Turnpike Properties for Dynamical Systems Determined by Differential Inclusions

: In this paper, we study the turnpike phenomenon for trajectories of continuous-time dynamical systems generated by differential inclusions, which have a prototype in mathematical economics. In particular, we show that, if the differential inclusion has a certain symmetric property, the turnpike possesses the corresponding symmetric property. If we know a ﬁnite number of approximate trajectories of our system, then we know the turnpike and this information can be useful if we need to ﬁnd new trajectories of our system or their approximations.


Introduction
In [1,2], A. M. Rubinov introduced a discrete disperse dynamical system which is generated by a set-valued self-mapping of a compact metric space. This dynamical system was investigated in [1][2][3][4][5][6][7]. It has a prototype in the economic growth theory [1,2,8,9]. In particular, it is an abstract extension of the classical von Neumann-Gale model [1,2,8,9]. This dynamical system is described by a compact metric space of states and a transition operator which is set-valued. Such dynamical systems correspond to certain models of economic dynamics [1,2,8,9]. More precisely, in [1][2][3], the description of global attractors for certain dynamical systems was obtained; the uniform convergence of trajectories to global attractors was studied in [4] and the behavior of trajectories under the presence of computational errors was analyzed in [5], while, in [6], analogous results were obtained for systems with a Lyapunov function. These results are collected in our recent book [7].
In the present paper, we study the convergence and structure of trajectories of the continuous-time analog of this dynamical system generated by a differential inclusion. In particular, we show that, if the differential inclusion has a certain symmetric property, its turnpike possesses the corresponding symmetric property.
We introduce a global attractor (turnpike) for our dynamical system which is the closure of the set of all limit points of all trajectories; we show that all trajectories on an infinite interval converge to this set and that all trajectories on finite and sufficiently large intervals spend most of the time in a small neighborhood of the turnpike. If we know a finite number of approximate trajectories of our system, then we know the turnpike and this information can be useful if we need to find new trajectories of our system or their approximations. We believe that our results can be extended to the case of perturbed trajectories of our system.
Let R n be the n-dimensional Euclidean space equipped with the inner product x i y i , x = (x 1 , . . . , x n ), y = (y 1 , . . . , y n ) which induces the Euclidean norm · and let X be a nonempty closed set in R n equipped with the relative topology. Let us denote, by N , the set of all natural numbers. For every point x ∈ R n and every positive real number r, let us put Let us suppose that F : The mapping F is upper semicontinuous if it is upper semicontinuous at each point For the proof of the following result see Proposition 2 of [17].

Proposition 1.
Let us assume that F is upper semicontinuous and that F(z) is closed for every z ∈ X. Then, graph(F) is closed in X × R n .
It is easy to see that the next result is true.

Proposition 2.
Let us assume that the set graph(F) is closed in X × R n , x 0 ∈ X, V is a neighborhood of the point x 0 in the space X and that the set F(V) is bounded. Then, the mapping F is upper semicontinuous at the point x 0 .
A function x : R 1 → X is a trajectory if, for each pair of numbers T 2 > T 1 , its restriction to the interval [T 1 , T 2 ] belongs to A(T 1 , T 2 ). We denote, by A(−∞, ∞), the collection of all trajectories x : R 1 → X.
In the sequel, we suppose that the following assumption holds. (A1) The mapping F is upper semicontinuous and F(x) is a compact, convex set for every point x ∈ X.
In view of Proposition 1, the set graph(F) is closed in X × R n .

Proposition 3.
The mapping F is bounded on bounded sets. In other words, for every positive number M 0 , there is a positive number M 1 such that Proof. Let M 0 > 0. We show that there exists M 1 > 0 such that (2) holds. Let us assume the contrary. Then, for every k ∈ N , there is such that In view of (3), we may assume, without loss of generality, that there is a limit Since the mapping F is upper semicontinuous, there is an open neighborhood U of the point x in the space X such that, for each point x ∈ U, we have This contradicts (4) and completes the proof of Proposition 3.
In our study, we apply the following two theorems (see Theorem 4 on page 13 of [17] and Theorem 1 on page 60 of [17], respectively). Theorem 1. Let I ⊂ R 1 be an interval, for every k ∈ N , x k : I → R n be an a. c. function such that, for each real number t ∈ I, the sequence {x k (t)} ∞ k=1 is bounded and let a positive function for a. e. real numbers t ∈ I and every k ∈ N . Then, there are a subsequence {x k i } ∞ i=1 and an a. c. function x : I → R n such that x k i converges to the function x uniformly over compact subsets of the interval I and that the functions x k i converges weakly to the function x in L 1 (I; R n ) as i → ∞.

