pair of functions {x, u}: [0, + ~ ) - ~ R ' x ~[~mwill be called an admissible pair if the conditions below are satisfied: (i) (ii)

the functions x: [0, + oo ) --, R ~ is absolutely continuous on each compact subset of [0, + oo), henceforth denoted x ~ ACloc; the function u: [0, + o o ) - , R" is Lebesgue measurable;

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(iii) the constraints given by Eq. (2) are satisfied; (iv) the map t~g(t,x(t), u(t)) is locally Lebesgue integrable on

[0, +oo). We follow the usual terminology by calling, for an admissible pair {x, u }, x an admissible trajectory and u an admissible control. In addition, we denote the set of all admissible pairs by £20. To discuss the existence of optimal solutions for the case where J(x, u) is not finite for all admissible pairs, several notions of optimality have appeared in the literature. Here, we restrict our attention to the ones given below. For a more detailed list of the others, we refer the reader to Stern (Ref. 6) or to the monograph of Carlson, Haurie, and Leizarowitz (Ref. t). Defintion 2.3. For the infinite-horizon optimal control problem described by Eqs. (t) and (2), a pair {x*, u*} eF2 o is said to be:

(i)

strongly optimal if J(x*, we have

u*) is finite and if, for any {x, u} ~ o ,

T

lim inf T ~ +oo

(ii)

fo g(t, x(t), u(t)) dt >~J(x*, u *),

(3)

overtaking optimal if, for any pair {x, u} e flo and ~ > 0, there exists a it= it(e, x, u) >/0 such that, for all T>~ T,

fog(t, x(t), u(t)) dt+ ~ >~ g(t, x*(t), u*(t)) dt; (iii) weakly overtaking optimal if, for any pair {x, u}, T > O, there exists ~ = ~(e, x, u) 7> T such that

f f g(t,x(t),u(t))dt+~ fog(t,x*(t),u*(t))dt;

(4)

e>O, and (5)

(iv) uniformly overtaking optimal if, for any r > 0 , there exists a iff= T(~) >/0 such that, for all T~> iff,

f~g(t, x(t), u(t)) dt + ~>if~g(t, (v)

X * ( I ) ~ U*(/))

dt;

(6)

uniformly weakly overtaking optimal if, for any e > 0 and T > 0, there exists T = T(e) <~T such that

[ glt, x~t), u~t)) cn +~ >1[ g~t, x*~t), u*~t)) dr. JO

aO

/7)

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From the above definition, it is clear that the notion of strong optimality coincides with the usual concept of a minimum, while the other concepts are weaker. The notions of overtaking and weakly overtaking optimality were first introduced in the 1965 paper of Von Weiz~icker (Ref. 7), where they are referred to as catching up and sporadically catching up optimality, respectively. Since then, these concepts of optimality have received considerable attention; see (Ref. 1). On the other hand, the uniform overtaking optimality concepts were introduced only in 1979 in Hammond and Kennan (Ref. 8) and have received much less attention. Recently, in Carlson (Ref. 9), the uniform optimality concepts have been shown to provide a link between overtaking optimal solutions and a different optimality concept known as an agreeable pair. The questions of existence of an optimal solution (for both overtaking and strong optimality) for infinite-horizon optimal control problems have been treated by a variety of authors since the mid 1970's. For brevity, we refer the reader to Carlson, Haurie, and Liezarowitz (Ref. 1) for a summary of these results. All of these works require that at least some minimal convexity and regularity (often called seminormality) conditions be satisfied. These minimal hypotheses are conveniently described in terms of the set-valued mapping Q :A ~ 2 ~ +~, defibed as

(~(t,x)={(tl,~):q>~g(t,x,u), ~ = f ( t , x , u ) , u E U ( t , x ) } .

(8)

Specifically, these conditions are that the sets ~)(t, x) are convex, for almost all t >~0, (t, x ) s A, and that, for almost all t/> 0, the mapping ~) satisfies property (K) with respect to x on A. That is, for almost all t >~O, (t, x) ~ A, we have ~(t,x)-=

N fi>o

clf~){Q(t,y):lx-yl<6}],

(9)

in which I" 1 denotes the usual Euclidean metric on N~. The condition called property (K) is also referred to as upper semicontinuity in the sense of Kuratowski [see Cesari (Ref. 3)] and is equivalent to requiring the map (~(t, .) to have a closed graph for almost all t ~>0.

3. Relaxed Optimal Control Problem As indicated above, the existence of an optimal solution for the infinite-horizon optimal control problem described by Eqs. (1) and (2), henceforth called the original problem, is only assured when the sets

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(~(t, x) are convex. While many models of interest satisfy this convexity requirement, models in which it is not satisfied have been considered. Indeed, in Lewis and Schmalensee (Ref. 4), it is recognized that, in the area of renewable resource economics, this lack of nonconvexity "radically alters the form of optimal strategies in some simple models" and that "it thus may be very important to deal with real nonconvexities in order to formulate sound resource policies." To deal with these nonconvex models, we will replace the original problem with a convexified one through the introduction of Gamkrelidze's chattering states. This technique was apparently first applied to a specific infinite-horizon optimal control model in the work of Davidson and Harris (Ref. 10), In addition, their work was perhaps the first instance in which it was applied to an economics model, although the possibility of chattering states is alluded to in Clark (Ref. 1 1). We refer to this convexified model as the relaxed problem. To describe the chattering states, we proceed as in Cesari (Ref. 3) and introduce the following notation. For any positive integer q and any set S, we will denote the q-fold Cartesian product of S by IS] q, that is,

[s]q=SxSxSx

.-. xS

(q-times).

