The theory of controlled processes is one of the most recent mathematical theories to show very important applications in modern engineering, parti cularly for constructing automatic control systems, as well as for problems of economic control. However, actual systems subject to control do not admit a strictly deterministic analysis in view of random factors of various kinds which influence their behavior. Such factors include, for example, random noise occurring in the electrical system, variations in the supply and demand of commodities, fluctuations in the labor force in economics, and random failures of components on an automated line. The theory of con trolled processes takes the random nature of the behavior of a system into account. In such cases it is natural, when choosing a control strategy, to proceed from the average expected result, taking note of all the possible variants of the behavior of a controlled system. An extensive literature is devoted to various economic and engineering systems of control (some of these works are listed in the Bibliography). is no text which adequately covers the general However, as of now there mathematical theory of controlled processes. The authors ofthis monograph have attempted to fill this gap. In this volume the general theory of discrete-parameter (time) controlled processes (Chapter 1) and those with continuous-time (Chapter 2), as well as the theory of controlled stochastic differential equations (Chapter 3), are presented.
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Table of Contents1 Discrete-Parameter Controlled Stochastic Processes.- 1 Definitions.- 2 Optimization Problem.- 3 Construction of Optimal and ?-Optimal Controls.- 4 Control of Processes with Incomplete Observations.- 5 Optimal Stopping Problems.- 6 Controlled Markov Chains.- 7 Homogeneous Controlled Markov Chains.- 8 Optimal Stopping of Markov Chains.- 2 Continuous-Time Control Processes.- 1 General Definitions.- 2 Representation of the Controlled Objects and Construction of Controlled Processes.- 3 Optimization Problem; Approximation Theorem.- 4 Controlled Markov Processes.- 5 Jump Markovian Controlled Processes.- 3 Controlled Stochastic Differential Equations.- 1 Some Preliminaries.- 2 Stochastic Differential Equations.- 3 Controlled Stochastic Differential Equations.- 4 Evolutional Loss Functions.- 5 Linear Systems without an After-effect.- 6 Control Equations with Continuous Noise.- 7 Controlled Diffusion Processes.- Historical and Bibliographical Remarks.