INVARIANT IDEALS AND ITS APPLICATIONS TO THE TURNPIKE THEORY MUSA MAMMADOV AND PIOTR SZUCA

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INVARIANT IDEALS AND ITS APPLICATIONS TO THE

TURNPIKE THEORY

MUSA MAMMADOV AND PIOTR SZUCA

Abstract. In this paper the turnpike property is established for a non-convex

optimal control problem in discrete time. The functional is deﬁned by the no-

tion of the ideal convergence and can be considered as an analogue of the termi-

nal functional deﬁned over inﬁnite time horizon. The turnpike property states

that every optimal solution converges to some unique optimal stationary point

in the sense of ideal convergence if the ideal is invariant under translations.

This kind of convergence generalizes, for example, statistical convergence and

convergence with respect to logarithmic density zero sets.

1. Introduction

The turnpike theory investigates an important property of dynamical systems.

It can be considered as a theory that justiﬁes the importance of some equilib-

rium/stationary states. For example, in macroeconomic models the turnpike prop-

erty states that, regardless of initial conditions, all optimal trajectories spend most

of the time within a small neighborhood of some optimal stationary point when the

planning period is long enough. Obviously, in the absence of such a property, us-

ing some of optimal stationary points as a criteria for “good” policy formulation

might be misleading. Correspondingly, the turnpike property is in the core of many

important theories in economics.

Many real-life processes are happening in an optimal way and have the tendency

to stabilize; that is, the turnpike property is expected to hold for a broad class

of problems. It provides valuable insights into the nature of these processes by

investigating underlying principles of evolution that lead to stability. It can also

be used to assess the “quality” of mathematical modeling and to develop more

adequate equations describing system dynamics as well as optimality criteria.

The ﬁrst result in this area is obtained by John von Neumann ([30]) for discrete

time systems. The phenomenon is called the turnpike property after Chapter 12, [9]

by Dorfman, Samuelson and Solow. For a classiﬁcation of diﬀerent deﬁnitions for

this property, see [2, 22, 28, 36], as well as [6] for the so-called exponential turnpike

property. Possible applications in Markov Games can be found in a recent study

[16].

The approaches suggested for the study of the turnpike property involve con-

tinuous and discrete time systems. Some convexity assumptions are suﬃcient for

discrete time systems [22, 28]; however, rather restrictive assumptions are usually

Date: November 1, 2022.

2020 Mathematics Subject Classiﬁcation. Primary 40A35; Secondary 49J99, 54A20.

Key words and phrases. I-convergence, I-cluster set, statistical convergence, turnpike prop-

erty, optimal control, discrete systems.

arXiv:2210.12399v2 [math.OC] 30 Oct 2022

2 M. MAMMADOV AND P. SZUCA

required for continuous time systems. The majority of them deal with the (dis-

counted and undiscounted) integral functionals. We mention here the approaches

developed by Rockafellar [33, 34], Scheinkman, Brock and collaborators (see, for

example, [21, 35]), Cass and Shell [4], Leizarowitz [18], Mamedov [24], Montrucchio

[29], Zaslavski [37, 38, 39] (we refer to [2, 36] for more references).

In this paper we consider an optimal control problem in discrete time. It extends

the results obtained in [23] where a special class of terminal functionals is introduced

as a lower limit at inﬁnity of utility functions. This approach allowed to establish

the turnpike property for a much broader class of optimal control problems than

those involving integral functionals (discounted and undiscounted).

Later, this class of terminal functionals was used to establish a connection be-

tween the turnpike theory and the notion of statistical convergence [25, 32]; as a

result, the convergence of optimal trajectories is proved in terms of the statistical

(“almost”) convergence. These terminal functionals also allowed the extension of

the turnpike theory to time delay systems; the ﬁrst results in this area have been

established in several recent papers [13, 26]. Moreover, some generalizations based

on the notion of the A-statistical cluster points have been obtained in [7].

The main purpose of this paper is to formulate the optimality criteria by using the

notion of ideal convergence. As detailed in the next section, the ideal convergence is

a more general concept than the statistical convergence as well as the A-statistical

convergence. In this way the turnpike property is established for a broad class of

non-convex optimal control problems where the asymptotical stability of optimal

trajectories is formulated in terms of the ideal convergence.

Recently (and independently) Leonetti and Caprio in [19] considered turnpike

property for ideals invariant under translation in the context of normed vector

spaces. We discuss our approaches in Section 4.

The rest of this article is organized as follows. In the next section the deﬁni-

tion of the ideal, its properties and some particular cases, including the statistical

convergence, are provided. In Section 3 we formulate the optimal control problem

and main assumptions. The main results of the paper — the turnpike theorems are

provided in Section 4. The proof of the main theorem is in Section 5.

