EXISTENCE AND UNIFORM BOUNDEDNESS OF OPTIMAL SOLUTIONS OF VARIATIONAL PROBLEMS

Given an x0 ∈ Rn we study the infinite horizon problem of minimizing the expression ∫ T 0 f(t, x(t), x ′(t))dt as T grows to infinity where x : [0, ∞) → Rn satisfies the initial condition x(0) = x0. We analyse the existence and the properties of approximate solutions for every prescribed initial value x0. We also establish that for every bounded set E ⊂ Rn the C([0, T ]) norms of approximate solutions x : [0, T ] → Rn for the minimization problem on an interval [0, T ] with x(0), x(T ) ∈ E are bounded by some constant which does not depend on T . Introduction The study of variational and optimal control problems defined on infinte intervals has recently been a rapidly growing area of research. These problems arise in engineering (see Anderson and Moore [1], Artstein and Leizarowitz [2]), in models of economic growth (see Rockafellar [14], Zaslavski [20]), in infinite discrete models of solid-state physics related to dislocations in one-dimensional crystals which are under discussion in Aubry and Le Daeron [3], Zaslavski [16] and in the theory of thermodynamical equilibrium of materials (see Leizarowitz and Mizel [12], Coleman, Marcus and Mizel [7], Zaslavski [17,18]). We consider the infinite horizon problem of minimizing the expression ∫ T 0 f(t, x(t), x′(t))dt as T grows to infinity where a function x : [0,∞) → K is absolutely continuous (a.c.) and satisfies the initial condition x(0) = x0, K ⊂ R is a 1991 Mathematics Subject Classification. 49J99, 58F99.


Introduction
The study of variational and optimal control problems defined on infinte intervals has recently been a rapidly growing area of research.These problems arise in engineering (see Anderson and Moore [1], Artstein and Leizarowitz [2]), in models of economic growth (see Rockafellar [14], Zaslavski [20]), in infinite discrete models of solid-state physics related to dislocations in one-dimensional crystals which are under discussion in Aubry and Le Daeron [3], Zaslavski [16] and in the theory of thermodynamical equilibrium of materials (see Leizarowitz and Mizel [12], Coleman, Marcus and Mizel [7], Zaslavski [17,18]).
We consider the infinite horizon problem of minimizing the expression T 0 f (t, x(t), x (t))dt as T grows to infinity where a function x : [0, ∞) → K is absolutely continuous (a.c.) and satisfies the initial condition x(0) = x 0 , K ⊂ R n is a closed convex set and f belongs to a complete metric space of functions to be described below.
The following notion known as the overtaking optimality criterion was introduced in the economics literature by Gale [8] and von Weizsacker [15] and has been used in control theory by Artstein and Leizarowitz [2], Brock and Haurie [5], Carlson [6] and Leizarowitz [11].
An a.c.function x : [0, ∞) → K is called (f )-overtaking optimal if for any a.c.function y : [0, ∞) → K satisfying y(0) = x(0) lim sup Usually it is difficult to establish the existence of overtaking optimal solutions, and actually, in general they may fail to exist.Most studies that are concerned with their existence assume convex integrands ( [11], [5], [14]).
Another type of optimality criterion for infinite horizon problems (which is probably the weakest optimality concept) was introduced by Aubry and Le Daeron [3] in their study of the discrete Frenkel-Kontorova model related to dislocations in one-dimensional crystals.More recently this optimality criterion was used by Moser [13], Leizarowitz and Mizel [12] and Zaslavski [16].A similar notion was introduced in Halkin [9] for his proof of the maximum principle.
Clearly every (f )-overtaking optimal function is an (f )-minimal solution.
In the present paper we consider a functional space of integrands M described in Section 1 and analyze existence and properties of (f )-minimal solutions with f ∈ M.More exactly we will show that given f ∈ M and z ∈ R n there exists a bounded (f )-minimal solution Z : [0, ∞) → R n satisfying Z(0) = z such that any other a.c.function Y : [0, ∞) → R n is not "better" then Z.We will also establish that given f ∈ M and a bounded set E ⊂ R n the C([0, T ]) norms of approximate solutions x : [0, T ] → R n for the minimization problem on an interval [0, T ] with x(0), x(T ) ∈ E are bounded by some constant which depends only on f and E. These results which are valid for any f ∈ M have been applied in [19] to get more information about the existence of optimal solutions over an infinite horizon and about the structure of optimal solutions on finite intervals for a generic integrand f ∈ M.
The paper is organized as follows.In Section 1 we state our main theorems, Section 2 contains several preliminary results, in Section 3 we consider discrete-time control systems obtained by discretization of variational problems and in Section 4 we prove the main theorems.

