By Andrew N. Krasovskii, Nikolai N. Krasovskii

The mathematical thought of keep an eye on, basically constructed over the last many years, is used for fixing many difficulties of useful significance. The potency of its functions has elevated in reference to the refine ment of laptop ideas and the corresponding mathematical smooth ware. Real-time keep watch over schemes that come with computer-realized blocks are, for instance, attracting ever extra realization. the speculation of regulate presents summary types of managed structures and the techniques learned in them. This thought investigates those versions, proposes tools for solv ing the corresponding difficulties and exhibits how you can build keep watch over algorithms and the equipment in their laptop cognizance. the standard scheme of keep an eye on is the next: there's an item F whose kingdom at at any time when quick t is defined by means of a part variable x. the article is subjected to a regulate motion u. This motion is generated via a regulate machine U. the item is usually suffering from a disturbance v generated through the surroundings. the data at the country of the method is provided to the generator U through the informational variable y. The mathematical personality of the variables x, u, v and yare made up our minds by means of the character of the system.

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12) 41 1. 2). 4. The positional quality index We consider minimax-maximin (with respect to u and v) problems for some quality index that determines the quality of the process of control. 8) of control actions and disturbances respectively. 9 is a fixed terminal time moment. As in Sections 1 and 2 we denote the quality index by 'Y. 1). 2) and of finding disturbances v that are aimed at maximizing 'Y. We employ the feedback positional control [48], [55], [79], [88], [114]. 1) that is realized at this time moment t, or only using the information about some position {T, X[T]} that is realized not long before the moment t (for example, using the information about the position {ti' X [t i ]} , ti ::; t < ti+1' where ti, i = 1, ...

T ~ {)} of the extended random motion of the complex {x-object, y-model}. ] = {xu,v[t, w], t. 60) Yu,v [t. ] = {Yu,v[t, W], t. ~ t ~ {), WEn}. 61 ) Here the letter W denotes an elementary event that defines the corresponding realization. Such a designation will be used further. But here the symbol W is omitted in the corresponding designations. 62) is not guaranteed now for certain, but is ensured only with the probability v. However, this probability can be made as close to 1 as desired. In fact, it suffices to fix a suitable value of a parameter c > 0, choose a suitable initial state Y.

Let us emphasize, however, that at every time moment t, as in the case of pure positional strategies, the positional mixed strategy uses information only about the position {T,X[T]}. 24). 33) is not valid. 34) and the upper index T denotes transposition. 24) that the value of the optimal ensured result p~(t*, x*) in the class of pure positional strategies u(·) = u(t, x, €) (see Section 1) proves to be larger than the value ofthe optimal ensured result P~Jt*, x*) in the class of counter strategies uv (·) = u(t,X,€,v).