Elements INto] n+2 will be denoted by ~ [i.e., fi = (Ul, u a , . . . , un+ 1) where uke R m for k = 1, 2 , . . . , n + 2 ] , and we let P c N,+2 be the set

P = p = ( p l , p2,...,pn+2):pk>~O,k-=l, 2 , . . . , n + 2 ,

~ pk=l

.

(10)

k=l

Also, we define U:A --+ 2 ER~?°+2 by the relation

•r(t, x) = [U(t, x)] n+2 x P,

(11)

where U:A-+2 a~ denotes the control set for the original problem. Similarly, we will denote the graph of D by 2~, M = {(t, x, (t, p)'(t, x ) e A and (fi, p ) e ~'(t, x)}.

(12)

Finally, we define the functions ~:21~-, R and ~ : ~ - - + Rn by the formulas n+2

~(t,x,~, p ) = ~. p k g ( t , x , uk),

(13)

k~l

n+2

f(t, x, fi, p) = ~. pkf(t, x, uk), k=l

where g and f are as indicated in Section 2.

(14)

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47t

With the above notation, the relaxed infinite-horizon optimal control problem consists of minimizing the integral functional

JR(x, ~, p) =

fo

,Tg(t,x(t), ~(t), p(t)) dt,

(15)

over all admissible pairs {x, (~,p)} defined on [0, +~v) satisfying the control system

~(~) =Y(t, x(t), ~(t), p(t)),

a.e. on [0, + oo),

(16b)

x(0) = xo,

(t,x(t))EA,

(16a)

o n [ 0 , +oo),

(~(t), p(t)) e ~2(t, x(O),

a.e. on [0, + oo).

(16c) (16d)

Remark 3.1. The chattering-controls approach to relaxation considered here is limited to finite-dimensional problems. An alternative approach is to introduce generalized controls through the use of parametrized probability measures. This approach is related to the work of L.C. Young in the calculus of variations and is fully discussed, for the finite-horizon problem, in a variety of places; see, e.g., Gamkrelidze (Ref. 12). Additionally, one could also approach this problem via F-regularization of the augmented cost functional; see Buttazzo (Ref. 13). The above approach was chosen for this exposition in an effort to avoid unnecessary notation and technical details. Formally, the relaxed problem given above is a Lagrange problem of optimal control of the same type as that considered in Section 2; to distinguish between the admissible pairs {x, (~, p)} for the relaxed problem and those of the original problem, we refer to {x, (~, p)} as a relaxed admissible pair and, in keeping with standard terminology, we call x an admissible relaxed trajectory and (~, p) an admissible relaxed control. Also, we let ~oR denote the set of all admissible relaxed pairs. In addition, it is easy to see that, if g : M ~ ~ is a Lebesgue normal integrand and if f : M ~ En satisfies the Carath6odory conditions, then ~ and f respectively, satisfy the same conditions. Moreover, the relevant set-valued mapping, denoted here by QR:A ~ 2 e~+" and defined by

QR(t,x)={(tl,~): q>~g~(t,x,~t,p),~=~c(t,x,(t,p),(fi, p)~Y(t,x)},

(17)

is convex-valued since, as a consequence of Carath6odory's theorem on convex sets, it happens that, for (t, x)~ A,

OR(t, x) = co[0(t, x)],

(18)

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where c o [ S ] denotes the convex hull of S. Thus, the convexity hypothesis discussed in Section 2 is automatically satisfied and the existence of an optimal solution of the relaxed problem can be established with known existence theorems. With that in mind, we now state two existence results for the relaxed problem. As these results are simply a restatement of the corresponding results for the original problem, they will be stated without proof. Theorem 3.1. Let A ~ [0, + o o ) x Nn be a given set such that, for almost all t ~>0, the sets A(t)= {x:(t, x ) e A} are closed and nonempty. Let U:A ~ 2 ~'~ be a set-valued map with closed graph M = {(t, x, u):(t, x ) 6 A , ue U(t, x)}. Let g:M-~ R be a Lebesgue normal integrand, and let f : M - * R n satisfy the Carath6odory conditions. Assume that (i)

there exists b e N" such that the set f~b defined by •b= {(x, (fi, p))eno,:JR(x, (fi, p)) ~ b }

(ii)

(19)

is nonempty; the set

{y(., x(.), a(.), p(-))l~0.~: {x, (~, p)} end} has equiabsolutely continuous integrals for each T>~0; i.e., for every s > 0 and T>~ 0, there exists 6 > 0 such that, for that, for all measurable subsets B ~ [0, T] with Lebesgue measure m(B) < 6 and all {x, (fi, p)} ef~b,

fB t7(t, x(t), ~(t), p(t))t

clt < ~;

(iii)

the set-valued map ~R: A-~ 2 ~1+', defined by Eq. (17), satisfies property (K) with respect to x on A; (iv) there exists k ~ L l ( [ 0 , +oo); R) such that, for all

{x, (~, p)} e rib, g(t, x(t), fi(t), p(t)) >~k(t),

a.e. on [0, + oo 3.

Then, there exists an admissible relaxed pair {x*, (~*, p*)} that is a strongly optimal solution for the relaxed problem described by Eqs. (15) and (16).

JOTA: VOL. 78, NO. 3, SEPTEMBER 1993 Proof. (Ref. 1 ).

