A Turnpike Property of Trajectories of Dynamical Systems with a Lyapunov Function

Abstract

In this paper, we study the structure of trajectories of discrete disperse dynamical systems with a Lyapunov function which are generated by set-valued mappings. We establish a weak version of the turnpike property which holds for all trajectories of such dynamical systems which are of a sufficient length. This result is usually true for models of economic growth which are prototypes of our dynamical systems.

1. Introduction

In [1,2] A. M. Rubinov introduced a discrete disperse dynamical system determined by a set-valued mapping acting on a compact metric space, which was studied in [1,2,3,4,5,6,7]. This disperse dynamical system has prototype in the mathematical economics [1,8,9]. In particular, it is an abstract extension of the classical von Neumann–Gale model [1,8,9]. Our dynamical system is determined by a compact metric space of states and a transition operator. In [1,2,3,4,5,6,7] and in the present paper, this transition operator is set-valued. Such dynamical systems correspond to certain models of economic dynamics [1,8,9].

Assume that $(X,\rho )$ is a compact metric space and that $a:X\to 2^X\backslash \{\varnothing \}$ is a set-valued mapping whose graph

$$\mathrm{graph}\left(a\right)=\left\{\right(x,y)\in X\times X:y\in a(x\left)\right\}$$

is a closed set in $X\times X$. For every nonempty set $E\subset X$ define

$$a\left(E\right)=\cup \{a\left(x\right):x\in E\}\mathrm{and}{a}^0\left(E\right)=E.$$

By induction we define $a^n\left(E\right)$ for every integer $n\ge 1$ and every nonempty subset $E\subset X$ as follows:

$$a^n\left(E\right)=a\left(a^{n-1}\left(E\right)\right).$$

In the present paper, we analyze the structure of trajectories of the dynamical system determined by a which is called a discrete dispersive dynamical system [1,2].

We say that a sequence ${\left\{x_t\right\}}_{t=0}^{\infty }\subset X$ is a trajectory of a (or just a trajectory if a is understood) if

$$x_{t+1}\in a\left(x_t\right),t=0,1,\dots .$$

Let $T_2>T_1$ be integers. We say that ${\left\{x_t\right\}}_{t=T_1}^{T_2}\subset X$ is a trajectory of a (or just a trajectory if a is understood) if

$$x_{t+1}\in a\left(x_t\right),t=T_1,\dots ,T_2-1.$$

Define

$$\Omega \left(a\right)=\{\xi \in X:\mathrm{for}\mathrm{every}\mathrm{positive}\mathrm{number}ϵ\mathrm{there}\mathrm{exists}\mathrm{a}\mathrm{trajectory}$$

$${\left\{y_t\right\}}_{t=0}^{\infty }\mathrm{for}\mathrm{whicht}\underset{t\to \infty }{lim\; inf}\rho (\xi ,y_t)\le ϵ\}.$$

(1)

Evidently, $\Omega \left(a\right)$ is a nonempty closed set in the metric space $(X,\rho )$. In the literature, the set $\Omega \left(a\right)$ is called a global attractor of a. Note that in [1,2] $\Omega \left(a\right)$ is called a turnpike set of a. This terminology is motivated by mathematical economics [1,8,9].

For every point $x\in X$ and every nonempty closed set $E\subset X$ define

$$\rho (x,E)=inf\left\{\rho \right(x,y):y\in E\}.$$

Let $\varphi :X\to R^1$ be a continuous function satisfying

$$\varphi \left(z\right)\ge 0\mathrm{for}\mathrm{every}z\in X,$$

(2)

$$\varphi \left(y\right)\le \varphi \left(x\right)\mathrm{for}\mathrm{every}x\in X\mathrm{and}\mathrm{every}y\in a\left(x\right).$$

(3)

It is clear that $\varphi$ is a Lyapunov function for the dynamical system determined by the map a. It should be mentioned that in mathematical economics usually X is a subset of the finite-dimensional Euclidean space and $\varphi$ is a linear functional on this space [1,8,9]. Our goal in [7] was to study approximate solutions of the problem

$$\varphi \left(x_T\right)\to max,$$

$${\left\{x_t\right\}}_{t=0}^T\mathrm{is}a\mathrm{program}\mathrm{satisfying}{x}_0=x,$$

where $x\in X$ and $T\in \{1,2,\dots \}$ are given.

The following result was obtained in [7].

Theorem 1.

