Infinite Horizon Optimal Control : Deterministic and Stochastic Systems

This monograph deals with various classes of deterministic and stochastic continuous time optimal control problems that are defined over unbounded time intervals. For these problems the performance criterion is described by an improper integral and it is possible that, when evaluated at a given admissible element, this criterion is unbounded. To cope with this divergence new optimality concepts, referred to here as overtaking optimality, weakly overtaking optimality, agreeable plans, etc. , have been proposed. The motivation for studying these problems arises primarily from the economic and biological sciences where models of this type arise naturally. Indeed, any bound placed on the time hori­ zon is artificial when one considers the evolution of the state of an economy or species. The responsibility for the introduction of this interesting class of problems rests with the economists who first studied them in the modeling of capital accumulation processes. Perhaps the earliest of these was F. Ramsey [152] who, in his seminal work on the theory of saving in 1928, considered a dynamic optimization model defined on an infinite time horizon. Briefly, this problem can be described as a Lagrange problem with unbounded time interval. The advent of modern control theory, particularly the formulation of the famous Maximum Principle of Pontryagin, has had a considerable impact on the treat­ ment of these models as well as optimization theory in general. This book presents a systematic account of the development of optimal control problems defined on an unbounded time interval - beginning primarily with the work of the early seventies to the present. The first five to six chapters provide a good introduction to infinite horizon control theory and require only a minimal knowledge of mathematical control theory. The remainder of the book considers extensions of the previous chapters to a variety of control systems, including Distributed Parameter Systems, Stochastic Control Systems and Hereditary Systems. Consequently these chapters require more sophistication. Throughout the book it is possible to distinguish three categories of research: - The extension of the classical necessary conditions to various weaker types of optimality (e.g., overtaking optimality); - The discussion of various sufficient conditions and verification theorems for the various types of optimality; - The discussion of existence theorems for the various types of optimality. The common link between these categories is the "Turnpike Property" and the notion of "Reduction to Finite Costs". Once these properties are established for a given control system, it is possible to begin investigating the issues described in the above three categories.