Infinite Horizon Discrete Time Control Problems for Bounded Processes

Advances in Difference Equations Volume 2008, Article ID 654267,14pages doi:10.1155/2008/654267

Research Article

Infinite Horizon Discrete Time Control Problems

for Bounded Processes

Jo ¨el Blot1 _{and Na¨ıla Hayek}1, 2

1_{Laboratoire Marin Mersenne, Universit´e Paris I, Panth´eon Sorbonne, Centre Pierre-Mend`es-France,}

90 rue de Tolbiac, 75013 Paris, France

2_{Universit´e de Franche-Comt´e, 45 Avenue de l’Observatoire, 25030 Besanc¸on, France}

Correspondence should be addressed to Na¨ıla Hayek,[email protected]

Received 15 October 2008; Accepted 27 December 2008

Recommended by Leonid Shaikhet

We establish Pontryagin Maximum Principles in the strong form for infinite horizon optimal control problems for bounded processes, for systems governed by difference equations. Results due to Ioffe and Tihomirov are among the tools used to prove our theorems. We write necessary conditions with weakened hypotheses of concavity and without invertibility, and we provide new results on the adjoint variable. We show links between bounded problems and nonbounded ones. We also give sufficient conditions of optimality.

Copyrightq2008 J. Blot and N. Hayek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The first works on infinite horizon optimal control problems are due to Pontryagin and his school 1. They were followed by few others 2–6. We consider in this paper an infinite horizon Optimal Control problem in the discrete time framework. Such problems are fundamental in the macroeconomics growth theory7–10and see references of11. Even in the finite horizon case, the discrete time framework presents significant differences from the continuous time one. Boltianski12 shows that in the discrete time case, a convexity condition is needed to guarantee a strong Pontryagin Principle while this last one can be obtained without such condition in the continuous time setting. We study our problem in the space of bounded sequences∞,which allows us to use Analysis in Banach spaces instead of

converge. Therefore we present other notions of optimality that are currently used, notably in the economic literature, see3,4,9and we show how our problem can be related to these other problems. We end the paper by establishing sufficient conditions of optimality.

Now we briefly describe the contents of the paper. In Section 2 we introduce the notations and the problem, then we state Theorems 2.1 and 2.2 which give necessary conditions of optimality namely the existence of the adjoint variable in the space1satisfying the adjoint equation and the strong Pontryagin maximum principle. InSection 3we prove these theorems through some lemmas and using results due to Ioffe-Tihomirov 17. In Section 4we introduce some other notions of optimality for problems in the nonbounded case and we show links with our problem. For example, we show that when the objective function is positive then a bounded solution is a solution among the unbounded processes. Finally we give sufficient conditions of optimality for problems in the bounded and unbounded cases adapting for each case the approprate transversality condition.

2. Pontryagin maximum principles for bounded processes

We first precise our notations. Let Ω be a nonempty open convex subset of Rn _{and U a} nonempty compact subset ofRm_{.LetΦ:Ω×U → Rand, for allt∈N, f}

t:Ω×U → Rn.We setx, u xt t,ut t .

Define domJ{x, u ∈ΩN×UN:_t0∞βt_Φx

t, ut converges inR}.

For everyx ∈ Rn _{defineCx as the closure of the set of terms of the sequencex.If}

x∈∞N,Rn , Cx is compact. We setX{x xt t∈∞N,Rn , such thatCx ⊂Ω}.Xis thus the set of the bounded sequences which are in the interior ofΩ.Note thatXis a convex open subset of∞N,Rn _{sinceΩis open and convex. We setU {u u}

t t ∈UN}.Define Admη {x, u ∈ΩN× U:xt1 ftxt, ut , t∈N, andx0 η}; it is the set of admissible processes with respect to the considered dynamical system.

Letβ∈0,1 .We consider first the following problemP1:

MaximizeJx, u

∞

t0

βt_Φx

t, ut

xt1ft

xt, ut

, t∈N

x0η

x, u∈ X × U

P1

which can be written as follows.