Statements of main results
Let K ⊂ R n be a closed convex set.Denote by | • | the Euclidean norm in R n and denote by M the set of continuous functions f : [0, ∞) × K × R n → R 1 which satisfy the following assumptions: (A) (i) for each (t, x) ∈ [0, ∞) × K the function f (t, x, •) : R n → R 1 is convex; (ii) the function f is bounded on [0, ∞) × E for any bounded set here a and ψ are independent on f ); (iv) for each M, > 0 there exist Γ, δ > 0 such that and there exists an increasing function For the set M we consider the uniformity which is determined by the the following base where N > 0, > 0, λ > 1.
Clearly, the uniform space M is Hausdorff and has a countable base.Therefore M is metrizable.We will show that the uniform space M is complete (see Proposition 2.2).
We consider functionals of the form Here we follow Leizarowitz [10] in defining "good functions" for the variational problem.
Let f ∈ M.An a.c.function x : [0, ∞) → K is called an (f )-good function if for any a.c.function y : [0, ∞) → K there is a number M y such that (1.5) In this paper our goal will be to study the set of (f )-good functions.We will establish the following results.
Theorem 1.1.For each f ∈ M and each z ∈ K there exists an (f )-good function 2. For each f ∈ M and each number M > inf{|u| : u ∈ K} there exist a neighborhood U of f in M and a number Q > 0 such that 3. For each f ∈ M and each number M > inf{|u| : u ∈ K} there exist a neighborhood U of f in M and a number Q > 0 such that for each g ∈ U , each z ∈ K satisfying |z| ≤ M , each T 1 ≥ 0, T 2 > T 1 and each a.c.function y : [T 1 , T 2 ] → K satisfying |y(T 1 )| ≤ M the following relation holds:

. function. Then y is an (f )-good function if and only if condition (ii) of Assertion 1 holds.
Theorem 1.2.For each f ∈ M there exists a neighborhood U of f in M and a number M > 0 such that for each g ∈ U and each (g)-good function In this paper we prove the following result which establishes that for every bounded set E ⊂ K the C([0, T ]) norms of approximate solutions x : [0, T ] → K for the minimization problem on an interval [0, T ] with x(0), x(T ) ∈ E are bounded by some constant which does not depend on T .Theorem 1.3.Let f ∈ M and M 1 , M 2 , c be positive numbers.Then there exist a neighborhood U of f in M and a number S > 0 such that for each g ∈ U , each T 1 ∈ [0, ∞) and each T 2 ∈ [T 1 + c, ∞) the following properties hold: (i) for each x, y ∈ K satisfying |x|, |y| ≤ M 1 and each a.c.function the following relation holds:

