By Ya. Z. Tsypkin
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18) 1, . . 14). It should be obvious by now that the stochastic processes differ from one another, and in particular, from the deterministic processes, but only by the form of probabilistic characteristics-probability density functions. The volume of a priori information for deterministic processes is usually larger than for stochastic processes since the probability density functions for deterministic processes are known in advance, while for stochastic processes they are to be determined. If the probability distribution is determined in advance, and if we can manage to write the functional and the constraint equations in an explicit form, then regardless of the basic differences between the deterministic and stochastic processes, it is difficult to establish any prominent dissimilarities in the formulation and the solution of optimization problems for these processes.
The speleologist who finds a certain low place in the desert cannot be certain that a lower place does not exist close by. 12 Multistage Algorithms of Optimization All the algorithms of optimization considered thus far are related to singlestage algorithms. They are described by a vector difference equation of the first order, and therefore they can be called the algorithms of the first order. If the functional J(c) has several extrema, the single-stage algorithms can determine only local extrema.
Here, we have attempted to emphasize the generality of the formulation and the solution of the problem of optimality for these processes, and thus their differences were not extensively discussed. The treatment found here is identical to one presented earlier in a paper by the author (Tsypkin, 1966). The relationship between the stochastic problem of synthesizing a linear system which is optimal with respect to the variance of the error, and the deterministic problem of synthesizing a linear system which is optimal according to the integral square error was shown by Kalman (1961),who formulated this relationship as the principle of duality.
Adaptation and Learning in Automatic Systems by Ya. Z. Tsypkin