A GENERALIZATION OF FIRST DIRECT METHOD OF PURSUIT FOR DIFFERENTIAL INCLUSIONS

Volume 67, Issue 5, 2022, pages 32-41

DOI: http://doi.org/10.34229/2786-6505-2022-5-3

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Ikromjon Iskanadjiev, Tashkent Chemical-Technological Institute, Uzbekistan, kaltatay@gmail.com


ABSRACT

To solve the problem of pursuit in linear differential games, L.S. Pontryaginsuggested two direct methods. Direct methods are of great importance in the development of the theory of differential games and in control theory under theconditions of uncertainty. It turned out to be useful also in solving the problemof control synthesis. Pontryagin direct methods have proved themselves as an effective means for solving problems of pursuit- evasion and control. These useintegrals, having a number of significant differences from the classical integral.One of the differences consists in the use of multivalued mapping. Pontryaginʼssecond direct method, based on concept of the alternating integral, which has noanalogs in integration of real function. In definition of alternating integral participate of integration of setvalued mappings and geometric difference (Minkovskidifference) of sets. These operations make difficulties for computation of alternating integral. From this point of view, the integral used by the first directmethod has a simpler construction. Therefore, the question naturally arises ofgeneralization the first direct method of pursuit. In this paper it will be studied ageneralization of the first direct method for pursuit games, being described bydifferential inclusionsz F t v  ( , ),where F is a continuous multivalued mapping. This method will be called the modified first direct method of pursuit fordifferential inclusions. In particular, the class of stroboscopic strategies, the trajectory of the system are determined. For these classes games, it is proved that ifthe starting point belongs to the modified first integral (the integral from themultivalued mapping, which is present in the definition of the modified fist direct metod), then this is necessary and sufficient condition for completing thegame in a fixed time instant in the class of stroboscobic strategies. The problemof computation this integral is important. In the present article it has also beenproved that the union operations in the definition of the modified first integralcan be narrowed down to the class of compact-valued mappings

Keywords: differential inclusion, differential games, crossection, stroboscopicstrategy, admissible control, evader, pursuer, pursuit partition, nearly stroboscopic strategy.


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