The following properties are equivalent:

(1) If a sequence ${\left\{x_t\right\}}_{t=-\infty }^{\infty }\subset X$, $x_{t+1}\in a\left(x_t\right)$ and $\varphi \left(x_{t+1}\right)=\varphi \left(x_t\right)$ for every integer t, then

$${\left\{x_t\right\}}_{t=-\infty }^{\infty }\subset \Omega \left(a\right).$$

(2) For every positive number ϵ there exists an integer $T\left(ϵ\right)\ge 1$ such that for every trajectory ${\left\{x_t\right\}}_{t=0}^{\infty }\subset X$ which satisfies $\varphi \left(x_t\right)=\varphi \left(x_{t+1}\right)$ for every nonnegative integer t the relation $\rho (x_t,\Omega \left(a\right))\le ϵ$ is valid for every integer $t\ge T\left(ϵ\right)$.

Put

$$\parallel \varphi \parallel =sup\left\{\right|\varphi \left(z\right)|:z\in X\}.$$

We denote by Card$\left(A\right)$ the cardinality of a set A and suppose that the sum over the empty set is zero.

In this paper, we establish a weak version of the turnpike property which hold for all trajectories of our dynamical system which are of a sufficient length and which are not necessarily approximate solutions of the problem above. This result as well as the turnpike results of [7] is usually true for models of economic growth which are prototypes of our dynamical system [1,8,9].

Namely, we prove the following result.

Theorem 2.

Let property (1) of Theorem 1 hold and let ϵ be a positive number. Then there exists an integer $L\ge 1$ such that for every natural number $T>L$ and every trajectory ${\left\{x_t\right\}}_{t=0}^T$ the inequality

$$Card\left(\{t\in \{0,\dots ,T\}:\rho (x_t,\Omega \left(a\right))>ϵ\}\right)\le L$$

is valid.

This result is proved in Section 3. Its proof is based on an auxiliary result which is proved in Section 2.

Assume that ${\left\{x_t\right\}}_{t=0}^{\infty }$ is a trajectory. By (3), there exists

$$c=\underset{t\to \infty }{lim}\varphi \left(x_t\right).$$

Evidently, the sequence ${\left\{x_t\right\}}_{t=0}^{\infty }$ converges to the set $\Omega \cap {\varphi }^{-1}\left(c\right)$. This fact is well-know in the dynamical systems theory as LaSalle’s invariance principle [10,11,12,13]. In the present paper, we are interested in the structure of trajectories on finite intervals of a sufficiently large length and their turnpike property established in Theorem 1.2, which was not considered in [10,11,12,13].

It should be mentioned that turnpike properties are well known in mathematical economics. The term was first coined by Samuelson in 1948 (see [14]), where he showed that an efficient expanding economy would spend most of the time in the vicinity of a balanced equilibrium path (also called a von Neumann path and a turnpike). This property was further investigated for optimal trajectories of models of economic dynamics. See, for example, [2,8,9] and the references mentioned there. Recently it was shown that the turnpike phenomenon holds for many important classes of problems arising in various areas of research [15,16,17,18,19,20,21,22,23]. For related infinite horizon problems see [9,24,25,26,27,28,29,30,31].

2. An Auxiliary Result

Lemma 1.

Let property (1) of Theorem 1 hold and ϵ be a positive number. Then there exist a positive number δ and an integer $L\ge 1$ such that for every natural number $T>2L$ and every trajectory ${\left\{x_t\right\}}_{t=0}^T$ satisfying

$$\varphi \left(x_0\right)\le \varphi \left(x_T\right)+\delta$$

the inequality

$$\rho (x_t,\Omega \left(a\right))\le ϵ,t=L,\dots ,T-L$$

is valid.

Proof. 

Assume the contrary. Then for every integer $n\ge 1$ there are a natural number $T_n>2n$ and a trajectory ${\left\{x_t^{\left(n\right)}\right\}}_{t=0}^{T_n}$ which satisfy

$$\varphi \left(x_0^{\left(n\right)}\right)\le \varphi \left(x_{T_n}^{\left(n\right)}\right)+1/n,$$

(4)

$$max\{\rho (x_t^{\left(n\right)},\Omega \left(a\right)):t=n,\dots ,T_n-n\}>ϵ.$$

(5)

By of (5), for every $n\in \{1,2,\dots \}$ there is

$$S_n\in \{n,\dots ,T_n-n\}$$

(6)

for which

$$\rho (x_{S_n}^{\left(n\right)},\Omega \left(a\right))>ϵ.$$

(7)

