Integral Representation and Non-Uniqueness of Solutions for Impulsive Right-Sided Riemann–Liouville Fractional-Order Systems

This paper investigates the equivalent integral equations (EIEs) of two impulsive right-sided Riemann–Liouville fractional-order systems (IRRFOSs). The limit properties of one IRRFOS are employed to establish the linear additivity of impulsive effects. A computational approach based on fractional calculus for piecewise functions is then employed to construct the EIE corresponding to a single impulse. With the aid of this linear additivity, the EIE of the considered IRRFOS is obtained, and through the relationship between the two IRRFOSs, the EIE of the other IRRFOS is further derived. The results indicate that the solutions of both EIEs consist of linear combinations of $\varphi \left(t\right)$ and ${\Phi }_j\left(t\right)$$(j=1,2,\dots ,N)$ containing an arbitrary constant, which implies the non-uniqueness of solutions to the two IRRFOSs. Finally, the computational procedure for deriving the EIEs of the two IRRFOSs is presented, and the non-uniqueness of solutions is illustrated through two numerical examples.