Infinite Dimensional Multipliers and Pontryagin Principles for Discrete-Time Problems
aa r X i v : . [ m a t h . O C ] J un INFINITE DIMENSIONAL MULTIPLIERS AND PONTRYAGINPRINCIPLES FOR DISCRETE-TIME PROBLEMS
MOHAMMED BACHIR AND JO¨EL BLOT
Abstract.
The aim of this paper is to provide improvments to Pontryaginprinciples in infinite-horizon discrete-time framework when the space of statesand of space of controls are infinite-dimensional. We use the method of re-duction to finite horizon and several functional-analytic lemmas to realize ouraim.
Key Words: Pontryagin principle, infinite horizon, difference equation, differenceinequation, Banach spaces, Baire category.M.S.C. 2010: 49J21, 65K05, 39A99, 46B99, 54E52.1.
Introduction
We treat of problems of Optimal Control in infinite horizon which are governedby discrete-time controlled dynamical systems in the following forms x t +1 = f t ( x t , u t ) , t ∈ N , (1.1)or x t +1 ≤ f t ( x t , u t ) , t ∈ N . (1.2) X and U are real Banach spaces, the state variable is x t ∈ X t ⊂ X , the controlvariable is u t ∈ U t ⊂ U and f t : X t × U t → X t +1 is a mapping. For (1.2), X isendowed with a structure of ordered Banach space, and its positive cone, X + := { x ∈ X : x ≥ } , is closed, convex and satisfies X + ∩ ( − X + ) = { } . An initial state, σ ∈ X is fixed. We define Ead( σ ) the set of the processes (( x t ) t ∈ N , ( u t ) t ∈ N ) whichbelong to Q t ∈ N X t × Q t ∈ N U t , which satisfy x = σ and (1.1) for all t ∈ N . We defineIad( σ ) the set of the processes (( x t ) t ∈ N , ( u t ) t ∈ N ) which belong to Q t ∈ N X t × Q t ∈ N U t ,which satisfy x = σ and (1.2) for all t ∈ N .To define criteria, we consider functions φ t : X t × U t → R for all t ∈ N . Fromthese functions we build the functional J (( x t ) t ∈ N , ( u t ) t ∈ N ) := + ∞ X t =0 φ t ( x t , u t ) . (1.3)Notice that J is not defined for all the processes (( x t ) t ∈ N , ( u t ) t ∈ N ) ∈ Q t ∈ N X t × Q t ∈ N U t . And so we define the set of the processes (( x t ) t ∈ N , ( u t ) t ∈ N ) ∈ Ead( σ )(respectively in Iad( σ )) such that the series P + ∞ t =0 φ t ( x t , u t ) converges into R , andwe denote this set by Edom( σ ) (respectively Idom( σ )).We also consider other criteria to define the following problems. Date : June 28th 2016. ( PE ( σ )) : Find ((ˆ x t ) t ∈ N , (ˆ u t ) t ∈ N ) ∈ Edom( σ ) such that J ((ˆ x t ) t ∈ N , (ˆ u t ) t ∈ N ) ≥ J (( x t ) t ∈ N , ( u t ) t ∈ N ) for all (( x t ) t ∈ N , ( u t ) t ∈ N ) ∈ Edom( σ ).( PE ( σ )) : Find ((ˆ x t ) t ∈ N , (ˆ u t ) t ∈ N ) ∈ Ead( σ ) such thatlim sup T → + ∞ P Tt =0 ( φ t (ˆ x t , ˆ u t ) − φ t ( x t , u t )) ≥ x t ) t ∈ N , ( u t ) t ∈ N ) ∈ Ead( σ ).( PE ( σ )) : Find ((ˆ x t ) t ∈ N , (ˆ u t ) t ∈ N ) ∈ Ead( σ ) such thatlim inf T → + ∞ P Tt =0 ( φ t (ˆ x t , ˆ u t ) − φ t ( x t , u t )) ≥ x t ) t ∈ N , ( u t ) t ∈ N ) ∈ Ead( σ ).We also consider similar problems when the system is governed by (1.2) instead of(1.1).( PI ( σ )) : Find ((ˆ x t ) t ∈ N , (ˆ u t ) t ∈ N ) ∈ Idom( σ ) such that J ((ˆ x t ) t ∈ N , (ˆ u t ) t ∈ N ) ≥ J (( x t ) t ∈ N , ( u t ) t ∈ N ) for all (( x t ) t ∈ N , ( u t ) t ∈ N ) ∈ Idom( σ ).( PI ( σ )) : Find ((ˆ x t ) t ∈ N , (ˆ u t ) t ∈ N ) ∈ Iad( σ ) such thatlim sup T → + ∞ P Tt =0 ( φ t (ˆ x t , ˆ u t ) − φ t ( x t , u t )) ≥ x t ) t ∈ N , ( u t ) t ∈ N ) ∈ Iad( σ ).( PI ( σ )) : Find ((ˆ x t ) t ∈ N , (ˆ u t ) t ∈ N ) ∈ Iad( σ ) such thatlim inf T → + ∞ P Tt =0 ( φ t (ˆ x t , ˆ u t ) − φ t ( x t , u t )) ≥ x t ) t ∈ N , ( u t ) t ∈ N ) ∈ Iad( σ ).These problems are classical in the theory of infinite-horizon discrete-time OptimalControl, [9], [5].A way to establish necessary optimality conditions in the form of Pontryaginprinciples for the above-mentionned problems is the method of reduction to finitehorizon which appears in [4]. In [5] several variations of this method are givenin the setting of the finite dimension, and in [1] we find the use of this methodin the setting of the infinite dimension. The basic idea of this method is thatwhen ((ˆ x t ) t ∈ N , (ˆ u t ) t ∈ N ) is optimal for one of the previous problems, its restrictionto { , ..., T } is an optimal solution of a finite-horizon optimization problem. Usingon these finite-horizon optimization problems a Karush-Kuhn-Tucker theorem or aMultipliers Rule, we obtain multipliers indexed by the finite horizon T . The secondstep is to build, from these multipliers sequences, multipliers which are suitable forthe infinite-horizon problems.When X and U have an infinite dimension, several difficulties arise, notably dueto the closure of the ranges of linear operators, and due to the fact that in infinitedimensional dual Banach space, the origine is contained in the weak-star closure ofits unit sphere.Now we briefly describe the contents of the paper. In Section 2, we give thestatements of the main results which are Pontryagin principles. In Section 3, weestablish results of Functional Analysis which are useful in the sequel. In Section 4,we establish results on Lagrange and Karush-Khun-Tucker multipliers. In Section5, we give the proofs of the results of section 2. In Section 6, we give some additionalapplications of results that we use in the proof of the main theorems.2. The pontryagin principles.