Assume that $n\in \{1,2,\dots \}$. Set

$$y_t^{\left(n\right)}=x_{t+S_n}^{\left(n\right)},t=-S_n,\dots ,T_n-S_n.$$

(8)

In view of (8), ${\left\{y_t^{\left(n\right)}\right\}}_{t=-S_n}^{T_n-S_n}$ is a trajectory. By (4) and (8),

$$\varphi \left(y_{T_n-S_n}^{\left(n\right)}\right)-\varphi \left(y_{-S_n}^{\left(n\right)}\right)=\varphi \left(x_{T_n}^{\left(n\right)}\right)-\varphi \left(x_0^{\left(n\right)}\right)\ge -1/n.$$

(9)

Equations (3) and (9) imply that for every integer $t\in \{-S_n,\dots ,T_n-S_n-1\}$, we have

$$\varphi \left(y_{t+1}^{\left(n\right)}\right)-\varphi \left(y_t^{\left(n\right)}\right)\ge \varphi \left(y_{T_n-S_n}^{\left(n\right)}\right)-\varphi \left(y_{-S_n}^{\left(n\right)}\right)\ge -1/n.$$

(10)

Equations (7) and (8) imply that

$$\rho (y_0^{\left(n\right)},\Omega \left(a\right))=\rho (x_{S_n}^{\left(n\right)},\Omega \left(a\right))>ϵ.$$

(11)

Clearly, there is a strictly increasing sequence of positive integers ${\left\{n_j\right\}}_{j=1}^{\infty }$ such that for every integer t there exists

$$y_t=\underset{j\to \infty }{lim}{y}_t^{\left(n_j\right)}.$$

(12)

By Equations (11) and (12),

$$\rho (y_0,\Omega \left(a\right))\ge ϵ.$$

(13)

By (12) and the closedness of the graph of a, we have

$$y_{t+1}\in a\left(y_t\right)\mathrm{for}\mathrm{all}\mathrm{integers}t.$$

(14)

By (10) and (12), for all integers t,

$$\varphi \left(y_{t+1}\right)-\varphi \left(y_t\right)=\underset{j\to \infty }{lim}\varphi \left(y_{t+1}^{\left(n_j\right)}\right)-\underset{j\to \infty }{lim}\varphi \left(y_t^{\left(n_j\right)}\right)\ge \underset{j\to \infty }{lim}(-n_j^{-1})=0.$$

Combining with (3) this implies that

$$\varphi \left(y_{t+1}\right)=\varphi \left(y_t\right)\mathrm{for}\mathrm{all}\mathrm{integers}t.$$

(15)

Property (1) of Theorem 1, (14), (15) imply the inclusion

$$y_t\in \Omega \left(a\right)$$

for every integer t. This inclusion contradicts Equation (13). The contradiction we have reached completes the proof of Lemma 1. □

3. Proof of Theorem 2

Lemma 1 implies that there are a positive number $\delta <ϵ$ and $L_0\in \{1,2,\dots \}$ for which the following property holds:

(a) for every integer $T>2L_0$ and every trajectory ${\left\{x_t\right\}}_{t=0}^T$ satisfying

$$\varphi \left(x_0\right)\le \varphi \left(x_T\right)+\delta$$

we have

$$\rho (x_t,\Omega \left(a\right))\le ϵ,t=L_0,\dots ,T-L_0.$$

Choose an integer

$$L>2L_0+2+(4L_0+7)(1+2{\delta }^{-1}\parallel \varphi \parallel ).$$

(16)

Suppose that $T>L$ is a natural number and that a sequence ${\left\{x_t\right\}}_{t=0}^T$ is a trajectory. By induction we define a strictly increasing finite sequence $t_i\in \{0,\dots ,T\}$, $i=0,\dots ,q$. Set

$$t_0=0.$$

(17)

If

$$\varphi \left(x_T\right)\ge \varphi \left(x_0\right)-\delta ,$$

then set

$$t_1=T$$

and complete to construct the sequence.

Assume that

$$\varphi \left(x_T\right)<\varphi \left(x_0\right)-\delta .$$

Evidently, there is an integer $t_1\in (t_0,T]$ satisfying

$$\varphi \left(x_{t_1}\right)<\varphi \left(x_0\right)-\delta$$

(18)

and that if an integer S satisfies

$$t_0<S<t_1,$$

then

$$\varphi \left(x_S\right)\ge \varphi \left(x_0\right)-\delta .$$

(19)

If $t_1=T$, then we complete to construct the sequence.

