econometrics
Turnpike Properties of Optimal
Control Systems
by: Alexander J. Zaslavski
1. Introduction
the turnpike theory we study the structure of solutions
when an objective function (an optimality criterion) is
In this paper we discuss recent progress in the turnpike
fixed while T1,T2 and the data vary. To have turnpike
theory which is one of our primary areas of research.
properties means that the solutions of a problem are
Turnpike properties are well known in mathematical
determined mainly by the objective function (optimality
economics. The term was first coined by Samuelson (see
criterion), and are essentially independent of the choice
[1]) who showed that an efficient expanding economy
of time interval and data, except in regions close to the
would for most of the time be in the vicinity of a balanced
endpoints of the time interval. If a real number t does not
equilibrium path. These properties were studied by many
belong to these regions, then the value of a solution at
researches for optimal paths of models of economic
the point t is closed to a “turnpike” - a trajectory (path)
dynamics determined by set-valued mappings with
which is defined on the infinite time interval and depends
convex graphs. In our recent book [5] we present a
only on the objective function (optimality criterion). This
number of turnpike results in the calculus of variations,
phenomenon has the following interpretation. If one
optimal control, the game theory and in economic
wishes to reach a point A from a point B by a car in an
dynamics obtained by the author. The results collected
optimal way, then one should enter onto a turnpike, spend
in [5] demonstrate that the turnpike properties are a
most of one’s time on it and then leave the turnpike to
general phenomenon which holds for various classes
reach the required point.
of variational problems and optimal control problems
P.A. Samuelson discovered the turnpike phenomenon
arising in engineering and in models of economic growth.
in a specific situation in 1948. In further studies
Turnpike properties are studied for optimal control
turnpike results were obtained under certain rather
problems on finite time intervals [T1,T2] such that T1 <
strong assumptions on an objective function (optimality
T2. Here T1,T2 are real numbers in the case of continuouscriterion). Usually it was assumed that an objective
time problems and are integers in the case of discretefunction is convex, as a function of all its variables and
time problems. Solutions of such problems (trajectories
does not depend on the time variable t. In this case it
or paths) depend on an optimality criterion determined
was shown that the “turnpike“ is a stationary trajectory
by an objective function (integrand), the time interval
(a singleton).
[T1,T2], and on data which is some initial conditions. In
Since convexity assumptions usually hold for models
of economic growth, turnpike theory
has many applications in mathematical
Alexander J. Zaslavski
economics. There are several turnpike
Alexander J. Zaslavski received his doctorate at the
results for nonconvex (noncocave)
Institute of Mathematics of the Siberian branch of
problems but for these problems
the Soviet Academy of Sciences in Novosibirsk.
convexity (concavity) was replaced by
He is a senior researcher at the Department of
other restrictive assumptions which
Mathematics of the Technion - Israel Institute
hold for narrow classes of problems.
of Technology. His research interests are the
Therefore experts considered the
optimization theory, the calculus of variations,
turnpike phenomenon as an interesting
optimal control, the dynamical systems theory,
and important property of some very
nonlinear analysis, the game theory and models
particular optimal control systems with
of economic dynamics. He is an author of 400
origin in mathematical economics and
research papers and three monographs.
for which a “turnpike” was usually a
singleton or a half-ray. This situation has
changed in the last period of time which
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AENORM vol. 20 (77) December 2012
econometrics
began at 1995 when the works [2, 3, 4] appeared. In [2]
we studied a general class of unconstrained discrete-time
optimal control problems and established that a turnpike
property holds for a typical (generic) problem and that a
turnpike is a set which is not necessarily a singleton. For
this class of problems the turnpike can be a singleton but
rather seldom. In [3, 4] we studied the turnpike properties
of extremals of one-dimensional second order variational
problems arising in the theory of thermodynamical
equilibrium for materials. We showed that for this class
of problems the turnpike is a periodic curve which is not
necessarily a singleton.
In our book [5] we collected turnpike results which
were obtained after 1995 and for which we do not
need convexity of an objective function and its time
independence. The results of [5] allow us today to think
about turnpike properties as a general phenomenon which
holds for various classes of optimal control problems.
It was my great pleasure to receive on October
2000 the following letter from Paul A. Samuelson, the
discoverer of the turnpike phenomenon.