First we specify some notation.We denote by
Int ( A ) the topological interior of a set A and by A its closure . When Z is a vector normed space, z ∈ Z and r ∈ (0 , + ∞ ), B Z ( z, r ) denotes the closedball with z as center and r as ray. The set co( A ) (respectively co( A )) stands for the convex hull (respectively the closed convex hull ) of a subset A in Z , Aff( A ) stands NFINITE DIMENSION 3 for the affine hull of a subset A in Z . The relative interior of A ⊂ Z , denoted ri( A ),is the topological interior of A in the topological subspace Aff( A ).In the paper, the assumptions that we will use in our results belong to thefollowing list of conditions.(A1) For all t ∈ N , X t is a nonempty open subsets of X , and U t is a nonemptysubsets of U .(A2) X is separable.And when ((ˆ x t ) t ∈ N , (ˆ u t ) t ∈ N ) is a given process and when σ ∈ X is given, weconsider the following conditions.(A3) For all t ∈ N , φ t is Fr´echet differentiable at (ˆ x t , ˆ u t ), f t is continuouslyFr´echet differentiable at (ˆ x t , ˆ u t ).(A4) for all t ∈ N ∗ , we have 0 ∈ Int [ Df t (ˆ x t , ˆ u t )(( X × T U t (ˆ u t )) ∩ B X × U )], where B X × U denotes the closed unit ball of X × U .(A5) For all t ∈ N , the range of D f t (ˆ x t , ˆ u t ) is closed and its codimension (in X )is finite.(A6) There exists s ∈ N such that A s := D f s (ˆ x s , ˆ u s )( T U s (ˆ u s )) contains a closedconvex subset K with ri( K ) = ∅ and such that Aff( K ) is of finite codimen-sion in X .Recall that T U t (ˆ u t ) := { α ( u t − ˆ u t ) : α ∈ [0 , + ∞ ) , u t ∈ U t } . We have not assumethat the sets U t are open, but when we speak of the differentiability of a mapping f on X t × U t at (ˆ x t , ˆ u t ), the meaning is that there exists a differentiable function ˜ f defined on an open neighborhood of (ˆ x t , ˆ u t ) which is equal to f on the intersectionof this neighborhood and X t × U t . When we speak of tangent cone, we considerthe case where U t is convex. Remark 2.1.
Note that the condition (A6) is satisfied and is included in (A5),whenever there exists an s ∈ N such that T U s (ˆ u s ) = X , in particular, if ˆ u s belongsto the interior of U s . The first main result concerns the problems governed by (1.1).
Theorem 2.2.
Let ((ˆ x t ) t ∈ N , (ˆ u t ) t ∈ N ) be an optimal process of ( PE k ( σ )) when k ∈{ , , } . Under [(A1)-(A6)], we assume moreover that U t is convex for all t ∈ N .Then, there exist λ ∈ R and ( p t ) t ≥ ∈ ( X ∗ ) N ∗ such that the following conditionshold. (1) ( λ , p t ) = (0 , , for all t ≥ s . (2) λ ≥ . (3) p t = p t +1 ◦ D f t (ˆ x t , ˆ u t ) + λ .D φ t (ˆ x t , ˆ u t ) for all t ∈ N ∗ . (4) h λ .D φ t (ˆ x t , ˆ u t ) + p t +1 ◦ D f t (ˆ x t , ˆ u t ) , u t − ˆ u t i ≤ for all t ∈ N and for all u t ∈ U t . The second main result concerns the problems governed by (1.2).
Theorem 2.3.
Let ((ˆ x t ) t ∈ N , (ˆ u t ) t ∈ N ) be an optimal process of ( PI k ( σ )) when k ∈{ , , } . Under [(A1)-(A6)], we assume moreover that U t is convex for all t ∈ N and that Int ( X + ) = ∅ . Then, there exist λ ∈ R and ( p t ) t ≥ ∈ ( X ∗ ) N ∗ such thatthe following conditions hold. (1) ( λ , p t ) = (0 , , for all t ≥ s . (2) λ ≥ , and p n ≥ for all n ∈ N ∗ . (3) p t = p t +1 ◦ D f t (ˆ x t , ˆ u t ) + λ .D φ t (ˆ x t , ˆ u t ) for all t ∈ N ∗ . BACHIR AND BLOT (4) h λ .D φ t (ˆ x t , ˆ u t ) + p t +1 ◦ D f t (ˆ x t , ˆ u t ) , u t − ˆ u t i ≤ for all t ∈ N and for all u t ∈ U t . The proofs of Theorem 2.2 and Theorem 2.3 are based on the following two ideas:the first one is the reduction to finite horizon given by Lemma 4.1 and the secondone is to find criteria ensuring that the multipliers are not trivial in the infinitehorizon. This criteria will be given by Lemma 3.3.3.
Preliminary results of Functional Analysis
It is well known from Josefson-Nissenzweig theorem, (see [[6], Chapter XII]) thatin infinite dimensional Banach space Z , there always exists a sequence ( p n ) n in thedual space Z ∗ that is weak ∗ null and inf n ∈ N k p n k >
0. In this section, we look aboutreasonable and usable conditions on a sequence of norm one in Z ∗ such that thissequence does not converge to the origin in the w ∗ -topology. This situation has theinterest, when we are looking for nontrivial multipliers for optimization problems,and was encountered several times in the literature. See for example [1] and [3].The key is Lemma 3.3 which permits to provide a solution to this problem. Wesplit this section in two subsections. The first is devoted to establish an abstractresult (Lemma 3.3) which permits to avoid the Josefson-Nissenzweig phenomenon.The second is devoted to the consequences of this abstract result which are usefulfor our optimal control problem.We need the following classical result. Proposition 3.1.
Let C be a convex subset of a normed vector space. Let x ∈ Int ( C ) and x ∈ C . Then, for all α ∈ (0 , , we have αx + (1 − α ) x ∈ Int ( C ) . We deduce the following useful proposition.
Proposition 3.2.
