Stability of solutions in impulsive integro-differential equations with applications to fading memory systems

In this paper, we analyze the stability of a class of nonlinear integro-differential equations in Banach spaces, incorporating both integral terms with fading memory effects and impulsive actions occurring at fixed moments in time. These impulses lead to piecewise continuous solutions, thus broadening the range of phenomena that can be modeled. Here we present results on uniform stability on the whole space and on the uniform either asymptotic or exponential stability of solutions over bounded sets of initial data. Further, also the global exponential stability of the null solution is analyzed. The presence of a Volterra integral term allows to consider physical systems which are influenced by their past states. To illustrate the applicability of our theoretical findings, we discuss models arising in population dynamics and robotics, where the memory effect plays some role in shaping the system’s evolution.