Assume that $k\in \{1,2,\dots \}$ and that we defined a strictly increasing sequence $t_0,\dots ,t_k\in \{0,\dots \}$ such that

$$t_0=0,t_k\le T$$

and that for each $i\in \{0,\dots ,k-1\}$,

$$\varphi \left(x_{t_{i+1}}\right)<\varphi \left(x_{t_i}\right)-\delta$$

and if an integer S satisfies $t_i<S<t_{i+1}$, then

$$\varphi \left(x_S\right)\ge \varphi \left(x_{t_i}\right)-\delta .$$

(In view of (18) and (19), the assumption is true with $k=1$).

If $t_k=T,$ then we complete to construct the sequence. Assume that $t_k<T$. If

$$\varphi \left(x_T\right)\ge \varphi \left(x_{t_k}\right)-\delta ,$$

then we set $t_{k+1}=T$ and complete to construct the sequenced.

Assume that

$$\varphi \left(x_T\right)<\varphi \left(x_{t_k}\right)-\delta .$$

(20)

Evidently, there is a natural number

$$t_{k+1}\in (t_k,T]$$

for which

$$\varphi \left(x_{t_{k+1}}\right)<\varphi \left(x_{t_k}\right)-\delta$$

and that if an integer S satisfies

$$t_k<S<t_{k+1},$$

then

$$\varphi \left(x_S\right)\ge \varphi \left(x_{t_k}\right)-\delta .$$

Evidently, the assumption made for k is true for $k+1$ too. Therefore by induction, we constructed the strictly increasing finite sequence of integers $t_i\in [0,T]$, $i=0,\dots ,q$ such that

$$t_0=0,t_q=T$$

and that for every i satisfying $0\le i<q-1$,

$$\varphi \left(x_{t_{i+1}}\right)<\varphi \left(x_{t_i}\right)-\delta$$

(21)

and for each $i\in \{0,\dots ,q-1\}$ and each integer S satisfies $t_i<S<t_{i+1}$, we have

$$\varphi \left(x_S\right)\ge \varphi \left(x_{t_i}\right)-\delta .$$

(22)

By (21),

$$2\parallel \varphi \parallel \ge \varphi \left(x_{t_0}\right)-\varphi \left(x_{t_{q-1}}\right)$$

$$\sum \{\varphi \left(x_{t_i}\right)-\varphi \left(x_{t_{i+1}}\right):i\mathrm{is}\mathrm{an}\mathrm{integer},0\le i\le q-2\}\ge \delta (q-1)$$

and

$$q\le 1+2{\delta }^{-1}\parallel \varphi \parallel .$$

(23)

Set

$$E=\{i\in \{0,\dots ,q-1\}:t_{i+1}-t_i\ge 2L_0+4\}.$$

(24)

Let

$$i\in E.$$

(25)

By (24) and (25),

$$t_{i+1}-1-t_i\ge 2L_0+3.$$

(26)

Equations (22) and (26) imply that

$$\varphi \left(x_{t_{i+1}-1}\right)\ge \varphi \left(x_{t_i}\right)-\delta .$$

(27)

Equations (26), (27) and property (a) applied to the program ${\left\{x_t\right\}}_{t=t_i}^{t_{i+1}-1}$ imply that

$$\rho (x_t,\Omega \left(a\right))\le ϵ,t=t_i+L_0,\dots ,t_{i+1}-1-L_0.$$

(28)

Equation (28) implies that

$$\{t\in \{0,\dots ,T\}:\rho (x_t,\Omega \left(a\right))>ϵ\}$$

$$\subset \cup \{\{t_i,\dots ,t_{i+1}\}:i\in \{0,\dots ,q-1\}\backslash E\}$$

$$\cup \{\{t_i,\dots ,t_i+L_0-1\}\cup \{t_{i+1}-L_0,\dots ,t_{i+1}\}:i\in E\}.$$

(29)

By (23), (24) and (29),

$$Card\left(\{t\in \{0,\dots ,T\}:\rho (x_t,\Omega \left(a\right))>ϵ\}\right)$$

$$\le q(2L_0+5)+(2L_0+2)q=q(4L_0+7)$$

$$(4L_0+7)(1+2{\delta }^{-1}\parallel \varphi \parallel )\le L.$$

Theorem 2 is proved.