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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Chapter I. Pure strategies for positional functionals

Abstract
This chapter begins with two model examples of the problems considered and solved in this book. The main content of this chapter is the introduction of the basic notions that enable us to formulate the problems of control under lack of Information. We obtain the problem of control with the purpose of optimizing the ensured result. This gives the game problem on minimax of a chosen index.
A. N. Krasovskii, N. N. Krasovskii

Chapter II. Stochastic program synthesis of pure strategies for a positional functional

Abstract
The basic result of this chapter is a new effective calculation procedure for the value of the game p0(t, x) and also the corresponding construction of the optimal strategies u0(·) = u0(t,x,e) and = v0(t, x,ε). These constructions were suggested and developed in [61], [64]-[66], [80], [98]. They are connected with the idea of stochastic program synthesis, which was developed in Ekatherinburg [57], [60], [61], [75], [77], [80], [144].
A. N. Krasovskii, N. N. Krasovskii

Chapter III. Pure strategies for quasi-positional functionals

Abstract
In this chapter we consider the same problem as in Chapters I and II but for functionals γ(i) combining typical estimates of the motion of x-system and also the actions u and v. We call these functionals γ(i) quasi-positional because the optimal strategies u0(i)(·) = u0(i)(y(i),ε) and u0(i)(·) = (y(i), ε) are based on the information images y(i) that can include now not only current position {t, x} of the controlled x-system but also other variables. We give some Classification of these functionals γ(i) and describe an effective computation of the values of the game ρ(i)(y(i)) and constructions of the pure optimal strategies u0(i)(·) and u0(i)(·) which form the saddle point in the corresponding differential games.
A. N. Krasovskii, N. N. Krasovskii

Chapter IV. Mixed strategies for positional and quasi-positional functionals

Abstract
In this chapter we consider the same feedback control problem on the minimax of the quality index γ. However, now we will consider the case when the saddle point condition in a small game (see (5.1) and (10.2)) for the controlled System (3.1) or (10.1) is not valid. In this case the problem can be solved effectively within the framework of the mixed strategies (see the particular case in Section 2). The description of this Solution is the subject of the following sections.
A. N. Krasovskii, N. N. Krasovskii

Backmatter

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