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Abstracts

Any system of the big complexity possesses ability to pass some condition set in some interthreshold condition space, as though probing extreme points, bounds of the solution existence. This system scans set of conditions in searches of a safe condition, following the organic law of systems of the big complexity. However if the system is complex, dynamically changing the conditions, than there are dangerous opportunity of system tend directly to existence thresholds, an opportunity of ocurrence in a zone of a precatastrophic condition. Danger of reception of the rough estimate consists not only in an error opportunity, but also in that opportunity of entropy processes will give system a superfluous push which can lead to catastrophe. To find guaranteed bounds of dangerous conditions zones and threshold values of system parameters which correspond to bounds of these zones, the class of guaranteed methods [1] - [12] is offered. These methods determine solution set bounds and value sets of functionans constructed on these solution. In work it is offered to estimate solution areas at final, permanent perutrbations. We shall allocate problems of check of guaranteed conditions of safety and a problem{task} of construction of sets of approachibility. Let there is a control system dy/dt = f (t, y, v,w). (1) It is required to check up performance of conditions y (t) in V for any movement y(.,) starting with points of allowable initial positions at taking every perturbation satisfying restrictions. So the problem of guaranteed safety conditions check is put. For its check it is necessary to construct reliable sets frequently. For system (1) with restrictions it is required to construct reachable set G (t) as set of all solution of a problem (1) at perturbations (2) for any fixed moment of time t. For many problems the restriction on revolting perturbations can have only geometrical character. It means, that during each moment of time t perturbation w (t) can be any of some convex compact set Q. By applying bilateral and interval estimations of ODE solutions the differential inequalities being solved, that results to qualitative changes (perturbation of structure) ODE systems. The main feature of classical stability concepts consists in that they concern to concrete system and behaviour of its trajectories in a vicinity of a point of balance (an attraction or pushing away). Completely other approach is demanded with the analysis of behaviour of trajectories family, including singular points, and limiting cycles, which relative positioning determines structure of trajectories family. This analysis arises by consideration of all systems "close" to standard system. According to the definition, entered by Pontryagin and Andronov, the system (1) refers to structurally rough system, if topological character of trajectories of all systems close to system (1). The certain mathematical difficulties are connected to specification of concepts "close system ", and also with a concrete definition of sense which is meant when speak that the trajectory is equivalent, or topological similar to other trajectory. But the basic idea remains clear, small enough changes of structurally rough system should result in accordingly small changes in dynamics of its behaviour. For research of safety of systems threshold values of parameters have great value. Threshold values are limiting sizes, the exit for which limits interferes with normal process of various elements, results in formation of negative, destructive tendencies in the field of economic safety. Approach to their maximum permissible size testifies to increase of instability threats, and excess limiting, values testifies about the introduction of system into a zone of instability and conflicts. The methods building guaranteed bounds of solution sets of ODE systems with the interval data [1] - [12], are based on symbolical representation of the formulas, approximating the operator of shift along a trajectory. There are resulted some examples of guaranteed methods application to bounds of stability areas of ODE systems describing dynamics of multimachine power systems, systems of economic growth and control systems of movement. 1. Rogalyov A.N., Shokin Ju.I. Research and an estimation of solutions of the ordinary differential equations by interval symbolical methods. Computational Technologies, 1999.- v.4, № 4.- pp. 51-76. 2. Rogalyov A.N. Research of practical stability at permanent perturbations. Computational technologies, 2002.- v.7, p.5. - pp 148-150. 3. Rogalyov A.N. Guaranteed methods of the systems of the ordinary differential equations solutions on the basis of transformation of symbolical formulas // Computational technologies.---2003. v. 8. № 5.--- pp. 102 - 116. 4. Rogalyov A.N. The behaviour of dynamic systems at extreme perturbations // Computational technologies.---2003. v. 8 - Joint release. On materials of the International conference " Computing and information technologies in a science, technics{technical equipment} and formation{education} "---Kazakhstan, Ust Kamenogorsk. - pp. 68 - 77. 5. Rogalyov A.N. guarantee of an estimation of safe functioning technical and power systems // Works of the All-Russia conference with the international participation " Modern methods of mathematical modelling of natural and anthropogenous accidents ".---Krasnoyarsk: ICM SBRAS.---2003, v.3.-pp. 42-48. 6. Rogalyov A.N. Inclusion of solution sets of the differential equations and guaranteed estimations of a global errors // Computational technologies.---2003. v. 8, № 6. - pp. 80 - 94. 7. Rogalyov A.N. Boounds of solution sets of ODE systems with the interval initial data // Computational technologies.-2004. v.9. № 1.-pp. 86 - 93. 8. Rogalyov A.N. The method of definition of the top and bottom bounds of the differential equations solutions and their application // Proceedings of the International conference on Computational mathematics МКВМ-2004./Eds. Mihajlov G.A., Iljin V.P., Laevskij Ju.E. - Novosibirsk: ICM and MG of the SBRAS-2004, p.2.- pp. 614 - 620. 9. Rogalyov A.N. Ansambli of systems of the differential equations with the interval data // Interval mathematics and methods of distribution of restrictions. Proceedings of the International conference on Computational mathematics МКВМ-2004./Eds. Mihajlov G.A., Iljin V.P., Laevskij Ju.E. - Novosibirsk: ICM and MG of the SBRAS-2004, - 2004.-pp. 240 - 254.

Note. Abstracts are published in author's edition

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