Let ( F, k . k F ) be a normed vector space and C be a closed convexsubset of F with non empty interior. Suppose that D ⊂ C is a closed subset of C with no empty interior in ( C, k . k F ) (for the topology induced by C ). Then, theinterior of D is non empty in ( F, k . k F ) .Proof. On one hand, there exists x such that x ∈ Int ( C ). On the other hand,since D has no empty interior in ( C, k . k F ), there exists x ∈ D and ǫ > B F ( x , ǫ ) ∩ C ) ⊂ D . Using Proposition 3.1, we obtain that for all α ∈ (0 , αx + (1 − α ) x ∈ Int ( C ). Since αx + (1 − α ) x → x when α →
0, there existsome small α and an integer number N ∈ N ∗ such that B F ( α x +(1 − α ) x , N ) ⊂ ( B F ( x , ǫ ) ∩ C ) ⊂ D . Thus D has a non empty interior in F . (cid:3) A key lemma.
A map p from a vector space Z into R is said to be subadditive if and only if, for all x, y ∈ Z one has p ( x + y ) ≤ p ( x ) + p ( y ) . A map p is said to be sublinear if it is subadditive and satisfies p ( λz ) = λp ( z ) forall λ ≥ z ∈ Z .We give now our principal lemma. This lemma is based on the Baire categorytheorem. Lemma 3.3.
Let Z be a Banach space. Let K be a non empty closed convex subsetof Z and suppose that ri( K ) = ∅ . Let T be any nonempty set and ( p n ) n ∈T be a NFINITE DIMENSION 5 collection of subadditive and lower semicontinuous functions on Z and let ( λ n ) n ∈T be a collection of nonnegative real number. Suppose that, for all z ∈ K , there exists C z ∈ R such that, for all n ∈ T , p n ( z ) ≤ C z λ n .Then, for all a ∈ K , there exists b a ∈ Aff( K ) such that for all bounded subset B of Aff( K ) there exists R B ≥ such that ∀ n ∈ T , sup h ∈ B p n ( h − a ) ≤ R B · ( λ n + p n ( b a − a )) . Proof.
For each m ∈ N , we set F m := { z ∈ K : ∀ n ∈ T , p n ( z ) ≤ mλ n } . The sets F m are closed subsets of K . Notice that F m = K ∩ \ n ∈T p − n (] − ∞ , mλ n ]) ! , where, for each n ∈ T , p − n (] −∞ , mλ n ]) is a closed subset of Z by the semicontinuityof p n . On the other hand, we have K = S m ∈ N F m . The inclusion ⊃ is trivial. Weprove the inclusion ⊂ . For each z ∈ K , there exists C z ∈ R such that p n ( z ) ≤ C z λ n for all n ∈ T . If C z ≤
0, we have that z ∈ F . If C z >
0, we put m := [ C z ] + 1where [ C z ] denotes the integer part of C z , then we have that z ∈ F m . We deducethen that for all m ∈ N , the sets F m − a are closed subset of K − a and that K − a = S m ∈ N ( F m − a ). Using the Baire category Theorem on the completemetric space K − a , we get an m ∈ N such that F m − a has a nonempty interiorin K − a . Since by hypothesis K − a has a nonempty interior in the normed vectorsubspace F := Aff( K ) − a of Z , then by using Proposition 3.2 we obtain that F m − a has a nonempty interior in F . So there exists z ∈ F m − a and someinteger number N ∈ N ∗ such that B F ( z , N ) := ( F ∩ B Z ( z , N )) ⊂ F m − a . Inother words, for all z ∈ B F ( b, N ) ⊂ F m where b := a + z ∈ F m ⊂ F and all n ∈ T , we have: p n ( z ) ≤ m λ n . (3.1)Now, let B be a nonempty bounded subset of F , there exists an integer number N B ∈ N ∗ such that B ⊂ B F (0 , N B ). On the other hand, for all h ∈ B , thereexists z h ∈ B F ( b, N ) such that h = N B N · ( z h − b ) (it sufficies to see that z h := b + hN B .N ∈ B F ( b, N )). So using (3.1) and the subadditivity of p n , we obtain that,for all n ∈ T : p n ( h ) = p n ( N B N · ( z h − b )) ≤ N B N · p n ( z h − b ) ≤ N B N · ( p n ( z h ) + p n ( − b )) ≤ N B N m λ n + N B N · p n ( − b ) ≤ N B N m λ n + N B N m · p n ( − bm )= N B N m (cid:18) λ n + p n ( − bm ) (cid:19) . Setting R B := N B N m and b := − bm ∈ F and by taking the supremum on B , weobtain for all n ∈ T , sup h ∈ B p n ( h ) ≤ R B · ( λ n + p n ( b )) . (3.2) BACHIR AND BLOT
Now, let ˜ B be any bounded subset of the closure F of F . There exists a boundedsubset of F , B , such that ˜ B = B . Hence for each z ∈ ˜ B , there exists a sequence( h k ) k in B such that h k → z when k → + ∞ . Thus, using the lower semicontinuityof p n for all n ∈ N and the inequality (3.2), we obtain p n ( z ) ≤ lim inf k → + ∞ p n ( h k ) ≤ sup h ∈ B p n ( h ) ≤ R B · ( λ n + p n ( b )) , and by taking the supremum on ˜ B , we obtain, for all n ∈ T ,sup z ∈ ˜ B p n ( z ) ≤ R B · ( λ n + p n ( b )) . Since F = Aff( K ) − a , by changing the bounded subsets ˜ B of F by B − a , where B is a bounded subset of Aff( K ) and by setting b a := b + a ∈ Aff( K ), we concludethe proof. (cid:3) We obtain the following corollary, which may be of interest in some cases.
Corollary 3.4.
Let Z be a Banach space and let A be a non empty subset of Z .Let T be any nonempty set and ( p n ) n ∈T be a collection of sublinear and lowersemicontinuous functions on Z and let ( λ n ) n ∈T be a collection of nonegative realnumber. Let C : Z −→ R be a upper semicontinuous function. Suppose that ∀ n ∈ T , ∀ z ∈ A, p n ( z ) ≤ C ( z ) λ n . (3.3) If ri(co( A )) = ∅ , then, for all a ∈ K , there exists b a ∈ Aff(co( A )) such that for allbounded subset B of Aff(co( A )) there exists R B ≥ such that ∀ n ∈ T , sup h ∈ B p n ( h − a ) ≤ R B · ( λ n + p n ( b a − a )) . Proof.
