Conflict-Controlled Processes Involving Fractional Differential Equations with Impulses

MSC 2010: 34A08, 34A37, 49N70

Here we investigate a problem of approaching terminal (target) set by a system of impulse differential equations of fractional order in the sense of Caputo. The system is under control of two players pursuing opposite goals. The first player tries to bring the trajectory of the system to the terminal set in the shortest time, whereas the second player tries to maximally put off the instant when the trajectory hits the set, or even avoid this meeting at all. We derive analytical solution to the initial value problem for a fractional-order system involving impulse effects. As the main tool for investigation serves the Method of Resolving Functions based on the technique of inverse Minkowski functionals. By constructing and investigating special setvalued mappings and their selections, we obtain sufficient conditions for the game termination in a finite time. In so doing, we substantially apply the technique of L £ B-measurable setvalued mappings and their selections to ensure, as a result, superpositional measurability of the first player's controls.