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Necessary conditions are proved for certain problems of optimal control of diffusions where hard end constraints occur. The main results apply to several dimensional problems, where some of the state equations involve Brownian motions, but not the equations corresponding to states being hard restricted at the terminal time. The necessary conditions are stated in terms of weak variations. Two versions of necessary conditions are given, one version involving solutions of variational equations, the other one involving first order adjoint equations.

The purpose of the present paper is to prove necessary conditions for the optimal control of certain types of control problems involving diffusions where hard end constraints on solutions occur. The books [

Let

Let

The following conditions are called the Basic Assumptions.

A_{1} The functions

A_{2} As a function of t, the functions f and

Write

The following assumptions are called the Global Assumptions.

B_{1}

B_{2} For some constant M, for some given

B_{3} A constant

B_{4}

Let

is denoted

Let

subject to

Below,

We have collected a few definitions that are going to be used in the sequel.

For

where

Finally, let

In the subsequent necessary conditions, the following local linear controllability condition (15) is needed. There exist numbers

A more “concrete” condition implying (15) is presented in Remark 3.

Theorem 1. Assume that

where

with^{1},

Remark 1. If (15) holds for

Remark 2. Let

by an

Let

vectors,

and such that, for all

In this case, we have that for all

When

Remark 3 The condition (15), with

The

systems with inequality constraints are stated in Remark 5 below, the local linear controllability condition just obtained suffices for the controllability condition in Remark 5 to apply to the original system).

The proof consists of three lemmas and six proof steps A.-F., and relies on an “abstract” maximum principle, Corollary I in Appendix.

Without loss of generality, from now on let

By general existence results and (1) and (2),

A) Growth properties.

When

(57) in Appendix, Lemma A, (a consequence of Gronwall’s inequality), with

Let

(the last inequality by (2)).

Let

The inequality (25) follows from (57) in Appendix, Lemma A, together with (2), for

Similarly^{2}, for

In (25) and (26),

We need to prove that

This follows from (24) and (2), because, in a shorthand notation, for all t,

We also need:

This follows from (26) and (2), because, in a shorthand notation, for

Note finally that, when

see Remark L in Appendix.

B) Local controllability of the linear perturbations.

Note that, by (2), in a shorthand notation,

when

where “co” can be omitted due to the concavity of U. To see this, apply Remark W in Appendix.

Next, let the number

To see this, note that by (25), for all t,

Then, using again Remark W in Appendix and (31), (29), (32), for all

C) Relations between exact and linear perturbations.

Let

Write

For any

by Lemmas B and C in Appendix.

Consider now

By (56) in Appendix, Lemma A, for

To see this, in Lemma A let

Next, given

Let

Now,

Because

D) Continuity of

Let

Now, in a shorthand notation,

Using the notation in Lemma A in Appendix, we write

(

Then, by (57),

where

for

and

for

Consider e.g the term

Next, assume that

where, in a shorthand notation,

Now,

Let us show that the four terms in square brackets have

E) Dense subsets of

Let

to the linear space

Note finally that if

F) Final proof steps.

Define

Using Jensens inequality, for an arbitrary function

For

Furthermore, let

set

so, in particular,

so

so

for all

Finally, if

where

and where

Let

Now, let_{(2)} be closure in

Now, for

Hence, by (33), using

where

(note that

Using Remark W,

Observe that, given

And, by (39) and

To obtain the conclusion in Theorem 1, we will now apply Corollary I in Appendix, and for this end, we make the following identifications:

By Hahn-Banach’s theorem,

Proof of Remark 1. Let

(16) implies

From (16), in a shorthand notation, we get, for

Fix any

By (25), for some constant K,

Thus, on this subspace of

Note that for

means that

Proof of Remark 2.

Let

It is easily proved (using Lemma A in Appendix and

Assume now

where

In case of

Let now

Then, consider the pair of equations

By Theorem 2.2, p. 349 in [

Let us show (19). For any

Define

So

Remark 4. (Exact attainability). In Theorem 1, drop the assumption that

(4) and _{2} and int = interior both corresponding to

and the space

Proof of Remark 4. Let

Assume that

Remark 5 If it is required that

(16) again holds, and with

Proof of Remark 5. A proof can be obtained (for

Remark 6 (The case

natural filtration, augmented by null sets as before, the necessary condition of Theorem 1 can be obtained for

a) for some

If

b): For some

Letting

-norm instead of

denoted

Now, in the manner required in Corollary I, (34) Þ (61) (with

In this paper, necessary conditions for optimal control of diffusions with hard end restrictions on the states have been obtained. The main case considered is the one where the states restricted at the terminal time correspond to differential equations not containing Brownian motions. Brownian motions only occur in differential equations for states unconstrained at the terminal time. A removal of this restriction is discussed in Remark 6.

The author is grateful to a referee for useful comments.

Seierstad, A. (2018) Necessary Conditions for Optimal Control of Diffusions with Hard Terminal State Restrictions. Open Journal of Optimization, 7, 1-40. https://doi.org/10.4236/ojop.2018.71001

The appendix contains, among other things, a number of wellknown results, included for the convenience of the reader. The first one concerns a result on comparison of solutions. Still

Lemma A. Assume that

and

where

(applied to matrices the index j indicates columns), and for some constant

Proof of (57). We shall use a shorthand notation. Using the algebraic inequality

The Burkholder-Davis-Gundy inequality yields, for a “universal” constant

Similar inequalities hold for the other terms involving

Note that, by Gronwall’s inequality, for any functions

Using the fact that the square root of a sum of positive numbers is £ the sum of square roots of the numbers, we get

Note that

Proof of (56).

Using Ito’s isometry,

Then, again using

Thus, for

so (56) follows.

Simple results om Gâteaux derivatives appear in the next two lemmas.

Lemma B. Let