Here T is a real number, y and z are points of the space Rn
and an integrand
is strictly convex
and differentiable function such that
If a reader is not familiar with absolutely continuous
functions, it is possible to assume that functions v are
continuously differentiable or at least piecewise C1 functions.
We intend to study the behavior of extremals of the
problem (P0) when the points y,z and the real number T
vary and T is sufficiently large.
In order to meet our goal let us consider the following
auxiliary minimization problem:
(P1)
By the strict convexity of f and the growth condition, the
problem (P1) possesses a unique solution . It is easy to
see that
Dear Professor Zaslavski:
Define
I note with interest your long paper “The Turnpike
Property ...Functions” in Nonlinear Analysis 42 (2000),
1465-98.
It may be of interest to report that this property and
name originated just over half a century ago when, as
a Guggenheim Fellow on a 1948-49 sabbatical leave
from MIT, I conjectured it in a memo written at the
RAND Corporation in Santa Monica, California. In
The Collected Scientific Papers of Paul A. Samuelson,
MIT Press, 1966, 1972, 1977, 1986, it is reproduced.
R. Dorfman, P.A. Samuelson, R.M. Solow, Linear
Programming and Economic Analysis, McGraw-Hill,
1958 gives a pre-Roy Radner exposition. I believe that
somewhere Lionel McKenzie has given a nice survey of
the relevant mathematical-economics literature.
With admiration,
Paul A. Samuelson
It is easy to see that the integrand
is a differentiable and strictly convex function such that
Since the functions
have
and L are both strictly convex we
and
Consider an auxiliary variational problem
2. Problems with convex integrands
Let | . | be the Euclidean norm in the n-dimensional Euclidean space Rn and let < . , . > be the scalar product in Rn.
We consider the variational problem
(P0)
(P2)
is an absolutely continuous function
such that v(0) = y, v(T) = z;
where T > 0 and
. Clearly, for any positive number T and any absolutely continuous function
we have
is an absolutely continuous function
such that
AENORM vol. 20 (77) December 2012
37
econometrics
It follows from the equations above that the problems (P0) and (P2) are equivalent. Namely, a function
is a solution of the problem (P0 ) if
and only if it is a solution of the problem (P2).
We claim that the integrand
possesses the following property (C):
Clearly, the integrals
do not exceed a positive constant c0(|y|,|z|) which depends
only on the norms |y|,|z| and does not depend on T. Therefore
then
Assume that a sequence
satisfies
= 0: The growth condition
implies that the sequence
is bounded. Let
(y, z) be its limit point. Then,
This implies that
, as claimed.
Assume that y,z are points of the space Rn, T > 2 is a
real number and that an absolutely continuous function
is an optimal solution of the problem
(P0). Since the problems (P0) and (P2) are equivalent the
function is also an optimal solution of the problem (P2).
We claim that
where a positive constants c0(|y|,|z|) depends only on |y|
and |z|.
Consider an absolutely continuous function
defined by
In the sequel we denote by mes(E) the Lebesgue
measure of a Lebesgue measurable set
.
If a reader is not familiar with the Lebesgue measure
theory, we can say, roughly speaking, that a set of real
numbers E is Lebesgue measurable if it is (in some sense)
the limit of a sequence of sets of real numbers
such that each Ei is a finite union of open intervals with no
intersections. In this case
where the Lebesgue measure of a set which is a finite
union of open intervals with no intersections is the sum
of their lengths.
Now let
be given. It follows from the property (C) that there exists a positive number
such
that for each point
which satisfies
In
view
of the choice of the constant
and the inequality
we have
and
By the definition of the functions
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December 2012
It is easy now to see that the optimal solution spends
most of the time in an - neighborhood of . By the inequality above, the Lebesgue measure of the set of all real
numbers t, such that (t) does not belong to this -neighborhood, does not exceed the constant
which depends only on |y|,|z| and and does not depend
on T. Following the tradition, the point is called the
turnpike. Moreover we can show that the set
econometrics
is
contained
in
the union
, where
of
two
intervals
3. Nonconvex nonautonomous integrands
We showed in the previous section that the structure of
optimal solutions of the problem (P0), under the assumptions posed on f, is rather simple and the turnpike is
calculated easily as a solution of the problem (P1). Nevertheless, the convexity of the integrand f and its time
independence are very essential for the proof of this turnpike result. In order to obtain a turnpike result for essentially larger classes of variational problems and optimal
control problems we need other methods and ideas. The
following example helps to understand what happens if
the integrand f is nonconvex and nonautonomous and
what kind of turnpike we have for general nonconvex
nonautonomous integrands.
Consider an integrand
defined by
and the family of the problems of the calculus of variations (P3):
where y, z, T1, T2 are real numbers and T2 > T1. It is clear
that the functions f depends on the time variable t, for
each real number t, the function f(t, . , . ) :
is
convex, and for each pair of numbers
the function
is nonconvex. Hence
the function
is also nonconvex
and depends on t.
Let y, z, T1, T2 be real numbers, T2 > T1 +2 and let a
function
R1 be an optimal solution of the
problem (P3). Note that the problem (P3) possesses a solution since the integrand f is a continuous function and the
function
is convex and grows superlinearly at infinity for each point
Consider a function
defined by
Clearly,
and
Therefore
where
It is easy to see that for any real number
following inequality holds:
the
Since the constant c1(|y|,|z|) does not depend on T2 and T1
it follows from the inequality above if the length of the
interval T2 - T1 is sufficiently large, then the function
is equal to cos(t) up to for most
. Again, as
in the case considered in the previous section, we can
show that
where > 0 is a constant which depends only on , |y|
and |z|.
This example demonstrates that there are nonconvex
time dependent integrands for which the turnpike property holds with the same type of convergence as in the case
of convex autonomous variational problems, but with the
the turnpike which is an absolutely continuous time dependent function defined on the infinite interval
.
This leads us to the following definition of the turnpike
property for general integrands.
Consider the problem of the calculus of variations
(P)
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econometrics
is an absolutely continuous function
such that v(T1) = y, v(T2) = z.
Here T1 < T2 are real numbers,
and the function
Rn is continuous.
We say that the integrand f possesses the turnpike property if there exists a locally absolutely continuous function
(called the “turnpike”) which depends only on f such that the following condition holds:
For each bounded subset K of the space Rn and each
positive number there exists a positive constant T(K, )
such that for each pair of real numbers T1 0 and T2 T1+
2T(K, ), each pair of points
and each optimal
solution
of the problem (P), we have
The turnpike property is very important for applications.
Assume that the integrand f possesses the turnpike property, the bounded set K and a small positive number
are given, and we know a finite number of “approximate”
solutions of the problem (P). Then we know the turnpike
Xf , or at least its approximation, and the positive constant
T(K, ) which is an estimate for the time period required
to reach the turnpike. We can use this information if we
need to find an “approximate” solution of the problem
(P) with a new time interval [T1, T2] and the new values
at the end points T1 and T2. More precisely, instead of solving this new problem on the “large”
interval [T1, T2] we can find an “approximate” solution
of problem (P) on the “small” interval [T1, T1 + T(K; )]
with the values y, Xf (T1 + T(K; )) at the end points and
an “approximate” solution of problem (P) on the “small”
interval [T2 - T(K; ), T2] with the values Xf (T2 - T(K;
)), z at the end points. Then the concatenation of the first
solution, the function Xf :[T1 + T(K; ), T2 + T(K; )] and
the second solution is an “approximate” solution of problem (P) on the interval [T1, T2] with the values y, z at the
end points.
In Chapter 2 of [5] we consider a general space
of continuous integrands
which is endowed with a natural complete metric. We
establish the existence of a set
which is a countable intersection of open everywhere dense sets in
such that for each
the turnpike property holds.
Moreover we show that the turnpike property holds for
approximate solutions of variational problems with a integrand
and that the turnpike phenomenon is stable under small perturbations of f.
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AENORM vol. 20 (77) December 2012
References
P. A. Samuelson, “A catenary turnpike theorem involving
consumption and the golden rule”,
American
Economic Review 55 (1965), 486-496
A. J. Zaslavski, “Optimal programs on infinite horizon
1,2“, SIAM Journal on Control and Optimization 33
(1995), 1643-1686
A. J. Zaslavski, “The existence and structure of extremals
for a class of second order infinite horizon variational
problems”, “Journal of Mathematical Analysis and
Applications 194 (1995), 459-476
A. J. Zaslavski, “Structure of extremals for onedimensional variational problems arising in continuum
mechanics“, Journal of mathematical Analysis and
Applications 198 (1996), 893-921
A. J. Zaslavski, “Turnpike properties in the calculus of
variations and optimal control”, Springer, New York,
2006