473

See Balder (Ref. 13) or Carlson, Haurie, and Leizarowitz

Before presenting the existence result for overtaking optimality, we make a few brief remarks concerning the hypotheses given in the above theorem. The existence of a nonempty set f~b assures us that the relaxed optimal control problem is nontrivial in that it guarantees there is some nonempty set of admissible pairs for which the functional JR is finite. The second condition is a weak compactness assumption placed on the set of relaxed trajectories, considered as a subset of AGo° endowed with the weak topology. For a discussion of this topology, see either Balder (Ref. 14) or Carlson, Haurie, and Leizarowitz (Ref. 1). Here, we merely remark that this condition is typically satisfied by showing that one of a vareity of growth conditions is satisfied, e.g., the growth condition (?) of Cesari, LaPalm, and Nishiura (Ref. 15); see also Cesari (Ref. 3). The conditions placed on the sets (~R(t, x) ensure the lower semicontinuity of the functional JR with respect to the topology placed on the set of admissible relaxed trajectories. Finally, the last condition ensures that, over f~b, the functional JR is bounded below and thus has a finite infimum. The next result is the relaxed analogue of the existence result for overtaking optimality presented in Carlson (Ref. 16). Theorem 3.2. Let U : A - , 2 ~", g:M--,~, and f : M - - . N n satisfy the hypotheses listed in Theorem 3.1. Let S:A--* ~ be a given differentiabte function such that the map (t, x, z) --, (O/&) S(t, x) + VxX(t, x) . z

satisfies the Carathhodory conditions on A x Nn Further assume that: (a)

h ( t , x , ~ , p ) = ~ ( t , x , Ft, p ) - ( ~ ? / & ) S ( t , x ) - V ~ S ( t , x ) . ~ c ( t , x , ~ p )

(20) (b)

is nonnegative a.e. in t >~0, (t, x, ~, p) e M; lim sup S( t, x( t ) ) < +0% for all admissible trajectories x e A Goo;

(c)

there exists a real number L such that if {x, (~,p)} is an admissible relaxed pair such that the improper integral

fo o~h(t, x(t), gt(t), p(t)) dt < +0%

(21)

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then we have lim S(t, x(t)) = L. Also assume that: (i)

there exists b E N such that the set f~b, defined by f~b = (x, (~, p) e f~o~:

(ii)

h(t,x(t),~(t),p(t))dt
is nonempty; the set {~(., x(.), ~(.), p ( ) ) l co.T1"(x, 07, p)} ~:~'-'~b}

has equiabsolutely continuous integrals for each T>~ 0; (iii) the set-valued map ~)R:A ~ 2 ~1+', defined by Eq. (17), satisfies property (K) with respect to x on A. Then, there exists a relaxed overtaking optimal solution {x*, (fi*, p)} for the relaxed optimal control problem described by Eqs. (15) and (16). Proof. (Ref. 1).

See Carlson (Ref. 16) or Carlson, Haurie, and Leizarowitz ©

In the above theorem, the hypotheses (i) to (iii), along with the nonnegativeness of the function h : M ~ R given in (a), guarantees that the optimal control problem of minimizing the improper integral given in (21) over f~0R satisfies the conditions of Theorem 3.1. The remaining hypotheses placed on S are then used to show that the strongly optimal solution guaranteed from Theorem 3.1 is an overtaking optimal solution of the relaxed optimal contral problem. We further remark that, if the inequality g(t, x, u) - (O/~tg) S(t, x) - V x S ( t , x ) . f ( t , x, u) >>.0

holds a.e.t>~0, (t, x, u)~ M, then condition (a) above holds. This fact shows that an appropriate candidate for S may be obtained via the Hamilton-Jacobi theory described in Carlson (Ref. 17); see also (Ref. t).

4. Approximation of Relaxed Trajectories by Ordinary Trajectories It is easy to see that the set of admissible pairs [2o for the original problem can be viewed as a subset of f20R, the set of relaxed admissible

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pairs. Indeed, it is easy to see that, if {x, u} eft0, then the relaxed pair {y, (~, p)}, defined by (22a)

y(t)=x(t), uk(t) = u(t),

(22b)

k = 1, 2 . . . . , n + 2,

p~(t) = 1/(n + 2),

k = 1, 2 , . . . , n + 2,

(22c)

is an element of f~oR- Thus, by formulating the relaxed problem, we are generally dealing with a larger set of admissible pairs, and so an optimal solution of the relaxed problem need not be an element of f~o. Also, an optimal solution of the original problem need not be an optimal solution for the relaxed problem. This leads us to investigate the question of when an admissible relaxed trajectory can be approximated by an admissible trajectory of the original problem. In the case of a finite interval, this question was first answered in Gamkrelidze (Ref. 5). The following generalization of Gamkrelidze's result can be found in either Cesari (Ref. 3) or Berkovitz (Ref. 18). Theorem 4.1. Let X c N" be a closed set, A = [0, T] x X, I = [0, T], and let f~ c N'~ be a given bounded set. Let f :A x f~ ~ N" be a given continuous function, and let U: [0, T] ~ 2 n be a given set-valued map such that the set M = {(t, x, u):(t, x ) e A , u ~ U(t)} is closed. Further, suppose that there exists a Lebesgue integrabte function mapping [0, T] into N such that i f ( t , x, u) - f ( t , z, u)[ ~< O ( t ) I x - z[,

(23)

for all ( t , x , u ) e M and (t,z, u ) s M . Let [to, tl] be a compact interval contained in the interior of [0, T], and let X1 be a compact set contained in the interior of X. Let (fi, p) = (ul, uz, . . . , u,+ 2, Pl, P2 . . . . , Pn+ z) be a relaxed control defined on I for the relaxed control system n+2

2 ( t ) = ~ p k ( t ) f ( t , x ( t ) , uk(t)),

a.e. on I,

(24a)

k=l

x(0) = Xo, (t, x ( t ) ) ~ A,

(24b) on/,

(u (t), p ( t ) ) ~ ~(t, x ( t ) ) ,

(24c) a.e. on L

(24d)

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corresponding to the control system

2(t) =f(t, x(t), u(t)),

a.e. on I,

(25b)

x(O) = Xo,

(t, x(t)) ~ A,

(25a)

on I,

u(t) ~ U(t, x(t)),

a.e. on I.

