STRUCTURE OF APPROXIMATE SOLUTIONS OF BOLZA VARIATIONAL PROBLEMS ON LARGE INTERVALS

. In this paper we study the structure of approximate solutions of autonomous Bolza variational problems on large ﬁnite intervals. We show that approximate solutions are determined mainly by the integrand, and are essentially independent of the choice of time interval and data.

In this paper we consider the following Bolza variational problems T 0 f (z(t), z (t))dt + h(z(0), z(T )) → min, (P B ) z : [0, T ] → R n is an absolutely continuous (a. c.) function, where T > 0 is sufficiently large, f : R n × R n → R 1 is an integrand and h : R n × R n → R 1 belongs to a space of functions to be described below.
For the set A we consider the uniformity which is determined by the following base: for all u, x ∈ R n satisfying |x|, |u| ≤ N } ∩{(f, g) ∈ A × A : (|f (x, u)| + 1)(|g(x, u)| for all x, u ∈ R n satisfying |x| ≤ N }, where N, > 0 and λ > 1. It is known [27] that the uniform space A is metrizable and complete.
A function x(·) defined on unbounded interval with the values in a finite-dimensional Euclidean space is called absolutely continuous (a. c.) if it is absolutely continuous on any bounded subinterval of its domain.
Let f ∈ A. For any a.c. function v : [0, ∞) → R n we set The real number µ(f ) = inf{J(v) : v : [0, ∞) → R n is an a.c. function} (1.4) is called the minimal long-run average cost growth rate of f . Clearly, −∞ < µ(f ) < ∞. By Theorems 3.6.1 and 3.6.2 of [27], for all x, y ∈ R n and all T ∈ (0, ∞), where π f : R n → R 1 is a continuous function and (T, x, y) → θ f T (x, y) ∈ R 1 is a continuous nonnegative function (1.6) defined for all T > 0 and all x, y ∈ R n , is an a.c. function satisfying v(0) = x}, x ∈ R n (1.7) and for every T > 0, every x ∈ R n there is y ∈ R n satisfying θ f T (x, y) = 0.
By Theorem 3.1.1 of [27], there exists a set F ⊂ A which is a countable intersection of open everywhere dense subsets of A such that each integrand f ∈ F posseses (ATP). In other words, (ATP) holds for a typical (generic) integrand f ∈ A.
By Theorem 1.1 for each integrand f ∈ A which possesses (ATP) there exists a compact set H(f ) ⊂ R n such that Ω(v) = H(f ) for each (f )-good function v : [0, ∞) → R n . In this case we say that the set H(f ) is the turnpike of f .
In [26] we established the following result which shows that for an integrand f ∈ M, (ATP) implies the turnpike property described above with the turnpike H(f ) (for its proof see also Theorem 5.1.1 of [27]). Theorem 1.3. Assume that an integrand f ∈ M has the asymptotic turnpike property and that , K > 0. Then there exist a neighborhood U of f in A and numbers M > K, l 0 > l > 0, δ > 0 such that for each g ∈ U, each T ≥ 2l 0 and each a.c. function v : [0, T ] → R n which satisfies Let k ≥ 1 be an integer. Denote by A k the set of all integrands f ∈ A ∩ C k (R 2n ). For any p = (p 1 , . . . , p 2n ) ∈ {0, . . . , k} 2n set |p| = 2n i=1 p i . For each f ∈ C k (R 2n ) and each p = (p 1 , . . . , p 2n ) ∈ {0, . . . , k} 2n satisfying |p| ≤ k define For the set A k we consider the uniformity which is determined by the following base: for all u, x ∈ R n satisfying |x|, |u| ≤ N and each p ∈ {0, . . . , k} 2n satisfying |p| ≤ k} for all x, u ∈ R n satisfying |x| ≤ N }, where N, > 0 and λ > 1. It is known (see Chapter 5 of [27]) that the uniform space A k is metrizable and complete. Set Let k ≥ 0 be an integer. Denote byM k the closure of M k in A k and consider the topological subspaceM k ⊂ A k equipped with the relative topology.
Denote by L the set of all f ∈ M ∩ C 2 (R 2n ) such that ∂f /∂u i ∈ C 2 (R 2n ) for i = 1, . . . , n.
