Instantaneous and Noninstantaneous Impulsive Hadamard Fractional Order System and Their Equivalent Integral Equalities

ABSTRACT Here, we will explore the equivalent integral equality (EIE) for instantaneous impulsive Hadamard fractional order system (IIHFrOS) and noninstantaneous impulsive Hadamard fractional order system (NIHFrOS). For one IIHFrOS, we apply the fractional property of segment function to construct the EIE of a special case of the IIHFrOS to deduce that the proposed EIE of the IIHFrOS in cited paper is incomplete. We combine the limit analysis with the limit properties of the IIHFrOS to find the correct IIHFrOS's EIE, and obtain another IIHFrOS's EIE by the connection between the two IIHFrOSs, which both IIHFrOSs' EIEs are an integral equation of the combination of and ( ) with an arbitrary real. Next, we deduce the EIE of one NIHFrOS by the corresponding IIHFrOS and obtain another NIHFrOS's EIE by the connection between the two NIHFrOSs, which both NIHFrOSs' EIEs are an integral equation of the combination of the similar items of and ( ) with an arbitrary real. Finally, we show the calculation process of the EIE and the nonuniqueness of solution for the IIHFrOSs and NIHFrOSs by applying some numerical examples.