(25c) (25d)

Let y :I ~ R" be a relaxed trajectory corresponding to the relaxed control (~, p) such that y(t)•X1 on L Then, for every ~> 0 satisfying 0 < ~ < dist(OX1, ~?X), where OX~ and cOX denote the boundaries of X1 and X respectively, there is an admissible control u~:I--. Em with corresponding trajectory x~:I-~ ~ such that x~(to)=y(t0) and such that, for all teI, we have

]x,(t)-y(t)] <~. Proof.

(26)

See Berkovitz (Ref. 18) or Cesari (Ref. 3).

From the above result, we easily obtain the following corollary for the infinite-horizon case. Corollary 4.1. Let X e R " be closed, and let A=[O, + o o ) x X . Let f~ c ~ " be a bounded set, and let U: [0, + o o ) ~ 2 ~ be a closed set-valued mapping such that the set M = {(t, x, u):(t, x)EA, ue U(t)}

(27)

is closed. Let f : M ~ N" be a given continuous function, and suppose that exists a locally integrable function ~b from [0, + oo) into ~ such that (23) holds with M as indicated above. Finally, let (fi, p) --- (ul, u2, • •., u,+2, Pl, P2 . . . . , P n + 2 ) be a relaxed control defined on [0, + oo) for the relaxed control system given by (24), corresponding to the control system given by (25); and let y:[0, + oo)-~ R" be a relaxed admissible trajectory corresponding to (fi, p) such that y(t) E X1, a compact subset of R" lying in the interior of X. Then, for any ~ s.t. 0 < s < d i s t ( S X 1 , 0X), and for S > 0 , T > 0 , with S < T, there exists an ordinary control US, T , e : [ S ~ T ] - ~ N " and a corresponding trajectory Xs.r,~: [S, T] ~ ~ such that

IXs, r,~(t) --y(t)[ < 5,

(28)

Us, z;~.(t) • U(t),

(29)

a.e. t • [& T].

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Proof. To establish this result, we merely apply Theorem 4.t with I = IS, T]. [7 Remark 4.1. By taking a positive null sequence ~.~ek j ~+~ and a k=l sequence of times { Tk } ~-~1 satisfying Tk ~ + ~ as k ---r + ~ , the above corollary generates a sequence of ordinary pairs {Xk, Uk} defined on I-T~-~, Tk] such that sequence of trajectories [ex k J~+~ converges to the k=l relaxed trajectory y uniformly on compacta. We now apply the above result to obtain a relationship between the optimal solutions of the original problem and the relaxed problem. Theorem 4.2.

Let I = [0, +co), let f 2 c ~m be bounded, and let

U : I ~ 2 ~ be a closed, measurable set-valued map such that the set M given by Eq. (27) is closed. Let F = ( g , f ) : M ~ f f ~ 1+~ be a given continuous vector-valued function, and let ~ be a locally integrable function from 1 into ~ such that

tf(t, x, u ) - F(t, z, U)l < ~ ( t ) t x - zt

(30)

holds for almost all t/> 0, (t, x, u) e M, and (t, x, z) e ?,4. Now, let to > 0 be fixed, and consider the infinite-horizon optimal control problem of minimizing the improper integral functional

](x, u) =

g(t, x(t), u(t)) dr, o

satisfying the control system

k(t) = f ( t , x(t), u(t)),

a.e. on [to, + ~ ) ,

X(to) = xo,

on [to, +oo),

u(t) e U(t),

a.e. on [0, + ~ ) .

Let {y*, (fi*, p*)} be a fixed admissible pair for the corresponding relaxed ~+~ of optimal control problem. Then, there exists a sequence {xk, u kJk=l admissible pairs for the above problem such that the sequence of admissible trajectories {xk} converges uniformly to y* on compact subsets of [to, + co). Moreover, we have the following results: (i)

If {y*, (~7", p)} is overtaking optimal for the relaxed optimal control problem, then for each e > 0 and relaxed admissible pair {y, (~, p)}, there exists : ~ ~(~, y, (£,, p ) ) > t o such that, for all

T>L T

lira k ~

_ g(t, x~(t), u(t~)) dt <

+ cto o tO

y(,),

fi(t), p(t)) dt + e.

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(ii)

If {y*, (fi*, p*)} is overtaking optimal for the relaxed optimal control problem, them for each e > 0 and relaxed admissible pair {y, (~, p)} and T > t o, there exists T - T(~,y, ( £ p ) ) > T such that

k~

lim

+co

f,o g(t, Xk(t), U(tk)) dt < f'o ~(t, y(t), ~t(t), p(t)) dt + e.

Proof. For brevity, we prove this result only for the case where {y*, (u*, p*)} is overtaking optimal, leaving the weakly overtaking case for the interested reader. We begin by observing that the hypotheses above correspond to those of Corollary 4.1 applied to the augmented control system

~°(t)=g(t, x(t), u(t)),

a.e. on [to, + oo),

2(t)=f(t, x(t), u(t)),

a.e. on [to, +oo),

(x°(to), X(to))=(0, Xo),

on [t o , +oo),

u(t)~U(t),

a.e. on [to, + oo).