We can apply Lemma 3.3, with K = co( A ). For this, it suffices to establishthat ∀ n ∈ T , ∀ z ∈ co( A ) , p n ( z ) ≤ C ( z ) λ n . The previous inequality is obtained by using (3.3), the sublinearity and semiconti-nuity of p n for all n ∈ N , together with the upper semicontinuity of the function C . (cid:3) Preliminaries for multipliers in infinite horizon.
As consequence ofLemma 3.3, we obtain the following proposition. The sequences ( λ n ) n ∈ ( R + ) N and ( f n ) n ∈ ( Z ∗ ) N in the following result, correspond to the multipliers. Proposition 3.5.
Let Z be a Banach space. Let ( f n ) n ∈ ( Z ∗ ) N be a sequence oflinear continuous functionnals on Z and let ( λ n ) n ∈ ( R + ) N such that λ n → when n → + ∞ . Let K be a non empty closed convex subset of Z such that ri ( K ) = ∅ .Suppose that (1) for all z ∈ K , there exists a real number C z such that, for all n ∈ N , wehave f n ( z ) ≤ C z λ n . (2) f n w ∗ → when n → + ∞ .Let a ∈ K and set X := Aff( K ) − a . Then, we have, (i) k ( f n ) | X k X ∗ → when n → + ∞ . (ii) If moreover we assume that the codimension of X in Z is finite, then k f n k Z ∗ → when n → + ∞ . NFINITE DIMENSION 7
Proof.
Using Lemma 3.3 with T = N , the linear continuous functions f n and thebounded set B := S X + a of Aff( K ) (where, S X denotes the sphere of X ), we geta point b depending only on X and a constant R B ≥ k ( f n ) | X k X ∗ = sup k h k X =1 f n ( h ) ≤ R B · ( λ n + f n ( b )) . Since f n w ∗ → n → + ∞ ) and λ n → n → + ∞ ) we obtain that k ( f n ) | X k X ∗ → n → + ∞ ). Suppose now that X is of finite codimension in Z , then there existsa finite-dimensional subspace E of Z , such that Z = X ⊕ E . Thus, there exists L > k f n k Z ∗ ≤ L (cid:0) k ( f n ) | E k E ∗ + k ( f n ) | X k X ∗ (cid:1) . Since f n w ∗ → n → + ∞ ) and since the weak-star topology and the norm topologycoincids on E since its dimension is finite, we have that k ( f n ) | E k E ∗ −→ n → + ∞ ). On the other hand, we proved above that k ( f n ) | X k X ∗ →
0. Thus, k f n k Z ∗ −→ n → + ∞ ). (cid:3) Remark 3.6.
Proposition 3.5 shows that under the condition (1) , we have that f n w ∗ , whenever k ( f n ) | X k X ∗ . If moreover, X is of finite codimension in Z ,then f n w ∗ , whenever k f n k Z ∗ . Thus, the condition (1) is a criterion ensuringthat a sequence of norm one in an infinite dual Banach space, does not convergesto in the weak ∗ topology. To ensure that the multipliers are nontrivial at the limit, the authors in [3] useda lemma from [[2], pp. 142, 135], which can be recovered by taking C ( z ) = 1 forall z ∈ Z in the following corollary. Definition 3.7.
A subset Q of a Banach space Z is said to be of finite codimensionin Z if there exists a point z in the closed convex hull of Q such that the closedvector space generated by Q − z := { q − z | q ∈ Q } is of finite codimension in Z and the closed convex hull of Q − z has a no empty interior in this vector space. Corollary 3.8.
Let Q ⊂ Z be a subset of finite codimension in Z . Let C : Z −→ R be a upper semicontinuous function. Let δ > , ( f k ) k ∈ ( Z ∗ ) and λ k ≥ , λ k → k → + ∞ ) such that ( i ) k f k k ≥ δ , for all k ∈ N and f k w ∗ → f ( k → + ∞ ) . ( ii ) For all z ∈ Q , and for all k ∈ N , f k ( z ) ≤ C ( z ) λ k .Then, f = 0 .Proof. First, note that from the condition ( ii ), the linearity and continuity of f k , k ∈ N and the upper semicontinuity of C , we have also that, for all z ∈ co( Q )and for all k ∈ N , f k ( z ) ≤ C ( z ) λ k . Suppose by contradiction that f = 0, thenusing Proposition 3.5 and the fact that Q is of finite codimension in Z , we get that k f k k Z ∗ → n → + ∞ ), which contredicts the condition ( i ). (cid:3) The following proposition is used in the proof of our main result Theorem 2.2.In Proposition 3.9, the sequence ( β ( n ) ) n ≥ in( R + ) and the list ( f ( n ) t ) ≤ t ≤ n +1 ∈ ( X ∗ ) n +1 , correspond to the non trivial multipliers at the finite horizon n , for all n ≥
2. The aim is to find conditions under which, these sequences have subsequenceswhich converge to non trivial multipliers at the infinite horizon.
BACHIR AND BLOT
Proposition 3.9.