P1 MaximizeJx, u whenx, u ∈Admη ∩X × U . Theorem 2.1. Letx, u be a solution of P1. Assume the following.

i For allu∈U,the mappingx→Φx, u is of classC1_{onΩand for allt∈N,the mapping}

x→ftx, u is Fr´echet-differentiable onΩ.

ii For allt∈N,for allxt∈Ω,for allu_t, u_t ∈U,for allα∈0,1,there existsut∈Usuch

that

Φxt, ut

≥αΦxt, ut

1−α Φxt, ut

ft

xt, ut

αft

xt, ut

1−αft

xt, ut

iii For any compact set C ⊂ Ω,there exists a constant KC such that for all t ∈ N,for all

x, x∈C,for allu∈U, ftx, u ≤KCandDxtftx, u −Dxtftx, u ≤KCx−x. iv There existsr >0such thatBx, r ⊂ Xand for allxt, ut ∈Bxt, r ×U,

sup t≥0

_Dx

tft

xt, ut<1. 2.2

Then there existspt1 t∈1N,Rn such that

a ptpt1◦Dxtftxt,ut βtDxtΦxt,ut , for allt,

b βtΦxt,ut pt1, ftxt,ut ≥βtΦxt, ut pt1, ftxt, ut , for allt, for allut∈U, c limt→∞pt0.

Comments

For continuous time problems, one does not need conditions to obtain a strong Pontryagin maximum principle, both in the finite horizon casesee, e.g.,18 and in the infinite horizon casesee, e.g.,5 . But for discrete time problems, strong Pontryagin principles cannot hold without an additional assumption namely a convexity condition, as Boltyanski shows in

12 for the finite horizon framework. Conditionii comes from the Ioffe and Tihomirov book17. It generalizes the usual convexity condition used to garantee a strong Pontryagin maximum principle. The usual condition is:Uconvex subset,Φconcave with respect to u

and for every t, ft affine with respect to u. It implies condition ii. In iii the condition

ftx, u ≤ KC is satisfied when ft is continuous since Uis compact and the condition

Dxtftx, u −Dxtftx, u ≤KCx−xis satisfied whenD2xtftexists and is continuous.

Conclusion a is the adjoint equation, conclusion b is the strong Pontryagin maximum principle and conclusionc is a transversality condition at infinity. In our case

c is immediately obtained sincept1 tis in1N,Rn ,but in generalnonbounded cases it is very delicate to obtain such a conclusion.9

In the next theorem we consider the autonomous case. Thus the hypotheses are simpler and easier to manipulate.

Theorem 2.2. Letft f for allt ∈N.Letx, u be a solution of P1 . Assume that the following

conditions are fulfilled.

i For allu∈U,the mappingsx→Φx, u andx→fx, u are of classC1_onΩ.

ii For allt∈N,for allxt∈Ω,for allu_t, u_t ∈U,for allα∈0,1,there existsut∈Usuch

that

Φxt, ut

≥αΦxt, ut

1−α Φxt, ut

,

fxt, ut

αfxt, ut

1−α fxt, ut

. 2.3

iii sup_t≥0Dxtfxt,ut <1.

Then there existspt1 t ∈ 1N,Rn such that the assertions (a), (b), and (c) of Theorem 2.1 are

3. Proofs of Theorems2.1and2.2

Proof ofTheorem 2.1

First part

The first part of the proof goes through several lemmas.

Lemma 3.1. J is well-defined and under hypothesis (i) ofTheorem 2.1, for allu,the mappingx →

Jx, u is of classC1_{and one has, for allδx∈}

∞N,Rn , DxJx, u δx _∞

t0βtDxtΦxt, ut δxt.

For the proof see19.

We setFx, u ftxt, ut −xt1 t≥0for allx, u ∈ X × U.

Lemma 3.2. Assume that hypothesis (iii) ofTheorem 2.1 holds. Then forx, u ∈ ∞N,Ω × U,

one hasFx, u ∈ ∞N,Rn .Moreover, if in addition hypotheses (i) and (iv) of Theorem 2.1hold,

then for allu,the mappingx → Fx, u is of classC1_{on the ballBx, r in}

∞N,Rn and for all

δx∈∞N,Rn ,one hasDxFx, u δx Dxtftxt, ut δxt−δxt1 t∈N.

Proof. Letx, u ∈ ∞N,Ω × U.Let KC be the constant of iii with C Cx . So for all

t∈N, ftxt, ut −xt1 ≤KCsupt∈Nxt.So one hasFx, u ∈∞N,Rn .

Assume now that hypothesesi andiv ofTheorem 2.1hold. Let us show thatx→ Fx, u is of classC0_{onBx, r .Takex}0_{, u ∈Bx, r × U.Let >0 be given. Letx∈Bx, r} be such thatx−x0_{<min{2r, /2}.Then, for allt∈N,f}

txt, ut −xt1ftxt0, ut −x_t0₁ ≤

ftx0t, ut −ftxt, ut xt1−xt01 ≤ supyt∈xt0,xtDxtftyt, ut · xt−x

0

txt1−x0t1 <

/2 sup_t≥0Dxtftyt, ut 1 < underiv , which implies thatFx, u −Fx

0_{, u}

∞< .