Theorem 2.
Let I ⊂ R 1 be an interval, for every k ∈ N , x k : I → X and y k : I → R n be Lebesgue measurable functions such that, for a. e. real numbers t ∈ I and every open neighborhood N of zero in the space R n × R n , there is k 0 (t, N) ∈ N such that, for every natural number k ≥ k 0 (t, N), Let us suppose that x k converges a. e. to the function x : I → R n and y k ∈ L 1 (I; R n ) converges to y weakly in L 1 (I; R n ) as k → ∞. Then, for a. e. real number t ∈ I, (x(t), y(t)) ∈ graph(F).

Proposition 3 implies the next proposition.
Proposition 4. Let us assume that T > 0 and x : [0, T] → X is an a. c. function which satisfies Then, the function x is Lipschitz on [0, T].
In addition to (A1), the following assumption (A2) is assumed to be satisfied everywhere below.
(A2) For every positive number M, there is a positive number M 0 such that, for every positive number T and each function x ∈ A(0, T) which satisfies x(0) ≤ M, the equation Note that (A2) holds for models of economic growth which are prototypes of our dynamical system [7,9].
The next result, which is deduced from Theorems 1 and 2, plays an important role in our study.
Theorem 3. Let us assume that −∞ < T 1 < T 2 < ∞, for each k ∈ N , x k ∈ A(T 1 , T 2 ) and that the set {x k (T 1 ) : k = 1, 2, . . . } is bounded. Then, there exist a subsequence {x i k } ∞ k=1 and a trajectory x ∈ A(T 1 , T 2 ) such that x i k converges to x as k → ∞ uniformly on [T 1 , T 2 ] and x i k converges to x as k → ∞ weakly in L 1 ([T 1 , T 2 ]; R n ).
Proof. There exists M 0 > 0 such that By (6) and (A2), there is a positive number M 1 for which It follows, from Proposition 3 and Equations (1) and (7), that there is a positive number Theorem 1 and Equation (8) imply that there exist a subsequence {x k i } ∞ i=1 and an a. c. function x : [T 1 , T 2 ] → R n such that x k i converges to x uniformly over [T 1 , Combined with Theorem 2, this convergence implies that Theorem 3 is proved.

The Results
We begin with the next theorem. It will be proved in Section 3.

Theorem 4.
Let C ⊂ X be a nonempty closed bounded set. Then, the following properties are equivalent: (1) There exists a function x ∈ A(0, ∞) such that x(0) ∈ C.
(2) For every k ∈ N , there exists a function x k ∈ A(0, T k ) such that lim k→∞ T k = ∞ and x k (0) ∈ C for every k ∈ N . Corollary 1. The following properties are equivalent: (1) There exists a function x ∈ A(0, ∞).
(2) For every k ∈ N , there is a function x k ∈ A(0, T k ) such that
(2) For every k ∈ N , there is a function x k ∈ A(0, T k ) such that lim k→∞ T k = ∞ and x k (0) = ξ for all k ∈ N .
In the sequel, we assume that there exists a function x ∈ A(0, ∞). We define Ω(F) = {z ∈ X : for every positive number there is a function x ∈ A(0, ∞) In view of (A2), Ω(F) = ∅. Evidently, Ω(F) is a closed subset of X. In the literature, the set Ω(a) is called a global attractor of a. Note that, in [1,2], Ω(a) is called a turnpike set of a.
For every point x ∈ R n and every nonempty set E ⊂ R n , we define The following proposition is proved in Section 3.

Proposition 5.
For every function x ∈ A(0, ∞), The following theorem is proved in Section 4. The following proposition is proved in Section 5.
The following theorem, which is proved in Section 6, is our main result. Theorem 6. The following properties are equivalent: Properties (1) and (2) usually hold for models of economic dynamics, which are prototypes of our dynamical system [1,2,8,9]. In particular, it holds for the von Neumann-Gale model generated by a monotone process of convex type which was studied in [18].
The next result is proved in Section 7.
The following theorem is our second main result, which is also proved in Section 7. It shows that, if the starting point of the trajectory is closed to the turnpike, then the trajectory leaves the turnpike only when t is closed to the right endpoint T of the domain. Now, we show that, if the set-valued mapping F has a certain symmetric property, then the turnpike Ω(F) possesses the corresponding symmetric property. Let us assume that A : R n → R n is a linear invertible mapping such that A 2 is the identity mapping in R n . Clearly, A = A −1 . Let us assume that A(X) = X and and Ax ∈ A(0, ∞). This implies that A(Ω(F)) ⊂ Ω(F).
Since A 2 is the identity mapping in R n , the inclusion above implies that Thus, A(Ω(F)) = Ω(F).

Proofs of Theorem 4 and Proposition 5
Proof of Theorem 4. Clearly, (1) implies (2). Let us assume that (2) holds and that, for every k ∈ N , x k ∈ A(0, T k ) satisfies lim k→∞ T k = ∞ and x k (0) ∈ C for all k ∈ N . By Theorem 3 and the relations above, extracting subsequences and using the diagonalization process, we obtain that there exist a subsequence {x k i } ∞ i=1 and x ∈ A(0, ∞) such that x k i converges to x as i → ∞ uniformly on [0, p] for every p ∈ N . Thus, (1) holds and (2) implies (1). Theorem 4 is proved.

Proof of Theorem 5
Let us assume that the theorem is not true. Then, for every k ∈ N , there exist for which and, for every a ∈ [0, In view of (A2) and (3.13), there exists a positive number M 0 for which By Theorem 3 and Equations (12) and (15), extracting subsequences and using the diagonalization process, we obtain that there exist a subsequence {x k j } ∞ i=1 and x ∈ A(0, ∞) such that, for every p ∈ N , Therefore, there is a positive number T 0 such that, for every real number t ≥ T 0 , By (16), there is a natural number k > T 0 + M for which Combined with (17), the equation above implies that This contradicts (14) and proves Theorem 5.

Proof of Proposition 6
By definition (9), for each k ∈ N , there is a function x k ∈ A(0, ∞) for which By (18), we may assume, without loss of generality, that Assumption (A2) and (19) imply that there is a positive number M 1 for which Let k ∈ N . By (18), there is a real number t k for which We define Clearly, Equations (20) and (22) imply that By (21)- (24), extracting subsequences and using the diagonalization process, we obtain that there exist a subsequence {y k j } ∞ i=1 and x ∈ A(−∞, ∞) such that, for every p ∈ N , Equations (21), (22) and (25) imply that By (24) and (25), x(t) ≤ M 1 for all t ∈ R 1 . Proposition 6 is proved.
Let us assume the contrary. Then, there are positive real numbers , M and, for each k ∈ N , there exist x k ∈ A(0, T k ) such that Assumption (A2) and (27) imply that there is a positive number M 1 for which Let k ∈ N . By (28), there is a number We define It follows, from (26), (29), (30) and (32), that By (33) and Theorem 3, extracting subsequences and using the diagonalization process, we obtain that there exist a subsequence {y k j } ∞ i=1 and x ∈ A(−∞, ∞) such that, for every p ∈ N , y k j converges to x as j → ∞ uniformly on [−p, p].
Property (i) and Equations (36), (40) and (41) imply that By (36), (39) and Theorem 3, extracting subsequences and using the diagonalization process, we obtain that there exist a subsequence {x k j } ∞ i=1 and x ∈ A(0, ∞) such that for every p ∈ N , x k j converges to x as j → ∞ uniformly on [0, p].
This contradicts (41) and completes the proof of Theorem 7.