Let Z be a separable Banach space and Z ∗ its topological dual.Let K be a closed convex subset of Z such that ri ( K ) = ∅ and that Aff( K ) is offinite codimension in Z . Let ( β ( n ) ) n ≥ be a sequence of nonegative real number and ( f ( n ) t ) ≤ t ≤ n +1 ∈ ( Z ∗ ) n +1 , for all n ≥ . Let s ∈ N ∗ be a fixed natural number.Suppose that: (1) for all n ≥ , β n + k f ns k Z ∗ = 1 , (2) there exists a t , b t ≥ such that k f nt k Z ≤ a t β n + b t k f ns k Z for all n ≥ andfor all ≤ t ≤ n + 1 , (3) for all z ∈ K , there exist a real number c z such that: f ns ( z ) ≤ c z β n for all n ≥ .Then there exist a strictly increasing map k n k , from N into N , β ∈ R + and ( f t ) t ≥ ∈ ( Z ∗ ) N such that: ( i ) β n k −→ β when k → + ∞ , ( ii ) for each t ∈ N , f n k t w ∗ −→ f t when k → + ∞ , ( iii ) ( β, f s ) = (0 , .Proof. From (1) and (2) we get that, for each t ≥
1, the sequences ( f nt ) ≤ t ≤ n +1 and ( λ n ) n ≥ are bounded. Hence, using the Banach-Alaoglu theorem and thediagonal process of Cantor, we get a strictly increasing map k n k , from N into N , a nonegative real number β ∈ R + , and a sequence ( f t ) t ≥ ∈ ( Z ∗ ) N ∗ satisfying( i ) and ( ii ). Suppose by contradiction that ( β, f s ) = (0 , β n k −→ f n k s w ∗ −→ k → + ∞ . Using the condition (3) and Proposition 3.5 we havethat k f n k s k Z ∗ −→ k → + ∞ . Since β n k −→ k → + ∞ , we have also β n k + k f n k s k Z ∗ −→ (cid:3) Multipliers
In this section, after the recall of the method of reduction to finite horizon, weestablish multiplier rules (Lemma 4.5 and Lemma 4.6), in the spirit of Fritz John’stheorem, for the problems of finite horizon.First we recall the method of reduction to finite horizon. When ((ˆ x t ) t ∈ N , (ˆ u t ) t ∈ N )is an optimal solution of ( PE k ( σ )), k ∈ { , , } , we build the following finite-horizon problem.( EF ( σ )) Maximize J T ( x , ..., x T , u , u..., u T ) := P Tt =0 φ t ( x t , u t )when ∀ t ∈ { , ..., T } , x t +1 = f t ( x t , u t ) x = σ, x T +1 = ˆ x T +1 . Similarly, when ((ˆ x t ) t ∈ N , (ˆ u t ) t ∈ N ) is an optimal solution of ( PI k ( σ )), k ∈ { , , } ,we build the following finite-horizon problem( IF ( σ )) Maximize J T ( x , ..., x T , u , u..., u T ) := P Tt =0 φ t ( x t , u t )when ∀ t ∈ { , ..., T } , x t +1 ≤ f t ( x t , u t ) x = σ, x T +1 = ˆ x T +1 . The proof of the following result is similar to the proof given in [4].
Lemma 4.1.
Let k ∈ { , , } . When ((ˆ x t ) t ∈ N , (ˆ u t ) t ∈ N ) is an optimal solution of( PE k ( σ ) ) (respectively ( PI k ( σ ) )), for all T ∈ N , T ≥ , then the restriction (ˆ x , ..., ˆ x T , ˆ u , ..., ˆ u T ) is an optimal solution of ( EF ( σ ) (respectively ( IF ( σ ))). NFINITE DIMENSION 9
To work on these problems, we introduce several notations. We write x T :=( x , ..., x T ) ∈ Q Tt =1 X t and u T := ( u , ..., u T ) ∈ Q Tt =0 U t . For all t ∈ { , ..., T } , wedefine the mapping g Tt : Q Tt =1 X t × Q Tt =0 U t → X t +1 by setting g Tt ( x T , u T ) := − x + f ( σ, u ) if t = 0 − x t +1 + f t ( x t , u t ) if t ∈ { , ..., T − }− ˆ x T +1 + f T ( x T , u T ) if t = T. (4.1)We define g T : Q Tt =1 X t × Q Tt =0 U t → Q Tt =0 X t by setting g t ( x T , u T ) := ( g T ( x T , u T ) , ..., g TT ( x T , u T )) . (4.2)And so the problem ( EF ( σ )) is exactly (cid:26) Maximize J T ( x T , u T )when g T ( x T , u T ) = 0 (4.3)and the problem problem ( IF ( σ )) is exactly (cid:26) Maximize J T ( x T , u T )when g T ( x T , u T ) ≥ g T is of class C at ( ˆx T , ˆu T ) as a composition of mappings of class C , and the calculation of its differential gives Dg T ( x T , u T ) · ( δ x T , δ u T ) = ( Dg T ( x T , u T ) · ( δ x T , δ u T ) , ..., Dg TT ( x T , u T ) · ( δ x T , δ u T ))and we have Dg T ( x T , u T ) · ( δ x T , δ u T ) = − δx + D f ( σ, u ) · δu , and when t ∈ { , ..., T − } , Dg Tt ( x T , u T ) · ( δ x T , δ u T ) = − δx t +1 + D f t ( x t , u t ) · δx t + D f t ( x t , u t ) · δu t , and Dg TT ( x T , u T ) · ( δ x T , δ u T ) = D f T ( x T , u Y ) · δx T + D f T ( x T , u T ) · δu T . Thus in order to study Im Dg T ( ˆx T , ˆu T ) we need to treat the equation Dg T ( ˆx T , ˆu T ) · ( δ x T , δ u T ) = ( b , ..., b T +1 ) . It is the following system b = − δx + D f ( σ, u ) · δu b = − δx + Df (ˆ x , ˆ u ) · ( δx , δu ) ....b T = − δx T + Df T − (ˆ x T − , ˆ u T − ) · ( δx T − , δu T − ) b T +1 = Df T (ˆ x T , ˆ u T ) · ( δx T , δu T ) . (4.5) Lemma 4.2.
Under (A1) and (A3), the set Im D g T ( ˆx T , ˆu T ) is closed into X T +1 .Proof. Suppose that a sequence (( b n , ..., b nT +1 )) n ∈ (Im D g T ( ˆx T , ˆu T )) N convergesto some ( b , b , ..., b T +1 ). We prove that ( b , b , ..., b T +1 ) ∈ Im D g T ( ˆx T , ˆu T ). In-deed, there exists ( δx n , δx n , ..., δx nT ) ∈ X T satisfying b n = − δx n b n = − δx n + Df (ˆ x , ˆ u ) · δx n ....b nT = − δx nT + Df T − (ˆ x T − , ˆ u T − ) · δx nT − b nT +1 = Df T (ˆ x T , ˆ u T ) · δx nT . (4.6) Since ( b n ) n converges to b , we get that ( δx n ) n converges to some δx and so( Df (ˆ x , ˆ u ) · δx n ) n converges to Df (ˆ x , ˆ u ) · δx by continuity. Since ( b n ) n converges to b , we get that ( δx n ) n converges to some δx and so b = − δx + Df (ˆ x , ˆ u ) · δx . We proceed inductively to obtain b − δx b = − δx + Df (ˆ x , ˆ u ) · δx ....b T = − δx T + Df T − (ˆ x T − , ˆ u T − ) · δx T − b T +1 = Df T (ˆ x T , ˆ u T ) · δx T . (4.7)This shows that ( b , b , ..., b T +1 ) ∈ Im D g T ( ˆx T , ˆu T ) and conclude the proof. (cid:3) The proof of the following result is similar to the proof of Lemma 3.10 in [1],replacing Lemma 3.5 in [1] by Lemma 4.2.
