Regularization of numerical estimates of solution regions of differential equations with disturbing effects

A. N. Rogalev

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 21 (2024) no. 2, pp. A57-A69

Résumé

The paper studies algorithms for solving applied problems of estimating the areas of solutions of differential equations. Such problems arise when assessing the practical stability of movements on a finite time interval, when determining the areas of attainability of controlled systems, when assessing the areas of survivability of controlled systems, when calculating guaranteed (not probabilistic) boundaries of zones of dangerous states of technical systems, when computing bounded values of parameters of technical systems that correspond to the boundaries of dangerous zones, when calculating the maximum deviations of movements in order to control whether the system's trajectory enters a dangerous zone. In the problems listed above, for some of the parameters, only estimates of the ranges of their values are known, which leads to the emergence of sets of solutions. The difficulty in solving these problems lies in the fact that most methods for estimating solution sets of ODE systems (or calculating upper and lower boundaries of solutions) lead to a strong increase in the boundaries of these solution sets. In this case, the problem of finding a solution to the same ODE system with numerical values of the parameters can be correctly posed and a convergent method can exist for it. To overcome such growth of boundaries of solution sets, it is useful to regularize the estimates of boundaries of solution sets by moving to a linear approximation of the original system. This regularization is specified by the values of compression/extension in given directions, displacement along the time axis, and rotation by a certain angle of the selected directions of the solution set.

Détail

Mots-clés : perturbations, solution set, compression
Keywords: interior solution, boundary solution, regularization, stretching, displacement, rotation by an angle.

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