Therefore, for each k e N, by letting S = to and T k = to + k, and by letting X1 be any compact interval in ~l +" containing the set

{ (y°*(t), y*(t)):t ~ [to, t o + k] }, where yO. : [to, + oo) ~ N is defined by

y°*(t) = f,t° ~( t, y*(t), fi*(t), p*(t)) dt, there exists a trajectory control pair {Xk, Uk} defined on [0, to + k] satisfying

(x~(to), x~(to)) = (y°*(to), y*(to)), such that

[(x°(t), xk(t)) - (y°*(t), y*(t))l < 1/k, for all t e [to, to + k]. The hypotheses placed on F and U(.) permit us to extend this pair to an admissible pair on [to, oo). Clearly, this sequence of admissible pairs {xk, Uk } is such that the trajectories converge uniformly to

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y* on compact subsets of [to, + ~ ) that, for any T > to, we also have

479

as desired. In addition, we observe

T ff ~(t, y*(t), ~*(t), p*(t)) dr. lim [ g(t, xk(t), u(tk)) dt = k ~

+oo

JtO

0

Thus, since {y*, (O*,p*)} is overtaking optimal, for each e > 0 and admissible relaxed pair {y, (fi, p)), there exists T-= T(e, y, (~, p)) > t o such that, for each T > T,

g(t, xF:(t), u~(t)) dt =

lim k ~oo

~(t, y*(t), ~*(t), p*(t)) dt

0 #, T

< | ~(t, y(t), ~(t), p(t)) dt + ~, 0

as desired.

[]

Remark 4.2. The above result presents the analogue of the finitehorizon result, which provides conditions under which the infimum of the original problem coincides with the infimum of the relaxed problem and when this infimum can be approximated by a sequence of ordinary admissible pairs. In particular, such a result allows us to view the relaxed optimal control problem as a limiting case for the original problem, since the relaxed optimal trajectory is the limit of a sequence of suboptimal trajectories for any finite interval problem defined on [to, T], T > to. The next result gives conditions under which an appropriate optimal solution of the original problem is optimal for the relaxed problem. As is seen, we must strengthen the overtaking optimality concepts to the corresponding uniform optimality concepts. Theorem 4.3.

Let I = [0, +oo), let t i e r m be bounded, and let

U:I-~ 2 a be a closed, measurable, set-valued map such that the set M = {(t, x, u):t ~>0, x~ ~", u~ U(t)} is closed. Let F = ( g , f ) : M - ~ l+n be a given continuous vector-valued function, and let ¢ be a locally integrable function from I to ~ such that IF(t, x, u) - F(t, z, u)[ ~< ~ , ( t ) l x - zl

(31)

holds for almost all t>~0, ( t , x , u ) ~ M , and ( t , z , u ) ~ M . Now, let to be given, and let {x*, u*} be a uniformly overtaking (respectively, uniformly

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weakly overtaking optimal) solution for the optimal control problem described as minimizing the functional

~+o~ g(t, x(t), u(t)) dt, o

over all admissible pairs {x, u} :[to, + oo) ~ Nl+n satisfying the control system 2(t) = f ( t , x(t), u(t)),

a.e. on [-to, + oo),

X ( t o ) = Xo,

on [-to, +oo),

u(t) e U(0,

a.e. on [0, + oo).

Then, {x*, u*} is a uniformly overtaking optimal solution (respectively, uniformly weakly overtaking optimal) for the corresponding relaxed optimal control problem. Proof. We prove only the case where {x*, u*} is uniformly overtaking optimal as the argument for the remaining case is completely analogous. To establish this result, let {x*, u*} be as indicated above, and let e > 0 be given. The optimality of {x*, u*} implies that there exists T-7"(e) > to such that, for all T > 7" and all admissible pairs {x, u}, now defined on [to, + ~ ) , we have T

Now, introduce the augmented control system 2°(t)

=g(t, x(t), u(t)),

a.e. on [to, +oo),

so(t) = f ( t , x(t), u(t)),

a.e. on [to, + oo),

(x°(to), X(to))= (0, Xo),

on [to, +oo),

u(t) e U(t),

a.e. on [to, + oo),

and observe that any admissible pair {x, u} defined on [to, + oo) generates an augmented pair through the introduction of the new coordinate x °" [to, + oo) -, E defined by t

x°(t)=

ft g(r, x(r), u(z))dr. o

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481

With e > 0 and T > ]'(e), as above let {y, (~, p)} be an arbitrary relaxed admissible pair for the corresponding relaxed optimal control problem. Observe that the augmented control system defined above satisfies the hypotheses of Theorem 4.1 with I= [to, T] c [0, T + 1] and XI taken as X1 = {(z °, z) ~ ~1 +,:(z 0, z) = (yOU), y(t)), for some to ~ t ~ T}, where for t t> to,

Y°(t) = ft ~,(z, y(z), ~(z), p(z)) dr. to

Thus, for any e > 0 , there exists a pair {x~, u~} defined on [to, T] such that t(y°(t), y(t)) - (x°(t), x~,(t))l < ~/2, for all t e [to, T]. In particular, we notice that this implies ¢, T

l" T

| g(t, x~(t), u~(t)) dt < | ~(t, y(t), tT(t), p(t)) dt + 8/2. ~t 0

~tO

In addition, the hypotheses placed on the functions (g, f ) imply that the pair {x~, u~} can be extended to the half-line [to, + o e). Therefore, this pair {x~, us} extended to [to, + oe), is admissible, and so by combining the above inequalities we obtain

g(t, x*(t), u*(t)) dt < 0

g(t, x(t), u(t)) dt + e/2 0

g(t, y(t), fi(t), p(t)) dt + ~.

