Optimal Control of Random Sequences in Problems with Constraints

Controlled stochastic processes with discrete time form a very interest ing and meaningful field of research which attracts widespread attention. At the same time these processes are used for solving of many applied problems in the queueing theory, in mathematical economics. in the theory of controlled technical systems, etc. . In this connection, methods of the theory of controlled processes constitute the every day instrument of many specialists working in the areas mentioned. The present book is devoted to the rather new area, that is, to the optimal control theory with functional constraints. This theory is close to the theory of multicriteria optimization. The compromise between the mathematical rigor and the big number of meaningful examples makes the book attractive for professional mathematicians and for specialists who ap ply mathematical methods in different specific problems. Besides. the book contains setting of many new interesting problems for further invf'stigatioll. The book can form the basis of special courses in the theory of controlled stochastic processes for students and post-graduates specializing in the ap plied mathematics and in the control theory of complex systf'ms. The grounding of graduating students of mathematical department is sufficient for the perfect understanding of all the material. The book con tains the extensive Appendix where the necessary knowledge ill Borel spaces and in convex analysis is collected. All the meaningful examples can be also understood by readers who are not deeply grounded in mathematics.

Klappentext

Controlled stochastic processes with discrete time form a very interest ing and meaningful field of research which attracts widespread attention. At the same time these processes are used for solving of many applied problems in the queueing theory, in mathematical economics. in the theory of controlled technical systems, etc. . In this connection, methods of the theory of controlled processes constitute the every day instrument of many specialists working in the areas mentioned. The present book is devoted to the rather new area, that is, to the optimal control theory with functional constraints. This theory is close to the theory of multicriteria optimization. The compromise between the mathematical rigor and the big number of meaningful examples makes the book attractive for professional mathematicians and for specialists who ap ply mathematical methods in different specific problems. Besides. the book contains setting of many new interesting problems for further invf'stigatioll. The book can form the basis of special courses in the theory of controlled stochastic processes for students and post-graduates specializing in the ap plied mathematics and in the control theory of complex systf'ms. The grounding of graduating students of mathematical department is sufficient for the perfect understanding of all the material. The book con tains the extensive Appendix where the necessary knowledge ill Borel spaces and in convex analysis is collected. All the meaningful examples can be also understood by readers who are not deeply grounded in mathematics.

Inhalt
1 Methods of Stochastic Optimal Control.- 1.1 Statement of the optimal control problem and examples.- 1.2 Markov decision processes.- 2 Optimal Control Problems with Constraints.- 2.1 Statement of the problem.- 2.2 Properties of the strategic measures space.- 2.3 Necessary and sufficient conditions for optimality.- 2.4 Essential and inessential constraints.- 2.5 Algorithm for solving the main convex programming problem.- 2.6 Example.- 3 Solvability of the main constrained problem and some extensions.- 3.1 Existence of solutions in constrained problems.- 3.2 Form of optimal control strategies.- 3.3 Example.- 3.4 Other constrained problems of optimal control.- 4 Linear-quadratic systems.- 4.1 Model with a finite horizon.- 4.2 Homogeneous discounted model.- 4.3 Homogeneous model with average losses.- 5 Some applications.- 5.1 Stochastic macroeconomic model of the Neumann type.- 5.2 Simplest ecological-economic system.- 5.3 Model of insurance.- 5.4 Stochastic stabilization problem.- 5.5 Queueing system.- 5.6 Optimization of publicity expenses.- 5.7 Simplest constrained game.- Conclusion.- A1 Borel spaces and their properties.- A1.1 Main concepts.- A1.2 Probability measures on Borel spaces.- A1.3 Semicontinuous functions and measurable selection.- A2 Elements of convex analysis.- A2.1 Certain definitions.- A2.2 Duality relation and Kuhn-Tucker theorem.- A2.3 Selected properties of convex sets.- A3 Proofs of auxiliary statements.- A4 Linear-quadratic systems: proofs of some statements.- References.- List of symbols.- List of the main statements.