For any k ∈ {0, 1, 2} denote by L k the closure of L in the space A k and consider the topological subspace L k ⊂ A k equipped with the relative topology.
In [26] we established the following generic turnpike result which shows that most integrands possess the turnpike property described above (for its proof see also Theorem 5.1.2 of [27]). Theorem 1.4. Let M be one of the following spaces: Then there exists a set F ⊂ M which is a countable intersection of open everywhere dense subsets of M such that each f ∈ F has (ATP) and the following property.
For each , K > 0 there exist a neighborhood U of f in A and numbers M > K, l 0 > l > 0, δ > 0 such that for each g ∈ U, each T ≥ 2l 0 and each a.c. function v : [0, T ] → R n which satisfies Note that in [26,27] the result stated above was proved in the case when M in any of the spacesM q , q ≥ 0. In the case when M is in any of the spaces L q , q = 0, 1, 2 Theorem 1.4 is proved with the same proof.
Our paper is organized as follows. Uniform boundedness of approximate solutions of problems (P 1 ), (P 2 ) and (P 3 ) is considered in Section 2. Section 3 contains preliminaries. Section 4 contains a turnpike result for problem (P 1 ). In Section 5 we begin to study Bolza problems (P B ). Uniform boundedness of approximate solutions of problems (P B ) is established in Section 6. Sections 7 and 8 contain two turnpike results for problems (P B ). Section 9 contains results on the the structure of approximate solutions of Bolza problems (P B ) in the regions close to the endpoints of time intervals (Theorems 9.2-9.4). Auxiliary results are collected in Section 10. In Section 11 we prove auxiliary results for Theorem 9.2 which is proved in Sections 12. Section 13 contains auxiliary results for Theorem 9.4 which is proved in Section 14. Auxiliary results for Theorem 9.3 are collected in Section 15. Theorem 9.3 is proved in Section 16.
2. Uniform boundedness of approximate solutions. For f ∈ A, x ∈ R n and a real number T > 0 set 2) The following result plays an important role in our study.
Theorem 2.1. Let f ∈ A and let M 1 , M 2 , c > 0. Then there exist a neighborhood U of f in A and S > 0 such that for each g ∈ U, each T 1 ∈ [0, ∞) and each T 2 ∈ [T 1 + c, ∞) the following properties hold: The properties (i) and (ii) were established in [25]. See also Theorem 1.2.3 of [27]. The property (iii) is proved analogously to the properties (i) and (ii).
It is easy to see that for each integer k ≥ 1,f ∈ A k for all f ∈ A k , for each integer k ≥ 0, f ∈ M k for all f ∈ M k and that the mapping f →f , f ∈ A k is continuous. This implies that for each integer k ≥ 0,f ∈M k for each f ∈M k . Evidently,f ∈ L for all f ∈ L and for any k ∈ {0, 1, 2} and any f ∈ L k ,f ∈ L k . Let f ∈ A. For any T > 0 and any a. c. function v : Clearly, for each T > 0 and each a. c. function v : The next result easily follows from (3.3).
Proposition 1. Let f ∈ A, T > 0, M ≥ 0 and v i : [0, T ] → R n , i = 1, 2 be a. c. functions. Then For each f ∈ A, each x ∈ R n and each real number T > 0 set Proposition 1 implies the following result.
The next result follows from Proposition 2, Theorem 2.1 and the continuity of the mapping f →f , f ∈ A.
Proposition 3. Let f ∈ A and let M 1 , M 2 , c > 0. Then there exist a neighborhood U of f in A and S > 0 such that for each g ∈ U, each T ≥ c and each a. c. function v : [0, T ] → R n which satisfies The following result is proved in [33]. The following result is proved in [27] (see Chapter 4, Proposition 4.2.1). (3.5) Since the function π f is continuous it follows from Proposition 5 that the set D(f ) is nonempty, bounded and closed. For each τ 1 ∈ R 1 , τ 2 > τ 1 , each r 1 , r 2 ∈ [τ 1 , τ 2 ] satisfying r 1 < r 2 and each a.c.