Lemma 4.3.
Under (A1), (A3) and (A5), the range Im Dg T ( ˆx T , ˆu T ) is closed in X T +1 . The following theorem was established in the book of Jahn [8] (Theorem 5.3 inp.106-111, and Theorem 5.6, p. 118).
Theorem 4.4.
Let Ξ , Y and Z three real Banach spaces, and ˆ ξ ∈ Ξ . We assumethat the following conditions are fulfilled. (1) Y is ordered by a cone C with a nonempty interior. (2) ˆ S is a convex subset of Ξ with a nonempty interior. (3) I : Ξ → R is a functional which is Fr´echet differentiable at ˆ ξ . (4) Γ : Ξ → Y is a mapping which is Fr´echet differentiable at ˆ ξ . (5) H : Ξ → Z is a mapping which is Fr´echet differentiable at ˆ ξ . (6) S := { ξ ∈ ˆ S : Γ( ξ ) ∈ − C, H ( ξ ) = 0 } is nonempty. (7) Im DH ( ˆ ξ ) is closed into Z .If ˆ ξ is a solution of the following minimization problem (cid:26) Minimize I ( ξ )when ξ ∈ S then there exist λ ∈ [0 , + ∞ ) , Λ ∈ Y ∗ a positive linear functional, Λ ∈ Z ∗ suchthat the following conditions are satisfied: (i) ( λ , Λ , Λ ) = (0 , , h λ D I ( ˆ ξ ) + Λ ◦ D Γ( ˆ ξ ) + Λ ◦ DH ( ˆ ξ ) , ξ − ˆ ξ i ≤ for all ξ ∈ S . Lemma 4.5.
Let ((ˆ x t ) t ∈ N , (ˆ u t ) t ∈ N ) be an optimal process of ( PE k )( σ )) when k ∈{ , , } . Under (A1), (A3) and (A5), we assume moreover that U t is convex for all t ∈ N . Then, for all T ∈ N , T ≥ , there exist λ T ∈ R and ( p Tt ) ≤ t ≤ T +1 ∈ ( X ∗ ) T +1 such that the following conditions hold. (a) λ T and ( p Tt ) ≤ t ≤ T +1 are not simultaneously equal to zero. (b) λ T ≥ . (c) p Tt = p Tt +1 ◦ D f t (ˆ x t , ˆ u t ) + λ T .D φ t (ˆ x t , ˆ u t ) for all t ∈ { , ..., T } . (d) h λ T .D φ t (ˆ x t , ˆ u t ) + p Tt +1 ◦ D f t (ˆ x t , ˆ u t ) , u t − ˆ u t i ≤ for all t ∈ { , ..., T } andfor all u t ∈ U t . NFINITE DIMENSION 11
Proof.
Using Lemma 4.1, we know that ( ˆx T , ˆu T ) = (ˆ x , ..., ˆ x T , ˆ u , ..., ˆ u T ) is anoptimal solution of ( EF ( σ )). We want to use Theorem 4.4 where the inequalityconstraints are absent, and so we don’t nee to the first assumption of Theorem 4.4,and among the conclusions we lost that the p t are positive. We have not inequalityconstraints and so we can delete Γ and conditions on the cone C , and we have H = g T . Using Lemma 4.3, we know that Im Dg T ( ˆx T , ˆu T ) is closed in X T +1 . Andso there exists λ ∈ [0 , + ∞ ) (that is the conclusion (ii)) and Λ ∈ ( X ∗ ) T +1 such( λ , Λ ) = (0 , p ( T ) t the coordinates of Λ in X ∗ , we obtain theconclusion (i). From conclusion (ii) of Theorem 4.4, using the partial differentialswith respect to u T and with respect to u T , and using the openess of Q Tt =1 X t , weobatin λ D J T ( ˆx T , ˆu T ) + Λ ◦ Dg T ( ˆx T , ˆu T ) = 0 h λ D J T ( ˆx T , ˆu T ) + Λ ◦ Dg T ( ˆx T , ˆu T ) , u T − ˆu T i ≤ u T ∈ Q Tt =0 U t . This gives the conclusions ( c ) and ( d ). (cid:3) Lemma 4.6.
Under (A1), (A3) and (A5), we assume moreover that U t is convexfor all t ∈ N and that Int ( X + ) = ∅ . Then, for all T ∈ N , T ≥ , there exist λ T ∈ R and ( p Tt ) ≤ t ≤ T +1 ∈ ( X ∗ ) T +1 such that the following conditions hold. (a) λ T and ( p Tt ) ≤ t ≤ T +1 are not simultaneously equal to zero. (b) λ T ≥ , and p t ≥ for all t ∈ { , ..., T + 1 } . (c) p Tt = p Tt +1 ◦ D f t (ˆ x t , ˆ u t ) + λ T D φ t (ˆ x t , ˆ u t ) for all t ∈ { , ..., T } . (d) h λ T D φ t (ˆ x t , ˆ u t ) + p Tt +1 ◦ D f t (ˆ x t , ˆ u t ) , u t − ˆ u t i ≤ for all t ∈ { , ..., T } andfor all u t ∈ U t .Proof. We procced as in the proof of Lemma 4.5 without deleting the inequalityconstraints, but deleting the equality constraints. (cid:3)
We need the following lemma for the proof of our main result Theorem 2.2.
Lemma 4.7.