Let us now show that x → Fx, u is Fr´echet-differentiable on Bx, r . Take

x0_{, u ∈ Bx, r × U. Let > 0 be given. Let x ∈ Bx, r be such that x −}

x0 _{< min{2r, /2}. Then, for all t ∈ N,f}

txt, ut − xt1 ftx0t, ut − x0_t1

Dxtftx0t, ut xt−x0t −xt1−x0t1 ≤supyt∈x0t,xtDxtftyt, ut −Dxtftx

0

t, ut xt−x0t ≤ sup_yt∈x0

t,xtDxtftyt, ut Dxtftx

0

t, ut xt − x0t < under iii . But this implies

that Fx, u −Fx0_{, u −D}

xtftx0t, ut xt−x0t −xt1−x0t1 t∈N∞ < . Thus x → Fx, u

is Fr´echet-differentiable atx0_andD

xFx0, u δx Dxtftx

0

t, ut δxt−δxt1 t∈N.

To show the continuity ofx → DxFx, u at x0 let KB be the constant of hypothesis iii corresponding to Bx, r . Let > 0 be given and let x ∈ Bx, r be such that x −

x0 _{< min{2r, /K}

B}·DxFx, u −DxFx0, u _∞ ≤ supt∈NDxtftx

0

t, ut − Dxtftxt, ut ≤

sup_t∈NKBxt0−xt< .SoFis of classC1.

Lemma 3.3. Under hypothesis (ii) ofTheorem 2.1, for allx∈ X,for allu, u∈ U,for allα∈0,1,

there existsu∈ Usuch that

Jx, u≥αJx, u 1−α Jx, u

Fx, uαFx, u 1−α Fx, u. 3.1

Φxt, ut bounded solutions of first-order linear difference equations.

Let Mt t≥0 ∈ ∞N,Rn,Rn .Assume that supt≥1Mt < 1. Then for all bt t≥0 ∈ functionals, see Aliprantis and Border20. In fact it consistsup to scalar multiples of all extensions of the “limit functional” to∞.

Lemma 3.5 _∞∗N,Rn

Our optimal control problem can be written as the following abstract static optimisation problem in a Banach space:

that satisfies all conditions ofTheorem 4.3, Ioffe-Tihomirov17. So we can apply this theorem and obtain the existence ofλ0∈R, P ∈_∞∗N,Rn ,not all zero,λ0 ≥0,such that:

AE denotes the adjoint equation of this problem and PMP the Pontryagin maximum principle. They can be written, respectively:

Since the inequality holds for everyu∈ U,we obtain

λ0βt

Φxt,ut

−Φxt, ut

pt1, ft

xt,ut

−ft

xt, ut ≥0, ∀ut∈U 3.18

usingθ,ftxt,ut −ftxt, ut t≥00 asftxt,ut −ftxt, ut t≥0is of finite support.

Lemma 3.6. (λ0/0).

Proof. Recall we obtained the existence ofλ0 ∈ R, P ∈ ∗∞N,Rn ,not all zero,λ0 ≥ 0,such that:

λ0DxJ

x,uDxF∗

x,uP 0. 3.19

Ifλ00,thenP0 since ImDxFx, u ∞N,Rn .

Henceλ0/0.We can set it equal to one.

FromLemma 3.6and the previous results, conclusionsa andb are satisfied. Conclusionc is a straightforward consequence of the belonging ofpt1 tto1N,Rn .

Lemma 3.7. (θ0).

Proof. Indeed we obtainedθ,Dxtftxt,ut δxt−δxt1 t≥00,for allδx∈∞N,R

n _.Using

ImDxFx, u ∞N,Rn one hasθ, h0,for allh∈∞N,Rn .Thusθ0.

Proof ofTheorem 2.2. Define F on X × U such that Fx, u fxt, ut − xt1 t≥0. Under hypothesisi of Theorem 2.2, for allu ∈ U,the mappingsx → Jx, u and x → Fx, u

are of classC1_{onX.The proof can be found in19.}

We consider the proof ofLemma 3.4and we setMtDxtfxt,ut .Then the proof goes

like that ofTheorem 2.1.

4. Results for unbounded problems

We study now problems of maximization over admissible processes which are not necessarily bounded when the optimal solution is bounded. So consider the following problems.

P2 MaximizeJx, u on domJ∩Admη .