Preliminary results
Proposition 2.1.Let f ∈ M, M and be positive numbers.Then there exist Γ, δ > 0 such that Proof.Fix a number By Assumption (Aiv) there exist Γ, δ > 0 such that To prove the proposition it is sufficient to show that f satisfies Assumption (Aiv).Let M, be positive numbers.Fix a number λ > 1 such that (2.5) Clearly there exists an integer j ≥ 1 such that (2.6) (f i , f j ) ∈ E(M, , λ) for any integer i ≥ j.
Then there exists a number and each a.c.function the following relation holds: Proof.By Assumption (Aiii) and the properties of the function ψ there exists a number c 0 > 0 such that (2.17) Let f ∈ M, T 1 , T 2 be numbers satisfying (2.14) and let x : [T 1 , T 2 ] → K be an a.c.function satisfying (2.15).We will show that (2.16) holds.
Proof.By Assumption (Aiii) and the properties of the function ψ there exists a number c 0 > 0 such that It follows from (2.25), the definition of c 0 , (2.23), (2.14) and Assumption (Aiii) that Together with (2.15), (2.14) and (2.24) this relation implies that This completes the proof of the proposition.
Then there exists a subsequence {x i k } ∞ k=1 and an a. c. function It is an elementary exercise to prove the following result.
and each the following relation holds: Proof.By Proposition 2.6 there exists a number By Proposition 2.3 there exists a number M 1 > 0 such that for each pair of numbers T 1 , T 2 ≥ 0 satisfying (2.26) and each a.c.function Choose a number δ 1 > 0 such that By Assumption (Aiv) there is a number There exists a positive number δ such that Assume that numbers T 1 , T 2 ≥ 0 satisfy (2.26) and y 1 , y 2 , z 1 , z 2 ∈ R n satisfy (2.27).By Corollary 2.1 there exists an a.c.function It follows from (2.26), (2.27), (2.39), (2.29) and the definition of (2.40), (2.27) and (2.26) imply that We have We Combining (2.45), (2.47), (2.48) and (2.31) we obtain that Together with (2.39) and (2.41) this implies that This completes the proof of the proposition.
Proof.By Proposition 2.3 there exists a number S > 0 such that for each There exist δ ∈ (0, 1), N > S and Γ > 1 such that and x : [T 1 , T 2 ] → K is an a.c.function satisfying (2.49).It follows from the definition of S that (2.50) holds.Set It follows from (2.50) and the definition of V and N that It follows from (2.50), (2.51), Assumption (Aiii) and the definition of The proposition is proved.
Proof.By Proposition 2.6 there exist a neighborhood V 1 of f in M and a number By Proposition 2.8 there exists a neighborhood To complete the proof it remains now to note that for g ∈ V , and y, z ∈ K satisfying |y|, |z| ≤ c 3 the following relation holds:

Discrete-time control systems
Let f ∈ M, z ∈ K and let 0 < c 1 < c 2 < ∞.By Proposition 2.6 there exists a neighborhood U 0 of f in M and a number (3.1) By Proposition 2.3 there exists a positive number M 1 such that Proposition 3.1.Assume that a positive number M 1 satisfies (3.2) and M 2 > 0. Then there exists a neighborhood U of f in M and an integer N > 2 such that: , each pair of integers q 1 , q 2 satisfying 0 ≤ q 1 < q 2 , q 2 − q 1 ≥ N and each sequence the following relation holds: where , each pair of integers q 1 , q 2 satisfying 0 ≤ q 1 < q 2 , q 2 − q 1 ≥ N and each sequence Proof.By Proposition 2.6 there exists a neighborhood U of f in M and a number M 3 > 0 such that Fix an integer N ≥ M 2 + 4M 3 + 4. The validity of the proposition now follows from the definition of U , M 3 , N and (3.1), (3.2).
Proposition 3.2.Assume that a positive number M 1 satisfies (3.2) and M 3 > 0. Then there exists a neighborhood V of f in M and a number , each pair of integers q 1 , q 2 satisfying 0 ≤ q 1 < q 2 and each sequence {z i } q 2 i=q 1 ⊂ K satisfying there is a sequence {y i } q 2 i=q 1 ⊂ K which satisfies y q j = z q j , j = 1, 2, (3.5)

