A TURNPIKE THEOREM FOR CONTINUOUS-TIME CONTROL SYSTEMS WHEN THE OPTIMAL STATIONARY POINT IS NOT UNIQUE

Here, x0 ∈Ω is an assigned initial point. The multivalued mapping a : Ω→ Πc(R) has compact images and is continuous in the Hausdorff metric. We also assume that at every point x ∈Ω the set a(x) is uniformly locally connected (see [2]). The function u : Ω→R1 is a given continuous function. In this paper, we study the turnpike property for problem (1.1) and (1.2). The term of turnpike property was first coined by Samuelson (see [17]) where it is shown that an efficient expanding economy would spend most of the time in the vicinity of a balanced equilibrium path. This property was further investigated by Radner [14], McKenzie [12], Makarov and Rubinov [7], and others for optimal trajectories of a von Neuman-Gale model with discrete time. In all these studies, the turnpike property was established under some convexity assumptions.


Introduction
Let x ∈ R n and let Ω ⊂ R n be a given compact set.Denote by Π c (R n ) the set of all compact subsets of R n .We consider the following problem: ẋ ∈ a(x), x(0) = x 0 , (1.1) Here, x 0 ∈ Ω is an assigned initial point.The multivalued mapping a : Ω → Π c (R n ) has compact images and is continuous in the Hausdorff metric.We also assume that at every point x ∈ Ω the set a(x) is uniformly locally connected (see [2]).The function u : Ω → R 1 is a given continuous function.
In this paper, we study the turnpike property for problem (1.1) and (1.2).The term of turnpike property was first coined by Samuelson (see [17]) where it is shown that an efficient expanding economy would spend most of the time in the vicinity of a balanced equilibrium path.This property was further investigated by Radner [14], McKenzie [12], Makarov and Rubinov [7], and others for optimal trajectories of a von Neuman-Gale model with discrete time.In all these studies, the turnpike property was established under some convexity assumptions.
In [11,13], the turnpike property was defined using the notion of statistical convergence (see [3]) and it was proved that all optimal trajectories have the same unique statistical cluster point (which is also a statistical limit point).In these works, the turnpike property is proved when the graph of the mapping a is not a convex set.
The turnpike property for continuous-time control systems was studied by Rockafellar [15,16], Cass and Shell [1], Scheinkman [6,18], and others where, besides convexity assumptions, some additional conditions are imposed on the Hamiltonian.To prove turnpike theorem without these kind of additional conditions became a very important problem.This problem was further investigated by Zaslavski [19,21], Mamedov [8,9,10], and others.
In [10], problem (1.1) and (1.2) is considered without convexity assumptions and the turnpike property is established assuming that the optimal stationary point is unique.In this paper, we consider the case when a turnpike set consists of several optimal stationary points.Definition 1.1.An absolutely continuous function x(•) is called a trajectory (solution) to system (1.1) on the interval [0,T] if x(0) = x 0 and almost everywhere on the interval [0,T] the inclusion ẋ(t) ∈ a(x(t)) is satisfied.
We denote the set of trajectories defined on the interval [0,T] by X T and we let (1.3) Since x(t) ∈ Ω and the set Ω is bounded, the trajectories of system (1.1) are uniformly bounded, that is, there exists a number L < +∞ such that (1.4) On the other hand, since the mapping a is continuous, then there is a number Note that in this paper we focus our attention on the turnpike property of optimal trajectories.So we did not study the existence of bounded trajectories defined on [0, ∞].This problem for different control problems has been studied by Leizarowitz [4,5], Zaslavsky [19,20], and others.

Musa A. Mamedov 633
Stationary points play an important role in the study of asymptotical behavior of optimal trajectories.We denote the set of stationary points by M: (1.7) We assume that the set M is nonempty.Since the mapping a(x) is continuous, then the set M is also closed.Therefore M is a compact set.
Definition 1.4.The point x * ∈ M is called an optimal stationary point if We denote the set of optimal stationary point by M op .Since the function u is continuous, then this set is not empty.In Turnpike theory, it is usually assumed that the optimal stationary point x * is unique.In this paper, we consider nonconvex problem (1.1) and (1.2) (i.e., the function u is not strictly concave and the graph of the mapping a is not convex) and therefore the optimal stationary point may be not unique.
We assume that the set Consider an example for which this assumption holds.
Example 1.5.Assume that the set M is convex and where the functions u i are continuous and strictly concave.For every i, there exists a unique point x i ∈ M for which Clearly, the function u is continuous and u * = max{u * i : i ∈ {1, 2,...,l}}.We also note that the function u may be not concave.In this example the number m and the points x * 1 ,x * 2 ,...,x * m in (1.9) can be chosen out of the points x i (i ∈ {1, 2,...,l}) for which u(x i ) = u * .

