Turnpike Properties of Optimal Control Systems

In this paper we discuss recent progress in the turnpike theory which is one of our primary areas of research. Turnpike properties are well known in mathematical economics. The term was first coined by Samuelson (see [1]) who showed that an efficient expanding economy would for most of the time be in the vicinity of a balanced equilibrium path. These properties were studied by many researches for optimal paths of models of economic dynamics determined by set-valued mappings with convex graphs. In our recent book [5] we present a number of turnpike results in the calculus of variations, optimal control, the game theory and in economic dynamics obtained by the author. The results collected in [5] demonstrate that the turnpike properties are a general phenomenon which holds for various classes of variational problems and optimal control problems arising in engineering and in models of economic growth. Turnpike properties are studied for optimal control problems on finite time intervals [T1,T2] such that T1 < T2. Here T1,T2 are real numbers in the case of continuoustime problems and are integers in the case of discretetime problems. Solutions of such problems (trajectories or paths) depend on an optimality criterion determined by an objective function (integrand), the time interval [T1,T2], and on data which is some initial conditions. In the turnpike theory we study the structure of solutions when an objective function (an optimality criterion) is fixed while T1,T2 and the data vary. To have turnpike properties means that the solutions of a problem are determined mainly by the objective function (optimality criterion), and are essentially independent of the choice of time interval and data, except in regions close to the endpoints of the time interval. If a real number t does not belong to these regions, then the value of a solution at the point t is closed to a “turnpike” a trajectory (path) which is defined on the infinite time interval and depends only on the objective function (optimality criterion). This phenomenon has the following interpretation. If one wishes to reach a point A from a point B by a car in an optimal way, then one should enter onto a turnpike, spend most of one’s time on it and then leave the turnpike to reach the required point. P.A. Samuelson discovered the turnpike phenomenon in a specific situation in 1948. In further studies turnpike results were obtained under certain rather strong assumptions on an objective function (optimality criterion). Usually it was assumed that an objective function is convex, as a function of all its variables and does not depend on the time variable t. In this case it was shown that the “turnpike“ is a stationary trajectory (a singleton). Since convexity assumptions usually hold for models of economic growth, turnpike theory has many applications in mathematical economics. There are several turnpike results for nonconvex (noncocave) problems but for these problems convexity (concavity) was replaced by other restrictive assumptions which hold for narrow classes of problems. Therefore experts considered the turnpike phenomenon as an interesting and important property of some very particular optimal control systems with origin in mathematical economics and for which a “turnpike” was usually a singleton or a half-ray. This situation has changed in the last period of time which Turnpike Properties of Optimal Control Systems