< 0

As ~ > 0 and T>I"(e) were chosen arbitrarily, it follows that {x*,u*} is uniformly overtaking optimal for the relaxed admissible control problem. []

5. Examples In this section, we provide several examples presenting both the utility of the above results as well as their limitations.

Example 5.1. The infimum for the relaxed problem may be less that the infimum of the original problem. This example is an adaptation of a

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finite-interval example found in Cesari (Ref. 3). Specifically, we consider the problem of minimizing the functional

J(x,y,z,u,v)=

f0

[2/rc(l +t2)](1-2lu(t)-x(t)[)dt,

over all pairs {((x, y, z), (u, v))} : [0, + oo) --, R 3 x N2 satisfying the control system 2(t) = [2/(1 + t2)(rca + 2 tan-l(t))](u(t) - x(t)), 9(t) -- [2/re(1 + tz)] v(t), ~(t) = [2/7r(I + t2)](x(t) - y ( t ) ) z, x(0) = y(0) = 1/2,

z(t)<<,O,

z(0) = 0,

for all t~>0,

u(t) ~ {0, 1/2, 1 }, v(t) ~ [ - 1/8, 1/8],

a.e. t~>0.

In the above, a is a positive constant satisfying a < 1. It is an easy matter to see that the integrand of the objective functional is a concave function of u. This, reinforced with the fact that the control u(.) takes on only a discrete set of values, allows one to easily conclude that the usual convexity hypotheses are not satisfied. We now demonstrate that the only feasible admissible pair for this problem is

u(t)=l/2,

x(t)=y(t)=l/2,

z(t)=0,

v(t)=0.

To see this, we observe that, since ~(t)t> 0, z ( 0 ) = 0, and z(t)<~ 0 must all hold, we necessarily have that z ( t ) - O. Thus, x(t)=y(t), giving us that 2(t)---2O(t) = [2/~(1 + t2)] v(t). This implies that x(. ) is Lipschitzian, since t2(01 ~< 1/41r(1 + t2), and so,

Ix(t)- 1/21 ~<

[1/4~(1 -l-S2) "1 ds

~< = 1/8.

[1/4~(1 +s2)] ds

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483

Therefore, we have 3/8 ~
fo [2/~(l+t~)] 1-2 E

dr,

j=l

over all pairs satisfying the control system

[4

2(0 = [2/(1 + t2)(~a + 2 tan-~(t))] j--~l (pj(t) uj(t))-x(t) 4

p(t) = [2/(~(1 + t2)] 2 (pj(t) vj(t)), j= i

~(t) = [2/n(1 + t2)](x(t) - y ( t ) ) 2, x(0) = y(0) = 1/2,

z(t)<~O,

z(0) = 0,

for all t>~0,

uj(t)e{O, 1/2,1},

t~(t)e[-1/8,1/8],

a.e. t ~>0,

4

pj(t)~ [0, 1],

~ p+(t)= 1. j=l

Proceeding exactly as in the original problem, it is easy to see that

z( t)-~ O, x(t) =--y(t), 3/8 ~
]

,

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484 and that 4

1/8 ~< ~ (&(t) Uj(t)) <~7/8. j=l

Observe however that, by making the following choices:

pl(t)=l/2,

pz(t)=l/2,

p3(t)=0,

pa(t)=0,

Ul(t ) = 0,

U2(t) = 1,

U3(/) = 0,

U4(t ) = 0,

vl(t) = 0,

vz(t) = 0,

v4(t)--0,

v4(t) = 0,

we have that 4

(&(t)~(t))=l/2, j=l

and so we generate the trajectory

x(t)=u(t)=l/2 as above. However, for these choices, we observe that, for any T > 0, I0r [2/7r(1+ tz)]

1-2

~ (pj(t) uj(t))- 1/2

dt

j=l

-- f o [2/~(1 + t2)][1 - 2{(1/2)l0 - 1/21 + (1/2)11 - 1/2] }] dt = 0,

whereas for the original problem, for [2/~z(1 + t2)](1 - 2 lu(t) - x(t)l) dt = ior [2/72(1 + t2)] dt = (2/re) t a n - I ( T ) > 0. From these two expressions, it is clear that the conclusion of Theorem 4.2 does not hold for this example. The difficulty arises as a result of the state constraint on the z-component, i,e., z(t)<.O. That is, Theorem4.2 is applicable only for the case where the state variables are unconstrained. Our next example is a simple example demonstrating the utility of Theorem 4.3.

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Example 5.2. The original problem in this model is briefly described as follows: minimize o ~ - y ( t ) dr,

over all pairs of functions { ( x , y ) , u } : [ O , + ~ ) ~ N 2 x N control system

satisfying the

2(t) =y(t), ) ( t ) = - x ( t ) + u(t),

[x(oq i-Ol

LoA' ,(t)~ {-1, 1}. y(O)J =

One can easily see that this example satisfies the hypotheses of Theorem 4.3. We shall see below that this problem has a uniformly weakly overtaking optimal solution, and so by an application of Theorem 4.3, this solution is also uniformly weakly overtaking optimal for the corresponding relaxed problem as well. The corresponding relaxed problem is formulated as minimizing o °° - y( t ) dt,

over all relaxed pairs {(x, y), (~, p)} : [0, + oe) ~ N2 x ~6 satisfying 2 ( 0 =y(t), 3

j~(t) = - x ( t ) + ~ pj(t) uj(t), j=l

Ix(O)] y(O)j=[°o] ' uj(t)e { - t , 1},

j = 1,2, 3,

3

pj(t) >~O,

~ pj(t) = l. j=,l

It is an easy matter to see that this relaxed problem is equivalent to mlmmlzlng

fo + ~ - y( t ) at,

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over all pairs of functions {(x, y), v} : [-0, + ov ) ~ R E × ~ satisfying

2(t) = y(t), ~(t) = - x ( t ) + v(t),

v(t)E [--1, 1]. One can easily show that the control v*:[O, + o o ) ~ [ - 1 , 1 ] defined as

2k~z <. t <. ( 2k + 1)n, ( 2 k + 1)Try< t < 2(k + 1)re,

v*(t) = sign(-sin(t)) = { - 1, 1, generates the admissible trajectory

Ix*(t)] , sin(t-s) y,(t)j=I0[cos(t_s)Jv

.

