On the errors of stochastic solution for equations of Boltzmann type: exact upper bounds

Abstract

In the paper it is theoretically ground and elaborated a special method for minimization of the states number and construct a minimal form of a generalized finite non-stationary minimax fuzzy automata which is based on the previously proven theorem about maximin and minimax fuzzy matrices product and elaborated matrix method for optimization of a generalized finite non-stationary maximin fuzzy automata. It is proved that from the given generalized finite non-stationary minimax fuzzy automaton may be turn to generalized finite non-stationary maximin fuzzy automata, which is an addition to the initial minimax automaton. It is also proved that if given the generalized finite non-stationary minimax and maximin fuzzy automata are addition of each other, their minimal forms have the same number of states, which allows first turn from the generalized finite non-stationary minimax fuzzy automaton to generalized finite non-stationary maximin fuzzy automaton, then to minimize this obtained generalized maxmin fuzzy automaton by known method of transform matrix and turn back to its addition, get a minimal form of initial generalized finite non-stationary minimax fuzzy automaton. As a result, the procedure and the corresponding algorithm of minimization of the number of states and construct a minimal form of a generalized finite non-stationary minimax fuzzy automaton worked out. Finally, an example of application of the proposed special method of minimization to the given generalized finite non-stationary minimax fuzzy automaton is given. Refs 7.

Keywords:

optimal control, Bellman equation, contour with minimal expenses, tropical mathematics