P3 Findx, u ∈domJ∩Admη such that, for allx, u ∈Admη , Jx, u ≥lim sup_{T→ ∞}T_t0βt_Φx

t, ut .

P4 Findx, u ∈Admη such that, for allx, u ∈Admη ,

lim infT→ ∞Tt0βtΦxt,ut −Tt0βtΦxt, ut ≥0. P5 Findx, u ∈Admη such that, for allx, u ∈Admη ,

lim sup_{T→ ∞}T_t0βtΦxt,ut −Tt0βtΦxt, ut ≥0.

Remark 4.1. Notice that x, u is an optimal solution of P3 implies x, u is an optimal solution ofP4 which impliesx, u is an optimal solution ofP5 .

Moreover ifx, u is a bounded optimal solution ofP4 thenP3 andP4 reduce to the same problem.

Lemma 4.2. The two following assertions hold.

a Ifx, u is an optimal solution of problem (P2), (P3), (P4) or (P5) andx, u ∈ X × Uthen x, u is an optimal solution of problemP1. ThereforeTheorem 2.1applies.

b Assume Φ ≥ 0 on Ω×U.If x, u is an optimal solution of problem (P3) or (P4) and x, u ∈ X × UthendomJAdmη .

Proof. a SinceX × U ∩Admη ⊂domJ∩Admη ⊂Admη ,a bounded optimal solution

ofP2 orP3 is an optimal solution ofP1 . Suppose now thatx, u is a bounded optimal solution of P4 that is lim infT→ ∞Tt0βtΦxt,ut −Tt0βtΦxt, ut ≥ 0, for all x, u ∈ Admη .Sincex, u ∈ X × Uthis can be writtenJx, u ≥lim sup_{T→ ∞}T_t0βt_Φx

t, ut ,for all x, u ∈Admηand so in particular for allx, u ∈ X × U.

In that case lim sup_{T→ ∞}T_t0βt_Φx

t, ut Jx, u .The proof is analogous forP5 . b If x, u is an optimal solution of problem P3 and x, u ∈ X × U, one has

∞> Jx, u ≥lim sup_{T→ ∞}T_t0βt_Φx

t, ut ,for allx, u ∈Admη.SinceΦ≥0,the sequence T

t0βtΦxt, ut _Tis increasing and since it is also upper bounded it converges inR. So lim sup_{T→ ∞}T_t0βt_Φx

t, ut limT→ ∞Tt0βtΦxt, ut Jx, u andx, u ∈domJ.

Theorem 4.3. Letftffor allt∈N.One assumes the following conditions fulfilled:

i Φ≥0onΩ×U.

ii For allx∈Ω,there existsv∈Usuch thatxfx, v .

Then one has

a sup_{x,u∈domJ∩Admη} Jx, u sup_{x,u∈X×U∩Admη}Jx, u .

b Ifx, u is an optimal solution of problemP1, then it is an optimal solution of problems (P3), (P4), and (P5) which all reduce to the same problem.

Remark 4.4. b shows that under a nonnegativity assumption, solving the problem in the space of bounded processes provides solutions for problems in spaces of admissible processes which are not necessarily bounded. This type of results is in the spirit of Blot and Cartigny

21where problems are studied in the continuous-time case.

Proof. a It is clear that the following inequality holds:

sup x,u∈domJ∩Admη

Jx, u≥ sup

x,u∈X×U∩Admη

Let x, u ∈ domJ ∩ Admη . Let > 0 be given and let T T be such that 0 ≤

t>Tβ

t_Φx

t,ut ≤.Set

x_t

⎧ ⎨ ⎩

xt, ift≤T

xT, ift > T,

u_t

⎧ ⎨ ⎩

ut, ift≤T

u_T, ift > T,

4.2

where u_T is such that x_T fx_T, u_T .x x_t tand u ut t are bounded and x, u ∈ X × U ∩Admη .

SinceΦ≥0 onΩ×Uone has

Jx, x

∞

t0

βtΦxt, ut

T

t0

βtΦxt,ut

∞ tT1

βtΦxT, uT

≥T

t0

βtΦxt,ut

sup x,u∈X×U∩Admη

Jx, u≥Jx, u≥

T

t0

βtΦxt,ut

4.3

so we obtain

sup

x,u∈X×U∩Admη

Jx, u≥Jx, u. 4.4

Since this is true fo all >0,letting → 0,we obtain

sup x,u∈X×U∩Admη

Jx, u≥ sup

x,u∈domJ∩Admη

Jx, u. 4.5

b SinceΦ≥0,for allx, u ∈Admη ,the sequenceT_t0βt_Φx

t, ut Tis nonnegative and nondecreasing so it converges in0,∞.

