On the equivalent integral equality of impulsive Hadamard fractional order system

Abstract

For the impulsive Hadamard fractional order system (IHFrOS), we apply the fractional property of segment function to construct the equivalent integral equality (EIE) of a special case of the IHFrOS to deduce that the proposed EIE of the IHFrOS in cited paper is incomplete. And next, we combine the limit analysis with the linear additivity of impulse effects to find the correct IHFrOS’s EIE that is an integral equation of the combination of ψ( t) and Ψ j ( t ) ( j=1 ,2 ,…,N) with an arbitrary constant, which reveal the nonuniqueness of the IHFrOS’s solution. Furthermore, we also give another IHFrOS’s EIE by the connection between two IHFrOSs. Finally, we show the calculation process of the EIE and the nonuniqueness of solution for two IHFrOSs by applying two numerical examples.