2. Convergence with respect to ideal vs statistical convergence

Let x= (xn)n∈Nbe a sequence of elements of Rm. For the sake of simplicity, we

will consider the Euclidean norm k·k .The classical deﬁnition of convergence of x

to asays that for every ε > 0 the set of all n∈Nwith kxn−ak ≥ εis ﬁnite, i.e.

it is “small” in some sense. If we understand the word “small” as “of asymptotic

density zero” then we obtain the deﬁnition of statistical convergence (Def. 6). The

same method can be used to formulate the deﬁnition of statistical cluster point.

The classical one says that ais a cluster point of xif for every ε > 0 the set of all

n∈Nwith kxn−ak< ε is inﬁnite, i.e. it has “many” elements. If “many” means

“not of asymptotic density zero” then we obtain the deﬁnition of statistical cluster

point (see e.g. [12]).

One of the possible generalizations of this kind of being “small” (having “many”

elements) is “belonging to the ideal” (“be an element of co-ideal”).

The cardinality of a set Xis denoted by #X.P(N) denotes the power set of N.

INVARIANT IDEALS AND ITS APPLICATIONS TO THE TURNPIKE THEORY 3

Deﬁnition 1. An ideal on P(N)is a family I ⊂ P(N)which is non-empty, hered-

itary and closed under taking ﬁnite unions, i.e. it fulﬁlls the following three condi-

tions:

(1) ∅∈I;

(2) A∈ I if A⊂Band B∈ I;

(3) A∪B∈ I if A, B ∈ I.

Example 2. By Fin we denote the ideal of all ﬁnite subsets of N={1,2, . . .}.

There are many examples of ideals considered in the literature, e.g.

(1) the ideal of sets of asymptotic density zero

Id=A⊂N:d(A)=0,

where d:P(N)→[0,1] is given by the formula

d(A) = lim sup

n→∞

#(A∩ {1,2, . . . , n})

is the well-known deﬁnition of upper asymptotic density of the set A;

(2) the ideal of sets of logarithmic density zero

Ilog =(A⊂N: lim sup

n→∞ Pk∈A∩{1,2,...,n}1

Pk≤n1

= 0);

(3) the ideal

I1/n =(A⊂N:X

n∈A

n<∞);

(4) the ideal of arithmetic progressions free sets

W={W⊂N:Wdoes not contain arithmetic progressions of all lengths}.

Ideals Idand Ilog belongs to the wider class of Erd˝os-Ulam ideal’s (deﬁned

by submeasures of special kind, see [14]). Ideal I1/n is an representant of the

class of summable ideals (see [27]). The fact that Wis an ideal follows from

the non-trivial theorem of van der Waerden (this ideal was considered by Kojman

in [15]). One can also consider trivial ideals I=P(N), I={∅}, or principal ideals

In={A⊂N:n /∈A}, however they are not interesting from our point of view. If

not explicitly said we assume that all considered ideals are proper (i.e. I 6=P(N))

and contain all ﬁnite sets (i.e. Fin ⊂ I). The inclusions between abovementioned

families are shown on Figure 1. The only non-trivial inclusions are: I1/n ⊂ Id(a

folklore application of Cauchy condensation test), W ⊂ Id(the famous theorem of

Szemer´edi), and Id⊂ Ilog (by well-known inequalities between upper logarithmic

density and upper asymptotic density). It is easy to observe that I1/n 6⊂ W, but

the status of the inclusion W ⊂ I1/n is unknown (“Erd˝os conjecture on arithmetic

progressions” says that the van der Waerden ideal Wis contained in the ideal I1/n.)

2.1. I-convergence and I-cluster points. The notion of the ideal convergence

is dual (equivalent) to the notion of the ﬁlter convergence introduced by Cartan

in 1937 ([3]). The notion of the ﬁlter convergence has been an important tool

in general topology and functional analysis since 1940 (when Bourbaki’s book [1]

appeared). Nowadays many authors prefer to use an equivalent dual notion of the

ideal convergence (see e.g. frequently quoted work [17]).

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INVARIANTIDEALSANDITSAPPLICATIONSTOTHETURNPIKETHEORYMUSAMAMMADOVANDPIOTRSZUCAAbstract.Inthispapertheturnpikepropertyisestablishedforanon-convexoptimalcontrolproblemindiscretetime.Thefunctionalisdenedbytheno-tionoftheidealconvergenceandcanbeconsideredasananalogueofthetermi-nalfunctionaldenedoverin...

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INVARIANT IDEALS AND ITS APPLICATIONS TO THE TURNPIKE THEORY MUSA MAMMADOV AND PIOTR SZUCA

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