So lim sup_{T→ ∞}T_t0βt_Φx

t, ut limT→ ∞Tt0βtΦxt, ut .HenceP3 ,P4 reduce to the same problem. SimilarlyP5 reduces to it. Letx, u be an optimal solution of problem

P1 and suppose it is not an optimal solution of problemP3 . So there existsx, u ∈Admη

such that limT→ ∞Tt0βtΦxt, ut ∞that is

∀R∈R, ∃TR∈N∗, ∀T ≥TR, T

t0

βtΦxt, ut

LetR ∈ RandT TR.Constructx x_t tandu ut tas ina . Thus _∞

t0βtΦxt, ut _T

t0βtΦxt, ut _∞

tT1βtΦxT, uT ≥R.

Hence we obtain sup_{x,u∈X×U∩Admη} Jx, u ≥R,so sup_{x,u∈X×U∩Admη} Jx, u ∞

which contradicts the hypothesis sox, u is an optimal solution of problemP3.

Following Michel,22, for allt∈Nand for allxt, xt1 ∈Ω×Ω,we defineAtxt, xt1 as the set of theνt, ζt ∈R×Rnfor which there existsut ∈Usatisfyingνt≤βtΦxt, ut , ζt

fxt, ut −xt1.We also defineBtxt, xt1 as the set of theνt, ζt ∈ R×Rnfor which there existsut, αt ∈U×Rnsatisfyingνt≤βtΦxt, ut , αkζkt fkxt, ut −xkt1for allk1, . . . , n.

Theorem 4.5. Letftffor allt∈N.Letx, u be an optimal solution of problemP1. One assumes

the following conditions fulfilled.

i Φ≥0onΩ×U.

ii For allx∈Ω,there existsv∈Usuch thatxfx, v .

iii For allt∈ N,the mappingsx→ Φx,ut andx →fx,ut are Fr´echet-differentiable at

xt.

iv For allt∈N,for allxt, xt1 ∈Ω×Ω,coAtxt, xt1 ⊂Btxt, xt1 wherecodenotes the

convex hull.

v For allt∈N, Dxtfxt,ut is invertible.

Then there existsλ0∈R, pt1 t∈Rn Nsuch thatλ0, p1 /0and

a ptpt1◦Dxtfxt,ut λ0βtDxtΦxt,ut , for allt,

b λ0βtΦxt,ut pt1, fxt,ut ≥λ0βtΦxt, ut pt1, fxt, ut ,for allt,for allut∈U.

Remark 4.6. Notice that condition iv is a convexity condition and that condition ii of Theorem 2.2implies this condition iv . Condition ii ofTheorem 2.2 is equivalent to the following condition: for allt,the setAtxt, xt1 is convex.

Proof. UseTheorem 4.3of this paper and apply Theorem 3 in Blot-Chebbi5.

5. Sufficient conditions of optimality

LetHtxt, ut, pt1 βtΦxt, ut pt1, ftxt, ut ,for allt∈N.

Theorem 5.1. Letx, u ∈ X × U ∩Admη where Uis convex. One assumes that there exists

pt1 t∈1N,Rn and that the following conditions are fulfilled.

i The mappings x, u → Φx, u and for allt ∈ N,x, u → ftx, u are of class C1 on Ω×U.

ii ptpt1◦Dxtftxt,ut βtDxtΦxt,ut , for allt≥1. iii βt_Φx

t,ut pt1, ftxt,ut ≥βtΦxt, ut pt1, ftxt, ut , for allt, for allut∈U. iv The mappingHtis concave with respect toxt, ut ,for allt.

One can weaken the hypothesis of concavity ofHtwith respect toxtandutand replace it by the concavity ofH_t∗with respect toxtas the following theorem shows.See23for a quick survey of sufficient conditions.

LetH_t∗xt, pt1 maxut∈UHtxt, ut, pt1 .

The maximum is attained sinceUis compact.

Theorem 5.3. Letx, u ∈ X × U ∩Admη .One assumes that there existspt1 t∈1N,Rn and

that the following hypotheses are fulfilled.

i For allu∈U,the mappingsx→Φx, u and for allt∈N, x→ftx, u are of classC1on Ω.

ii Alsoiii of the previous theorem.

iv The mappingHt∗is concave with respect toxt,for allt. concavity ofH_t∗with respect toxtgives

Notice that ifx, u ∈Admη with lim sup_T→∞pT1, xT1−xT10 we obtain that

x, u is a solution ofP5.

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