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Conflict-controlled processes involving fractional differential equations with impulses. (English) Zbl 1272.34062

Summary: 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 set-valued 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 \({\mathcal L}\times{\mathcal B}\)-measurable set-valued mappings and their selections to ensure, as a result, superpositional measurability of the first player’s controls.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
34A08 Fractional ordinary differential equations
34A37 Ordinary differential equations with impulses
49N70 Differential games and control
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