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Necessary conditions for optimality are proved for smooth infinite horizon optimal control problems with unilateral state constraints (pathwise constraints) and with terminal conditions on the states at the infinite horizon. The aim of the paper is to obtain strong necessary conditions including transversality conditions at infinity, which in many cases lead to a set of candidates for optimality containing only a few elements, similar to what is the case in finite horizon problems. However, strong growth conditions are needed for the results to hold.

The aim of this paper is, in a control problem with unilateral state constraints and terminal conditions at infinity, to obtain necessary conditions, with a full set of transversality conditions at infinity, which frequently make it possible to narrow down the set of candidates for optimality to only a few, or sometimes a single one. In infinite horizon problems without unilateral state constraints (pathwise constraints), with or without terminal conditions on the states at the infinite horizon, there exist various types of necessary conditions for optimality, and examples are [^{1} in [

The novelty of the results in this paper is hence the establishment of necessary conditions that include a full set of transversality conditions at infinity in an infinite horizon problem with both terminal constraints at the infinite horizon and unilateral state constraints (constraints of the form

For Michel-type necessary condition in the case of unilateral state constraints, sees [

The growth conditions used below, ((11), (12), (13)) are more demanding than the conditions applied in [

The results below are of especial interest in the case where not all states are completely constrained at infinity. In the opposite case, generalizations of Halkinâ€™s infinite horizon theorem in [

In certain cases there is a danger of degeneracy of multipliers. See the early review in [

Consider the problem

where

where we require that

to (2)-(4). When the solution

We assume that

and that for any x,

assumptions. At various points some strengthening of these assumptions are added.

The following definitions are needed: let

In Theorem 1, in addition to the basic assumptions, assumptions (5)-(15) below are needed. It is assumed that for all

We shall make use of some constraint qualifications, (6) and (8) below, related to

((6) holds vacuously if

Either^{2}

The following growth conditions are also needed: For some

and there exist some positive constants

where

Assume finally that, for all j

Define

Theorem 1. (Necessary condition, infinite horizon) Assume (5)-(8), (11)-(15) and the basic smoothness as

sumptions. There exist a number ^{3} satisfies, for

then

Moreover,

Finally,

Remark 1. For

In the sequel three trivial examples with rather obvious optimal controls will be presented, but to illustrate the use of the necessary conditions, we derive the form of the optimal controls from these conditions.

Example 1.

Solution:

Evidently

Consider first the case that we might have

see the expression for

the expression for

when

Remark 2. (Further non-triviality properties)

a) Replace (6) by the assumption that either ^{4}. Then

b) Assume in addition that, for any

For finite horizon normality conditions, see [

The main reason for including the next theorem is that it forms a basis for obtaining Theorem 1, but it has some interest of its own.

It contains necessary conditions for the case where (14) and/or (15) fail, in particular where

Theorem 2. In the situation of Theorem 1, with (5), (6), (8), (14), and (15) deleted, assume that the three conditions (25), (26), (27) below are satisfied. Then the following necessary conditions hold: for some

where

If (7) and (8) hold,

As before when

Assume, for some arbitrarily large

Let

Moreover, for any given number

As an example in which (25)-(27) hold, consider a case where

is concave in x, and where, for some positive

hand notation

where

Remark 3. For Theorem 2 to hold, we can weaken (7) and the basic assumptions on

Moreover in the growth conditions (12) and (13), roughly speaking, the inequalities need not hold for states x that cannot possibly occur, more precisely, the conditions can be modified as follows. Define for each t,

Finally, U can be replaced by a time dependent subset

Example 2.

and

Let ^{5}, (

argument would not work if we had replaced

stant on

possible, is impossible: Let

Example 3.

Again, assuming, for some

Remark 4. Assume in the problem (1)-(7), (11)-(15), that

negative finitely additive set functions

for

(

When for some vectors

mum condition (17) holds for

for all

and where, for some nonnegative

represented by a bounded nondecreasing right-continuous function

When, in addition, for some

Proof of Theorem 2. To simplify the notation, instead of the criterion (1), we can and shall assume that

Overview of the proof. A rough outline of the proof is as follows. We are going to make a number of strong (needleshaped) perturbations of

Detailed proof. To avoid certain problems connected with coinciding perturbation time points, the following construction is helpful (we then avoid coinciding perturbation time points). Let

such that for each

that

to

Let

and

where a sum over an empty set is put equal to zero. For

Let ^{6}. Define

where

The linear variations

Fix a pair

To see this, choose

For some positive

(

Now,

by (40) because

mula

^{7}), it is easily seen that

is small when

Now, for

Because

Next, it is well-known that

uniformly in

for all

both for all

bine (27) and (26)). Then, by (45), for some positive

for some positive

Hence, (44) has been shown, and in particular, (see (42))

So far, we have only used the basic assumptions and the first of the three conditions on

We want to show that

So far, we have proved (44) for

Finally, let us prove (44) for

Then the right derivative

consider the subcase where

Consider next the subcase where

Thus, when

Observation 1. Define

the moment we allow the pairs in

indices on

single index, with the time points in the pairs in increasing order. Let

The following result should surprise nobody, a proof however is given in Appendix.

Lemma 1. Assume that for

By optimality, for all

that for some

for some ^{8}: there exists a nonzero vector

Define

for all

finitely additive nonnegative set function

nishes on

where

Let us now choose a sequence

described for

(for some subsequence

equality holds). The cluster point

^{9} as well as for

Now ^{10}.

Finally, let us extract an additional property. If

for (say) ^{11}_{. }

Let us prove the results concerning the multipliers in the three last sentences in Theorem 2 in the case where

Define

Now, assume (9) and (10) (a), with

holds for this cluster point

Evidently,

When

i.e. (10) (g), see Appendix), we donâ€™t need the assumption that ^{12}. In fact,

when (8) holds, in Theorem 2, we can assume

Define

(see the end of Theorem 2). For

Assume for the moment that

Can

Finally, assume that both (58) and (8) are satisfied. By contradiction assume now that

Proof of Theorem 1.

We still keep the assumption that

on

Now,

Let

and ^{13}, for ^{14}. Evidently, (61) yields

In Theorem 1, we have written

Note that (6) implies that (58) holds, as

Proof of Remark 2. We give a proof for the case where

Proof of a)

Let

(the last term is

Proof of b) Assume by contradiction that

If

Using the vectors

Next, let

There exist ^{15}. From this, we finally get, by Lipschitz continuity of