Let f ∈ A and x ∈ R n . By Proposition 7 any function belonging to P(f, x) is bounded and (f )-good. 4. A turnpike result. In the sequel we use the following turnpike result which was proved in [36].
5. Bolza variational problems. Let a 1 > 0. Denote by A(R n × R n ) the set of all lower semicontinuous functions h : R n × R n → R 1 which are bounded on bounded subsets of R n × R n and satisfy For simplicity we set A = A(R n × R n ). We equip the set A with the uniformity which is determined by the following base: where N, > 0. (Here | · | is the Euclidean norm in the space R 2n .) It is not difficult to see that the uniform space A is metrizable and complete. We consider the following Bolza variational problems

6.
A uniform boundedness result for Bolza variational problems. We begin our study of Bolza variational problems with the following uniform boundedness result.
Proof. By Theorem 2.1, there exist a neighborhood U 1 of f in A and S 1 > 0 such that for each g ∈ U 1 , each T 1 ∈ [0, ∞) and each T 2 ∈ [T 1 + c, ∞) the following property holds: (i) if an a. c. function v : [T 1 , T 2 ] → R n satisfies In view of (5.2), there exist a neighborhood V of h in A and S 2 > 0 such that for all ξ ∈ V and each z 1 , z 2 ∈ R n satisfying |z i | ≤ S 1 , i = 1, 2, we have (6.1) By Theorem 2.1, there exist a neighborhood U ⊂ U 1 of f in A and S > S 1 + S 2 such that for each g ∈ U, each T 1 ∈ [0, ∞) and each T 2 ∈ [T 1 + c, ∞) the following property holds: and that an a. c. function v : There exists an a. c. function u : [T 1 , T 2 ] → R n such that It follows from (6.2), (6.4) and property (i) that By (6.2), (6.5) and the choice of V (see (6.1)), In view of (5.1), (6.3), (6.4) and (6.6), Property (ii), (6.2) and (6.7) imply that |v(t)| ≤ S, t ∈ [T 1 , T 2 ]. Theorem 6.1 is proved.
7. The first turnpike result for Bolza problems. Let f ∈ A have (ATP). By Theorem 2.1, there exist a neighborhood U f of f in A and S f > 0 such that the following properties hold: (P1) for each g ∈ U f , each T ≥ 1 and each a. c. function u : The following turnpike result for Bolza variational problems shows that the turnpike phenomenon, for approximate solutions on large intervals, is stable under small perturbations of the objective functions.
Theorem 7.1. Assume that an integrand f ∈ A has the asymptotic turnpike property and that M 1 , M 2 , > 0. Then there exist a neighborhood U of f in A, numbers l, S > 0 and integers L, and each a.c. function v : [0, T ] → R n which satisfies Proof. By Theorems 1.2 and 2.1, there exist a neighborhood U ⊂ U f of f in A, numbers l, S ≥ 1 and integers L ≥ 1, Q * ≥ 1 such that the following property holds: There exists an a. c. function u : [0, T ] → R n such that Property (P1), (7.3) and (7.6) imply that By (5.1) and (7.4)-(7.7), This completes the proof of Theorem 7.1.
8. The second turnpike result for Bolza problems. The following turnpike result for Bolza variational problems shows that the turnpike phenomenon, for approximate solutions on large intervals, is stable under small perturbations of the objective functions.
Theorem 8.1. Assume that an integrand f ∈ M has the asymptotic turnpike property and that , M 1 , M 2 > 0. Then there exist a neighborhood U of f in A and numbers S > 0, L > l > 0, δ > 0 such that for each g ∈ U, each h ∈ A satisfying Proof. Denote by mes(E) the Lebesgue measure of a Lebesgue measurable set E ⊂ R 1 .
By Theorem 4.1, there exist a neighborhood U 1 of f in A and numbers l 1 > l > 0, δ ∈ (0, 1] such that the following property holds: (i) for each g ∈ U 1 , each T ≥ 2l 1 + l and each a.c. function v : By Theorem 7.1, there exist a neighborhood U ⊂ U 1 of f in A and numbers S > 0, L 0 > 2l 1 + 1 such that the following property holds: the inequality |v(t)| ≤ S holds for all t ∈ [0, T ] and there exists a set E ⊂ [0, T ] which is a finite union of closed subintervals of [0, T ] such that mes(E) < L 0 and Property (ii) and (8.2)-(8.6) imply that |v(t)| ≤ S, t ∈ [0, T ] and that there exist such that 9. Structure of solutions of Bolza problems near the endpoints. For each nonempty set X and each η : Proposition 5, (5.1) and (9.1) imply the following result.