(s) ds.

This trajectory-control pair is admissible for both the original and relaxed problem. Moreover, in Ref. 9 it is shown that this pair is uniformly weakly overtaking optimal for the relaxed problem, so that we know it is also optimal for the original problem as well. In our last example, we consider a model utilized in fishery economics. This model is similar to the renewable resource models discussed in Ref. 4 as well as the continuous-time investment model treated in Ref. 10. Example 5.3.

We consider minimizing the functional

fo+ ~ [ C(u(t)) - rcqx(t) u(t)] dt, over all pairs of functions {x, u} : [0, + oo) ~ N 2 satisfying the control system ~(t) = F ( x ( t ) ) - qx(t) u(t),

a.e. o n [ 0 = oo ),

x(O) = Xo, O~x(t)~K,

t e [0, + o o ) ,

O<~u(t)<~ U,

a.e. o n [0, + oo).

In the above model, the state variable x(.) denotes the biomass of the fish population; the control variable u(-) denotes the fishing effort; F(. ) is

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487

the natural growth rate of the fishery; C(.) is the cost per unit effort; g is the price per unit of l~tnded fish; K is the carrying capacity of the fishery; U represents an upper bound on fishing effort; and q is a positive normalization constant. With this interpretation of the variables, the integrand of the objective functional represents the negative of profit. Thus, our objective is to maximize accumulated accumulated profit. For a detailed discussed of this model, we refer the reader to Clark (Ref. 11 ). The specific hypotheses placed on the above model are listed as follows: (H1)

C:[0, +oo)-~[R is a nonnegative, strictly increasing twice continuously differentiable function which satisfies

c(o) = 0,

(H2)

C"(u)<0,

0
C " ( u ) > 0,

(t
where u e [0, U] is a fixed constant; F: [0, + oo) ~ N is twice continuously differentiable with F(O)=F(K)=O, F(x) > 0 for 0 < x < K ,

and F(x) < 0 for x > K .

Remark 5.1. If ~ = 0 in the above, then the map u --* C(u) is strictly convex; in Carlson (Ref. 16), conditions are given under which this problem has an overtaking optimal solution. When ~ > 0, the result does not necessarily hold, since the requisite convexity hypotheses is not met. That is, the sets ~)(x), given here by Q(x) = {(a °, z):z° ~> C(u)-~zqxu, z = F ( x ) - q x u , ue [0, I]}, are not convex. As discussed in Ref. 4, this lack of convexity occurs if there are initially increasing marginal returns from harvesting. To describe the relaxed problem, we observe that the nonconvexity of the sets ~(x) occurs only as a result of the nonconvexity of C(-). Consequently, it is easy to see that the relaxed problem takes the form of minimizing the functional

fo+ co [8(v(0) - ~qy(t) v(t)] dr, over all pairs of functions { y , v } : [ 0 , + o o ) - , N 2 satisfying the control system

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f,(t)=F(y(t))-qy(t)

v(t),

a.e. on [0, + ~ ) ,

y ( 0 ) = Xo,

O<<.y(t)<~K,

t e l 0 , +os),

O<~v(t)<~U,

a.e. on [0, +oo),

where the function (~:[0, + ~ ) ~ function C(-). That is,

~ is the convex hull of the nonconvex

~(v) = inf{2C(ut) + (1 - 2) C(u2):2 ~ [0, 1 ], ~ 2 b / 1 + ( l -- )~)b/2 = v, b/l, l / 2 e [0, U ] ) .

This problem has the same form as the original problem, except that now the requisite convexity hypotheses are met. The existence of a relaxed overtaking optimal solution {y*, (~7", p*)} is assured [see Carlson (Ref. 16)] if we impose the following additional assumptions: (H3) The constant U is chosen sufficiently large enough to ensure that the equation 0 = Y ( y ) - qyv (H4)

has a solution v ~ [0, U] for each y ~ [0, k]. The optimal steady-state problem, described simply as min

(y,v)c [~o,K] × [o, u3

{~(v)-zqyv:O=F(y)-qyv},

has a unique solution (37, ~). Since ~'(-) is convex, it is an easy matter to see that the hypotheses of Theorem4.2 are satisfied. Thus, we can conclude that there exists a sequence of original admissible pairs, say {xg, ilk}, such that x ~ y * . Moreover, we also have that, for that, for each e > 0 and relaxed admissible pair {y, (fi, p)}, there exists ~ - T(e,y, (ft, p ) ) > 0 such that, for all T > 7", lim

~ C(fik(t) ) -- ~xk(t) dt <

C.(v(t)) - ~y(t) v(t) dt + ~,

JO

where 3

v(t) = ~. pi(t)ui(t). j~=l

To relate these conclusions to the works of Lewis and Schmatensee (Ref. 4) and Davidson and Harris (Ref. 10), we first remark that they concern

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themselves with a discounted model. That is, the objective functional includes an exponential weight exp(-rt), r > 0. This permits them to consider only strongly optimal solutions, and consequently they obtain the result that the infima of the relaxed and the original problems coincide and that they are attained by a chattering trajectory (i.e., a relaxed admissible pair), which can be approximated to any desired degree of accuracy by an admissible pair for the original problem. These results concur with ours, since we consider the weaker notion of overtaking optimality. In addition, we remark that the results of these researchers are obtained through a phase portrait analysis of the classical necessary conditions. Therefore, their arguments depend heavily on the fact that their model has only one state variable. This fact precludes any extension to higher dimensions.

