Linear differential holding game with a break

V. O. Anisov

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 14 (2022) no. 2, pp. 5-12

Résumé

A linear simple motion constraint differential game is considered. This game is considered from the part of the first player, who needs to keep the state of the system in a given convex terminal set throughout the game, despite the possible glitch and control of the second player. A glitch is understood as an instantaneous stop of the first player at a previously unknown point in time; after a certain time he will eliminate the glitch and will continue his motion. The player control vectograms are $n$-dimensional convex compacts that depend on time. To construct a $u$-stable bridge, the second method of L.S. Pontryagin is used. This is how a multi-valued mapping is constructed on the basis of the alternating integral of L.S. Pontryagin. After that, it is proved that the constructed mapping is a $u$-stable bridge for the game under consideration if a number of conditions are satisfied. At the end of the article, a simple example on the plane is considered, where the vectors of the players are circles centered at the origin and with a constant radius, while the radius of the circle of the first player is strictly greater than the second. In this example, a $u$-stable bridge is built according to the method proposed in the article, and an extremal strategy is found for the first player on the constructed $u$-stable bridge.

Détail

Keywords: differential game, alternating integral, stable bridge.
Mots-clés : constraint