For each
, each pair of integers q 1 , q 2 satisfying 0 ≤ q 1 < q 2 and each sequence {z i } q 2 i=q 1 ⊂ K satisfying there is a sequence {y i } q 2 i=q 1 ⊂ K which satisfies y q 1 = z q 1 and (3.5).Proof.There exist a neighborhood U of f in M and an integer N > 2 such that Proposition 3.1 holds with M 2 = 4(M 3 + 1) and U ⊂ U 0 .By Proposition 2.6 there exist a neighborhood V of f in M and a number r 1 such that By Proposition 2.3 there exists a positive number (recall a in Assumption (Aiii)).We will prove Assertion 1.
By Proposition 2.3 there exists a number By (4.1), (4.2) there exists a neighborhood U 1 of f in M and a number M 2 such that By Proposition 2.6 there exist a neighborhood U 2 of f in M and a number By Proposition 2.3 there exists a number (4.5) and each a.c.function By Proposition 2.6 there exist a neighborhood U of f in M and a number We may assume without loss of generality that there exists a number (4.8)
The obtained contradiction proves the lemma.
Then (4.29) Proof.There are two cases: Consider the case a).Set for all large i. (4.29) now follows from this relation and Lemma 4.1.
Consider the case b).By (4.28) there exists a subsequence It follows from (4.3), (4.30) and Proposition 3.2 that for any integer k ≥ 1 there exists a sequence {h j } Fix an integer q ≥ 4. By (4.30), Lemma 4.1 and (4.31) for an integer N > i q This completes the proof of the lemma.Proof.There are two cases: a) lim sup i→∞ |y(i)| > M 2 ; b) lim sup i→∞ |y(i)| ≤ M 2 where i is an integer.Consider the case a).It follows from Lemma 4.4, (4.12) that (4.34)I g (0, q, y) − I g (0, q, Z g ) → ∞ as an integer q → ∞.
Proof.By Lemma 4.5 we may assume that lim sup t→∞ |y(t)| ≤ Q 1 .There exists an integer i 0 > 0 such that (4.40) Fix an integer i > i 0 .By Corollary 2.1 there exists an a.c.function ȳ : (4.42) holds for each integer i > i 0 and each T > i.
Proof of Theorem 1.1.At the begining of Section 4 for each f ∈ M and each M > 2|z| we constructed a neighborhood U of f in M and for each g ∈ U and each z ∈ K satisfying |z| ≤ M we defined a.c.functions Proof of Theorem 1.3.Fix z ∈ K.By Proposition 2.6 there exists a neighborhood U 0 of f in M and a number (4.50) By Proposition 2.3 we may assume without loss of generality that There exists a neighborhood U 1 of f in M and a number S 1 such that (4.52) U 1 ⊂ U 0 , S 1 > M 1 and Proposition 3.2 holds with By Proposition 2.6 there exist a neighborhood U of f in M and a number There is a natural number p such that pc ≤ T Therefore property (i) holds.Analogously to this we can show that property (ii) holds.The theorem is proved.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

4 .
Proof of Theorems 1.1-1.3Construction of a neighborhood U .Let f ∈ M, z ∈ K, M > 2|z|.By Proposition 2.6 there exist a neighborhood U 0 of f in M and a number (4.1)

Lemma 4 . 6 .
Let g ∈ U , z ∈ K, |z| ≤ M and let y : [0, ∞) → K be an a.c.function.Then one of the relations below holds:

2 ,
. . .satisfying (4.9)-(4.15).Clearly an a.c.function Z f : [0, ∞) → K was defined for every f ∈ M and every z ∈ K.By Lemmas 4.5,4.6 for each f ∈ M and each z ∈ K the function Z f is (f )-good and Assertion 1 of Theorem 1.1 holds.Assertion 2 of Theorem 1.1 follows from (4.15) which holds for every g ∈ U (U is a neighborhood of f in M) and each z ∈ K satisfying |z| ≤ M .Assertion 3 of Theorem 1.1 follows from Lemma 4.3.Lemma 4.7 implies Assertion 4 of Theorem 1.1.Theorem 1.1 is proved.Theorem 1.2 follows from Lemma 4.5.

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation