Content.- I. Pure Strategies for Positional Functionals.- 1. A model problem.- 2. A model problem for mixed strategies.- 3. Equation of motion.- 4. The positional quality index.- 5. Pure positional strategies.- 6. Statement of the problem.- 7. An auxiliary w-object.- 8. Accompanying points. An extremal shift.- 9. Existence of the Solution.(Value of the game. Optimal strategies).- II. Stochastic Program Synthesis of Pure Strategies for a Positional Functional.- 10. Differential games for typical functionals ?(1) and ?(2).- 11. Approximating functionals.- 12. Stochastic program maximins ?*(1)(·) and ?*(2)(·).- 13. Recurrent construction for the program extremum (i)(·).- 14. Recurrent construction for the program extremum e*(2)(·).- 15. Condition of u-stability for the program extremum e*(1)(·).- 16. Condition of v-stability for the program extremum e*(1) (·).- 17. Approximating Solution of the differential game for ?(1).- 18. Construction of optimal approximating strategies.- 19. Conditions of w-stability and v-stability for the program extremum e*(2) (·).- 20. Approximating Solution of the differential game for ?(2).- III. Pure Strategies for Quasi-Positional Functionals.- 21. Quasi-positional functionals.- 22. The differential game for the functional ?(4).- 23. Calculation of the approximating value of the game for ?(4).- 24. Differential games with the functional ?(3).- 25. Constructions of approximating optimal strategies for ?(3).- 26. Differential games for the functionals ?(i), i = 5 , . . . , 8.- 27. Construction of approximating optimal strategies for games with functionals ?(i), i = 5 , . . . , 8.- 28. Example of constructing the optimal strategies for ?(i).- 29. Example with a quadratic quality index.- 30. Examples of constructing optimal strategies for positional functionals.- 31. Examples of constructing optimal strategies for quasi-positional functionals.- IV. Mixed Strategies for Positional and Quasi-Positional Functionals.- 32. Mixed strategies Su and Sv.- 33. The motions xsu,v [·] and xsv,u [·].- 34. Statement of the problem in the class of mixed strategies. Existence of the Solution.- 35. Constructions of the values e(s)*1(·) and e(s)*2(·).- 36. Construction of approximating optimal strategies Sua and Sva.- 37. Example with mixed strategies.- 38. The second example with mixed strategies.
The mathematical theory of control, essentially developed during the last decades, is used for solving many problems of practical importance. The efficiency of its applications has increased in connection with the refine ment of computer techniques and the corresponding mathematical soft ware. Real-time control schemes that include computer-realized blocks are, for example, attracting ever more attention. The theory of control provides abstract models of controlled systems and the processes realized in them. This theory investigates these models, proposes methods for solv ing the corresponding problems and indicates ways to construct control algorithms and the methods of their computer realization. The usual scheme of control is the following: There is an object F whose state at every time instant t is described by a phase variable x. The object is subjected to a control action u. This action is generated by a control device U. The object is also affected by a disturbance v generated by the environment. The information on the state of the system is supplied to the generator U by the informational variable y. The mathematical character of the variables x, u, v and yare determined by the nature of the system.
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