6. Conclusions

In the work reported above, we have considered infinite-horizon optimal control problems in which the standard convexity hypothesis, needed to discuss the existence of an optimal solution, are not satisfied. An additional disinguishing feature of these models is that we did not assume a priori that the objective functional (described by an improper integral) was finite. Thus, we additionally considered the weaker notions of overtaking and weakly overtaking optimality. To treat these models, we introduce the relaxed optimal control problem through the introduction of Gamkrelidze's chattering controls. This new optimal control problem now satisfies the standard convexity conditions, and the existence of optimal solutions can be established through the Use of known existence results. For completeness, two such results are reported in Section 3. With these two models in hand, we have demonstrated that, under conditions analogous to those used in finite-interval problems, one could approximate an optimal trajectory of the relaxed problem uniformly on compact subsets by a sequence of admissible trajectories for the original problem. In addition we demonstrated that the corresponding sequence of cost functions also converges when truncated to a finite interval [0, T]. These results allowed us to further investigate the relationships between the optimal solutions of the original problem with those of the relaxed problem. Specifically, we showed that, under fairly general hypotheses, a uniformly weakly overtaking optimal solution of the original problem enjoyed the same optimality property for the relaxed problem. In this way, we can view the relaxed optimal control problem as a limiting case for the original problem. We concluded our work with a few examples demonstrating the utility of our results. Of particular interest is the nonconvex fishery model similar to

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models discussed in the economics literature demonstrating the existence of a relaxed overtaking optimal solution which can be approximated as described above. We also observe that previous results in this direction were only applicable to the case of a single state variable, whereas the results reported here are established for n-dimensional models.

References 1. CARLSON,D. A., HAURIE, A., and LEIZAROWITZ,A., Infinite-Horizon Optimal Control." Deterministic and Stochastic Systems, 2nd Edition, Springer-Verlag, New York, New York, 1991. 2. MCSHANE, E. J., Relaxed Control and Variational Problems, SIAM Journal on Control, Vol. 5, pp. 438-485, 1967. 3. CESARI,L., Optimization-Theory and Applications, Springer-Verlag, New York, New York, 1983. 4. LEWIS, T. R., and SCHMALENSEE,R., Optimal Use of Renewable Resources with Nonconvexities in Production, Essays in the Economics of Renewable Resources, Edited by L.J. Morman and D.F. Spulber, North-Holland Publishing Company, Amsterdam, Holland, pp. 95-111, 1982. 5. GAMKRELIDZE,R. V., On Some Extremal Problems in the Theory of Differential Equations with Applications to the Theory of Optimal Control, SIAM Journal on Control, Vol. 3, pp. 106-128, 1965. 6. STERN, L. J., Criteria for Optimality in the Infinite-Time Optimal Control Problem, Journal of Optimization Theory and Applications, Vol. 44, pp. 497-508, 1984. 7. VON WEIZ~CKER, C. C., Existence of Optimal Programs of Accumulation for an Infinite-Time Horizon, Review of Economic Studies, Vol. 32, pp. 85-104, 1965. 8. HAMMOND,P. J., and KENNAN, J., Uniformly Optimal Infinite-Horizon Plans, International Economic Review, Vol. 20, pp. 283-296, 1979. 9. CARLSON, D. A., Uniformly Overtaking and Weakly Overtaking Optimal Solutions in Infinite-Horizon Optimal Control: When Optimal Solutions Are Agreeable, Journal of Optimization Theory and Applications, Vol. 64, pp. 55-69, 1990. 10. DAVIDSON, R., and HARRIS, R., Nonconvexities in Continuous-Time Investment Theory, Review of Economic Studies, Vol. 43, pp. 235-253, 1981. 11. CLARK, C., Mathematical Bioeconomies: The Optimal Management of Renewable Resources, John Wiley and Sons, New York, New York, 1976. 12. GAMKRELIDZE,R. V., Principles of Optimal Control Theory, Plenum Press, New York, New York, 1978. 13. BUTTAZZO, G., Semicontinuity, Relaxation, and Integral Representations in the Calculus of Variations, John Wiley and Sons, New York, New York, 1989, 14. BALDER,E. J., An Existence Result for Optimal Economic Growth, Journal of Mathematical Analysis and Applications, Vol. 95, pp. 195-213, 1983.

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15. CESARI, L., LAPALM, J. R., and NISHIURA, T., Remarks on Some Existence Theorems for Optimal Control, Journal of Optimization Theory and Applications, Vol. 31, pp. 397-416, 1969. 16. CARLSON, D. A., On the Existence of Catcking up Optimal Solutions Jbr Lagrange Problems Defined on Unbounded Intervals, Journal of Optimization Theory and Applications, Vol. 49, pp. 207-225, 1986. 17. CARLSON,D. A., A Caratkdodory-ttamilton-Jacobi Theory for Infinite-Horizon Optimal Control Problems, Journal of Optimization Theory and Applications, Vol. 48, pp. 265-287, 1986. 18. BERKOVITZ, L. D., Optimal Control Theory, Springer-Verlag, New York, New York, 1974.