Lemma 9.1. The function ψ f,h is lower semicontinuous, for every M > 0 the set is finite and the function ψ f,h has a point of minimum.
The next result is proved in Section 12.
Theorem 9.2. Suppose that an integrand f ∈ M has the asymptotic turnpike property and that h ∈ A. Let , L 0 > 0. Then there exist a neighborhood U of f in A, a neighborhood V of h in A and numbers δ ∈ (0, ) and T 0 > L 0 such that for each T ≥ T 0 , each g ∈ U, each ξ ∈ V and each a.c. function v : [0, T ] → R n which satisfies In the next theorem M is one of the spaces L k , k = 0, 1, 2,M q , q ≥ 3 is an integer and product M × A is equipped with the product topology.
and such that the following assertion holds. Let , M, τ 0 > 0. Then there exist a neighborhood U of (f, h) in M × A and numbers δ ∈ (0, ) and T 0 > τ 0 such that for each (g, ξ) ∈ U, each T ≥ T 0 and each a.c. function v : [0, T ] → R n satisfying Theorem 9.3 is proved in Section 16. The next result is proved in Section 14.
Theorem 9.4. Suppose that an integrand f ∈ M has the asymptotic turnpike property and that h ∈ A. Let , L 0 > 0. Then there exist a neighborhood U of f in A, a neighborhood V of h in A and δ ∈ (0, ) such that for each g ∈ U, each ξ ∈ V and each pair of a (g)-perfect function w 1 : [0, ∞) → R n and a (ḡ)-perfect function and that for all t ∈ [0,  [27]) Assume that f ∈ A has (ATP). Then f is a continuity point of the mapping g → (µ(g), π g ) ∈ R 1 × C(R n ), g ∈ A, where C(R n ) is the space of all continuous functions φ : R n → R 1 with the topology of the uniform convergence on bounded sets.
Lemma 10.5. (Corollary 1.3.1 of [27]) For each f ∈ A, each pair of numbers T 1 , T 2 satisfying 0 ≤ T 1 < T 2 and each z 1 , z 2 ∈ R n there is an a.c. function x : The following lemma is a particular case of Lemma 3.3 of [28]. The following lemma is a particular case of Lemma 5.1 of [28].
Lemma 10.11. (Lemma 6.12 of [33]) Let f ∈ A have (ATP) and S 0 > 0. Then there exist K 0 > 0 and a neighborhood U of f in A such that for each g ∈ U and each x ∈ R n satisfying |x| > K 0 the inequality π g (x) > S 0 holds.
By Lemmas 13.1-13.3, there exist a neighborhood U 0 of f in A, a neighborhood V 0 of h in A and δ ∈ (0, 4 −1 δ 0 ) such that the following properties hold: (ii) for each g ∈ U 0 , each ξ ∈ V 0 and each z 1 , z 2 ∈ R n satisfying
The following result was proved in Section 3 of Chapter 2 of [2] (see also Proposition 3.7.1 of [27]).

Lemma 15.4.
Let Ω be a closed subset of R s . Then there exists a bounded nonnegative function φ ∈ C ∞ (R s ) such that Ω = {x ∈ R s : φ(x) = 0} and for each sequence of nonnegative integers p 1 , . . . , p s , the function ∂ |p| φ/∂x p1 Denote by E 1 the set of all (f, h) ∈ E 0 × A such that the function ψ f,h has a unique point of minimum.
It is clear that for any r > 0, h r ∈ A, (z * ,1 , z * ,2 ) is a unique point of minimum of the function ψ f,hr and that (f, h r ) ∈ E 1 . Clearly, h = lim r→0 + h r in A. Thus there exists r > 0 such that (f, h r ) ∈ V. Lemma 15.8 is proved.