Under the assumptions of Lemma 4.5 or Lemma 4.6, suppose more-over that (A4) is satisfied. Then, for all T ∈ N , T ≥ , there exist λ T ∈ R and ( p Tt ) ≤ t ≤ T +1 ∈ ( X ∗ ) T +1 such that the following conditions hold. (1) For all T ≥ , for all s ∈ { , ..., T } and all ≤ t ≤ T + 1 , there exists a t , b t ≥ such that k p Tt k ≤ a t λ T + b t k p Ts k . (2) For all s ∈ { , ..., T } , ( λ T , p Ts ) = 0 . (3) For all s ∈ { , ..., T } , for all z ∈ A s := D f s − (ˆ x s − , ˆ u s − )( T U s − (ˆ u s − )) ,there exists C z ∈ R such that: ∀ T ≥ , p Ts ( z ) ≤ C z λ T .Proof. By adding ( c ) and ( d ) of Lemma 4.5 (respectively Lemma 4.6) we obtain forall t ∈ { , ..., T } , for all h ∈ X and for all u t ∈ U t h p Tt +1 , D f t (ˆ x t , ˆ u t )( h ) + D f t (ˆ x t , ˆ u t ) · ( u t − ˆ u t ) i + λ T . [ D φ t (ˆ x t , ˆ u t )( h ) + D φ t (ˆ x t , ˆ u t ) · ( u t − ˆ u t )] ≤ p Tt ( h ) . Equivalently, for all t ∈ { , ..., T } and for all ( h, k ) ∈ X × T U t (ˆ u t ) h p Tt +1 , Df t (ˆ x t , ˆ u t ) · ( h, k ) i ≤ p Tt ( h ) − λ T Dφ t (ˆ x t , ˆ u t )( h, k ) . (4.8)Thus we get for all t ∈ { , ..., T } and for all ( h, k ) ∈ X × T U t (ˆ u t ) h p Tt +1 , Df t (ˆ x t , ˆ u t ) · ( h, k ) i ≤ k p Tt kk h k X + λ T k Dφ t (ˆ x t , ˆ u t ) k · k ( h, k ) k X × U . (4.9) Since, for all t ∈ N ∗ , 0 ∈ Int ( Df t (ˆ x t , ˆ u t )(( X × T U t (ˆ u t )) ∩ B X × U )), there exists aconstant r t > B X (0 , r t ) ⊂ Df t (ˆ x t , ˆ u t )(( X × T U t (ˆ u t )) ∩ B X × U ). Thus,from (4 .
9) we obtain k p Tt +1 k ≤ r t ( k p Tt k + λ T k Dφ t (ˆ x t , ˆ u t ) k ) . (4.10)On the other hand, using ( c ) of Lemma 4.5 (respectively Lemma 4.6), we get, forall t ∈ { , ..., T } , k p Tt k ≤ k p Tt +1 k · k D f t (ˆ x t , ˆ u t ) k + λ T k D φ t (ˆ x t , ˆ u t ) |k . (4.11)Thus, by combining (4 .
10) and (4 .
11) for all T ≥
2, for all s ∈ { , ..., T } , and all1 ≤ t ≤ T + 1, there exist a t , b t ≥ k p Tt k ≤ a t λ T + b t k p Ts k . This gives the part (1). Suppose that there exists s ∈ { , ..., T } such that ( λ T , p Ts ) =(0 , λ T and ( p Tt ) ≤ t ≤ T +1 are simul-taneously equal to zero which contredicts the part ( a ) of Lemma 4.5 (respectivelyLemma 4.6). Thus, ( λ T , p Ts ) = (0 ,
0) which gives the part (2).Now, using ( d ) of Lemma 4.5 (respectively Lemma 4.6) for an arbitrary s ∈{ , ..., T } , for all T ≥
2, and for all u s ∈ U s , we have h p Ts ◦ D f s − (ˆ x s − , ˆ u s − ) , u s − − ˆ u s − i ≤ −h λ T D φ s − (ˆ x s − , ˆ u s − ) , u s − − ˆ u s − i . For all z ∈ A s := D f s − (ˆ x s − , ˆ u s − )( T U s − (ˆ u s − )), using the definition of theset T U s − (ˆ u s − ), there exist ( u y k s − ) k ∈ U N s − and ( α k ) k ∈ ( R + ) N such that y z :=lim k → + ∞ ( α k ( u y k s − − ˆ u s − )) and z = D f s − (ˆ x s − , ˆ u s − ) · y z . So, using the aboveinequality and doing k → + ∞ , we get p Ts ( z ) ≤ −h λ T D φ s − (ˆ x s − , ˆ u s − ) , y z i , and so there exists C z := −h D φ s − (ˆ x s − , ˆ u s − ) , y z i such that, for all T ≥
2, we have p Ts ( z ) ≤ C z λ T . This gives the part (3). (cid:3) The proof of the main results.
This section is devoted to the proofs of the Pontryagin principle for systemsgoverned by a difference equation, and of the Pontryagin principle for systemsgoverned by a difference inequation
Proof of Theorem 2.2.
Let us prove the existence of the sequences ( p t ) t ∈ N ∗ ∈ ( X ∗ ) N ∗ and λ ≥ T ∈ N , T ≥ λ T ∈ R and ( p Tt ) ≤ t ≤ T +1 ∈ ( X ∗ ) T +1 such that the following conditionshold.(a) λ T and ( p Tt ) ≤ t ≤ T +1 are not simultaneously equal to zero.(b) λ T ≥ p Tt = p Tt +1 ◦ D f t (ˆ x t , ˆ u t ) + λ T D φ t (ˆ x t , ˆ u t ) for all t ∈ { , ..., T } .(d) h λ T .D φ t (ˆ x t , ˆ u t ) + p Tt +1 ◦ D f t (ˆ x t , ˆ u t ) , u t − ˆ u t i ≤ t ∈ { , ..., T } andfor all u t ∈ U t . NFINITE DIMENSION 13
From ( A s ∈ N such that the set A s := D f s (ˆ x s , ˆ u s )( T U s (ˆ u s )) containsa closed convex subset K with ri( K ) = ∅ and Aff( K ) is of finite codimension in X .Since the set of the lists of multipliers of a maximization problem is a cone, usingthe above consequences of Lemma 4.5, we can normalize the pair ( λ T , p Ts ) = (0 , λ T + k p Ts k X ∗ = 1. By combining Lemma 4.7 andProposition 3.9 applied with K , we get a strictly increasing map k T k , from N into N , and λ ∈ R + and ( p t ) t ≥ ∈ ( X ∗ ) N such that:(i) λ T k −→ λ ≥ k → + ∞ ,(ii) for each t ∈ N , p T k t w ∗ −→ p t when k → + ∞ ,(iii) ( λ , p s ) = (0 , . Thus, by doing k → + ∞ in ( c ) and ( d ) we obtain (3) and (4). From ( i ) we get (2).Now, if there exists t > s such that ( λ , p t ) = (0 , λ , p s ) = (0 ,
0) which is a contradiction with ( iii ). Thus, for all t ≥ s , ( λ , p t ) = (0 ,
0) this gives the part (1). (cid:3)
Proof of Theorem 2.3.
We proceed as in the proof of Theorem 2.2, replacingthe use of Lemma 4.5 by the use of Lemma 4.6. (cid:3) Appendix: Some additional applications.
In this section we establish some additional consequences of the abstract result(Lemma 3.3). We begin by the following extension of [Theorem 2.5.4 [7]]. The[Theorem 2.5.4 [7]] can be obtained by taking K = Z , a = 0 and B = B Z (0 ,
1) inProposition 6.1.
Proposition 6.1.
Let Z be a Banach space, T be any nonempty set and ( p n ) n ∈T be a collection of lower semicontinuous and subadditive functions from Z into R .Let K be a closed convex subset of Z such that ri( K ) = ∅ . Suppose that for each z ∈ K we have sup n ∈T p n ( z ) < + ∞ . Then, for each a ∈ K and each boundedsubset B of Aff( K ) , we have sup n ∈T sup z ∈ B p n ( z − a ) < + ∞ . Proof.
The proof is immediat by using Lemma 3.3 with λ n = 1 for all n ∈ T . (cid:3) The above proposition is in fact an extention to subadditive functions of theclassical Banach-Steinauss theorem.
Corollary 6.2. (Banach-Steinauss) Let X be a Banach space and Y be a normedvector space. Let T be any nonempty set. Suppose that ( T n ) n ∈T is a collection ofcontinuous linear operators from X to Y . Suppose that for each x ∈ X one has sup n ∈T k T n ( x ) k Y < + ∞ , then sup n ∈T sup k x k =1 k T n ( x ) k Y = sup n ∈T k T n k B ( X,Y ) < + ∞ . Proof.
The proof follows immediately from Proposition 6.1 applied with: K = X , a = 0, the bounded set S X (the unit sphere of X ) and with the collection ofthe continuous subadditive functions p n ( x ) := k T n ( x ) k Y for all n ∈ T and all x ∈ X . (cid:3) We also have the following corollary.
Corollary 6.3.
Let A be a nonempty set and ( Z, k . k ) be a Banach space. Let ϕ : A × Z −→ R be a map such that: (1) For all x ∈ A , the map z ϕ ( x, z ) is lower semicontinuous and sublinear. (2) For all z ∈ Z , the map x ϕ ( x, z ) is bounded.Then, there exists a real number C ∈ R such that sup x ∈ A ϕ ( x, z ) ≤ C k z k , for all z ∈ Z .Proof. We apply Proposition 6.1 with T = A , p x := ϕ ( x, . ), using (1) and (2), thereexists C ∈ R such that sup x ∈ A sup k z k =1 ϕ ( x, z ) ≤ C. Thus, by the homogeneity of p x , we have ϕ ( x, z ) ≤ C k z k , for all x ∈ A and all z ∈ Z . (cid:3) Finally, we get the following proposition, which gives, a necessary and sufficientcondition such that the Dirac masses are continuous functionals.
Proposition 6.4.
Let X be a nonempty set and ( B ( X ) , k . k ∞ ) be the Banach spaceof all bounded real-valued functions. Let Y ⊂ B ( X ) be a subspace and k . k Y be anorm on Y such that ( Y, k . k Y ) is a Banach space. Let us denote by δ x the Diracmass or the evaluation at x ∈ X defined by δ x : f f ( x ) for all f ∈ B ( X ) . Then,the following assertions are equivalent. ( a ) δ x : ( Y, k . k Y ) −→ R is continuous for each x ∈ X , ( b ) there exists a constant α ∈ R + ∗ such that k . k Y ≥ α k . k ∞ .Proof. Indeed, suppose that δ x : ( Y, k . k Y ) −→ R is continuous for each x ∈ X .Consider the map ϕ : X × Y −→ R defined by ϕ ( x, f ) = f ( x ) for all ( x, f ) ∈ X × Y .This map satisfies the hypothesis of Corollary 6.3, so there exists C ∈ R suchthat sup x ∈ X f ( x ) = sup x ∈ X ϕ ( x, f ) ≤ C k f k Y for all f ∈ Y . Thus by symmetry,sup x ∈ X | f ( x ) | = k f k ∞ ≤ C k f k Y for all f ∈ Y . This implies that C > α := C . For the converse, we have | δ x ( f ) | = | f ( x ) | ≤ k f k ∞ ≤ α k f k Y whichshows that δ x is continuous on ( Y, k . k Y ) since it is linear. (cid:3) References
1. M. Bachir and J. Blot,
Infinite Dimensional Infinite-horizon Pontryagin Principles forDiscrete-time Problems , Set-Valued Var. Anal. 23 (2015) 43-54.2. X. Li and J.M. Yong,
Optimal Control Theory for Infinite Dimensional Systems , Birkhauser,Boston, 1995.3. M. McAsey and L. Mou,
A multiplier rule on a metric space , J. Math. Anal. Appl. 337(2008) 1064-1071.4. J. Blot and H. Chebbi,
Discrete time Pontryagin principle in infinite horizon , J. Math. Anal.Appl. (2000), 265-279.5. J. Blot and N. Hayek,
Infinite-horizon optimal control in the discrete-time framework ,Springer, New York, 2014.6. J. Diestel,
Sequences and series in Banach spaces , Graduate texts in Mathematics, SpringerVerlag, N.Y., Berlin, Tokyo, 1984.7. E. Hille and R.S. Phillips, Functional Analysis and Semigroups, American MathematicalSociety, Providence, RI, 1957.8. J. Jahn,
Introduction to the theory of nonlinear optimization , Third edition, Springer-Verlag,Berlin, 2007.9. P. Michel,
Some clarifications on the transversality conditions , Econometrica, (3) (1990),705-728. NFINITE DIMENSION 15
Mohammmed Bachir: Laboratoire SAMM EA4543,Universit´e Paris 1 Panth´eon-Sorbonne, centre P.M.F.,90 rue de Tolbiac, 75634 Paris cedex 13, France.
E-mail address : [email protected]
Jo¨el Blot: Laboratoire SAMM EA4543,Universit´e Paris 1 Panth´eon-Sorbonne, centre P.M.F.,90 rue de Tolbiac, 75634 Paris cedex 13, France.
E-mail address ::