Main conditions and Turnpike theorem
The turnpike theorem will be proved under two main conditions, Conditions 2.1 and 2.2.The first condition is about the existence of "good" trajectories starting from the initial state x 0 .The second is the main condition which provides the turnpike property.
Condition 2.1.There exists b < +∞ such that, for every T > 0, there is a trajectory x(•) ∈ X T satisfying the inequality (2.1) Note that the satisfaction of this condition depends in an essential way on the initial point x 0 , and in a certain sense it can be considered as a condition for the existence of trajectories converging to some points x * i , i = 1,2,...,m.Thus, for example, if there exists a trajectory that hits some optimal stationary point x * i in finite time, then Condition 2.1 is satisfied. Set We fix p ∈ R n , p = 0, and define a support function py. (2.3) Here, the notation py means the scalar product of the vectors p and y.By |c| we denote the absolute value of c.We also define the function Condition 2.2.There exists a vector p ∈ R n such that (H1) c(x) < 0 for all x ∈ Ꮾ and x = x * i , i = 1,2,...,m; (H2) there exist points xi ∈ Ω such that (H3) for all points x, y, for which the inequality ϕ(x, y) < 0 is satisfied; and also if then limsup k→∞ ϕ(x k , y k ) < 0.
Note that if Condition 2.2 is satisfied for any vector p, then it is also satisfied for all λp, (λ > 0).That is why we assume that p = 1.

Musa A. Mamedov 635
Condition (H1) means that derivatives of system (1.1) are directed to one side with respect to p; that is, if x ∈ Ꮾ and x = x * i , i = 1,2,...,m, then py < 0 for all y ∈ a(x).It is also clear that py ≤ 0 for all y ∈ a(x * i ) and c(x * i ) = 0, i = 1,2,...,m.The main condition here is (H3).It can be considered as a relation between the mapping a and the function u which provides the turnpike property.In [8] it is shown that conditions (H1) and (H3) hold if the graph of the mapping a is a convex set (in R n × R n ) and the function u is strictly concave.On the other hand, an example given in [10] shows that Condition 2.2 may hold for mappings a having nonconvex graphs and for functions u that are not strictly concave (in this example the function u is convex).
The main sense of the turnpike property is that optimal trajectories can stay just during a restricted time interval on the outside of the ε-neighborhood of the turnpike set M op .When the set M op consists of several different points, it is interesting to study a state transition of the trajectories from one optimal stationary point to another.We introduce the following definition.Take any number δ > 0 and let S δ (x) stands for the closed δ-neighborhood of the point x.
Clearly in Definition 2.3 a small number δ should be used.We take Now we formulate the main result of the present paper.
Theorem 2.4.Suppose that Conditions 2.1 and 2.2 are satisfied and there are m different optimal stationary points x * i .Then (1) there exists C < +∞ such that for every T > 0 and every trajectory x(•) ∈ X T ; (2) for every ε > 0, there exists K ε,ξ < +∞ such that for every T > 0 and every ξ-optimal trajectory x(•) ∈ X T ; (3) for every ξ > 0 and δ > 0 (satisfying (2.9)), there exists a number N δ,ξ < +∞ such that for every T > 0 and every ξ-optimal trajectory x(•) ∈ X T ; (4) if x(•) is an optimal trajectory and x(t . The proof of this theorem is given in Section 4. In Section 3, we present preliminary results.

Let x
By the condition (H2) we have c(x) < 0. Since the function c(x) is continuous, there is a number ε x > 0 such that c(x ) < 0 for all x ∈ V εx (x) ∩ Ω.We define the set Ᏸ as follows: It is not difficult to show that the following conditions hold: Here, and we recall that for every x ∈ Ω, x / ∈ intᏰ, and Assume on the contrary that for any ε > 0, there exists a sequence ∈ intᏰ, and also u(x ) = u * , which implies x ∈ Ꮾ.This contradicts property (a) of the set Ᏸ. Lemma 3.2.For every ε > 0, there exists Proof.Assume on the contrary that for any ε > 0, there exists a sequence ..,m), and c(x k ) → 0. Let x be a limit point of the sequence x k .Then x ∈ Ᏸ, x = x * i (i = 1,...,m), and c(x ) = 0.This contradicts property (b) of the set Ᏸ. (a) the set π can be presented as a union of two sets, π = π 1 ∪ π 2 , such that Note that the inclusion x(t) ∈ intᏰ means that u(x(t)) > u * whereas the condition where 638 A turnpike theorem for continuous-time control systems (c) for every δ > 0, there exists a number K(δ) < ∞ such that ) ..,m} and p * i = px * i , i = 1,...,m.The proof of this lemma is similar to the proof of [10,Lemma 5.4], so we do not give it.We also present the next two lemmas without proofs.Their proofs can be done in a similar way to the proofs of [10, Lemmas 6.6 and 6.7].
Lemma 3.6.Assume that x(•) ∈ X T is a continuously differentiable function.Then, the interval [0,T] can be divided into subintervals such that Here, we have ) (3) the set E such that (4) for every δ > 0, there is a number C(δ) such that where ) and the number C(δ) < +∞ does not depend on the trajectory x(•), on T, and on the intervals of (3.10).
Lemma 3.7.Assume that x(•) ∈ X T is a continuously differentiable function and the sets F i (i = 1,2,3) are defined in Lemma 3.6.Then, there is a number L < +∞ such that where the number L does not depend on the trajectory x(•), on T, and on the intervals in (3.10).

Proof of Theorem 2.4
From Condition 2.1, it follows that, for every T > 0, there is a trajectory x T (•) ∈ X T , for which (1) First we consider the case when x(t) is a continuously differentiable function.
In this case we can use the results obtained in Section 3. From Lemmas 3.6 and 3.7, we have Then from Lemma 3.5, we obtain (see, also, (3.10)) [0,T] that is,
Let ε > 0 and δ > 0 be given numbers and let x(•) be a continuously differentiable ξ-optimal trajectory.We denote First we show that there is a number Kε,ξ < +∞ (which does not depend on T > 0) such that the following inequality holds Assume that (4.13) is not true.In this case, there exist sequences T k → ∞ and K k ε,ξ → ∞, and sequences of trajectories {x k (•)} (every x k (•) is a ξ-optimal trajectory in the interval [0,T k ]) and {x Tk (•)} (satisfying (4.1) for every (4.15) Denote ν = min{ν ε ,δ 2 ε } > 0. From (4.6), it follows that Therefore, for sufficient large numbers k, we have which means that x k (t) is not a ξ-optimal trajectory.This is a contradiction.Thus (4.13) is true.Now, we show that, for every δ > 0, there is a number From (4.9) and (4.10), we have Here K(δ) = max{1,K(δ)}.Since Z δ ⊂ ᐄ δ , then taking into account (4.13) we obtain (4.18),where We denote Clearly, ᐄ 0 ε/2 is an open set and therefore it can be presented as a union of at most countable number of open intervals τk .Out of these intervals, we chose the intervals τ k , k = 1,2,..., which have nonempty intersections with ᐄ ε .Then we have Since a derivative of the function x(t) is bounded, it is not difficult to see that there is a number σ ε > 0 such that measτ k ≥ σ ε , ∀k. (4.23) But the interval [0,T] is bounded and therefore the number of intervals τ k is finite too.Let k = 1,2,3,...,N T (ε).We divide every interval τ k into two parts: From (4.8) and (4.22), we obtain and therefore from (4.13) it follows that Now we apply Lemma 3.2.We have
(2) Now we take any trajectory x(•) to system (1.1).It is known that (see, for example, [2]) for a given number δ > 0 (we take δ < ε/2), there exists a continuously differentiable trajectory x(•), to system (1.1), such that Since the function u is continuous, then there is η(δ) > 0 such that Let ξ > 0 be a given number.For every T > 0, we choose a number δ such that Tη(δ) ≤ ξ.Then, Since the function x(•) is continuously differentiable, then the second integral in this inequality is bounded (see the first part of the proof), and therefore the first assertion of Theorem 2.4 is proved.Now, we prove the second assertion of Theorem 2.4.We will use (4.55).Take a number ε > 0 and assume that x(•) is a ξ-optimal trajectory; that is, From (4.55), we have

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Thus x(•) is a continuously differentiable 2ξ-optimal trajectory.That is why (see the first part of the proof) for the numbers ε/2 > 0 and 2ξ > 0, there is which implies that the proof of the second assertion of the theorem is completed; that is, Now, we prove the third assertion of Theorem 2.4.We take any numbers ε > 0 and δ > 0 (satisfying (2.9)).Consider a ξ-optimal trajectory x(•) ∈ ᐄ T , T > 0, and let N = N T (δ,ξ,x(•)) be a number of state transitions.By Definition 2. The third assertion of the theorem is proved if we take N δ,ξ = K δ,ξ /η < ∞.
(4) Now, we prove the fourth assertion of Theorem 2.4.Let x(•) be an optimal trajectory and x(t 1 ) = x(t 2 ) = x * = x * i for some i ∈ {1, 2,...,m}.Consider a trajectory x * (•) defined by the formula Assume that the third assertion of Theorem 2.4 is not true; that is, there is a point t ∈ (t 1 ,t 2 ) such that x(t ) − x * = c > 0.