Nonlinear Analysis 74 (2011) 4653–4659 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Almost periodic solutions in the PC -space for uncertain impulsive dynamical systems Gani Tr. Stamov a , Jehad O. Alzabut b,∗ a Department of Mathematics, Technical University—Sofia, 8800 Sliven, Bulgaria b Department of Mathematics and Physical Sciences, Prince Sultan University, P.O. Box 66388, Riyadh 11586, Saudi Arabia article info Article history: Received 18 January 2011 Accepted 6 April 2011 Communicated by S. Ahmad MSC: 34C27 34K45 Keywords: Almost periodic solution Hamilton–Jacobi–Riccati inequality Lyapunov’s functions Uncertain impulsive dynamical system abstract This paper provides sufficient conditions for the existence of almost periodic solutions for an uncertain impulsive dynamical system. The investigation is carried out by utilizing the concept of uniformly positive definite matrix functions, Hamilton–Jacobi–Riccati inequality and piecewise continuous functions of the Lyapunov functions type. © 2011 Elsevier Ltd. All rights reserved. 1. Introduction The dynamics of many evolving processes are subject to abrupt changes such as shocks, harvesting and natural disasters. These phenomena involve short-term perturbations from continuous and smooth dynamics, whose duration is negligible in comparison with the duration of an entire evolution. In models involving such perturbations, it is natural to assume that these perturbations act instantaneously or in the form of impulses. As a consequence, impulsive differential equations have been developed in modeling the impulsive problems in physics, population dynamics, ecology, biological systems, biotechnology, industrial robotics, optimal control and so forth. Associated with this development, a theory of impulsive dynamical systems has been given extensive attention. Works recognized as landmark contributions include [1–3] with [1] devoted especially to periodic impulsive dynamical systems. Searching the literature, one can realize that there exists an intensive work regarding the study of periodic impulsive dynamical systems; see for instance the papers [4–10] in which the existence of periodic solutions has been the main concern of the authors. On the other hand, upon considering long-term dynamical behaviors, it has been noticed that periodic parameters often experience certain perturbations. Thus, almost periodic oscillatory behavior is considered to be more accordant with reality. Although it is recognized to be a natural generalization to periodicity, the notion of almost periodicity has been rarely considered. The reader can easily figure out that a few results exist in this direction. The theory of almost periodic solutions for impulsive differential equations goes back to the works of Samoilenko and Perestyuk [3]. The main results related to the study of the existence of almost periodic solutions for impulsive dynamical systems have been obtained in many areas of applications, for instance, Lotka–Volterra models [11,12], biological and ecological models [13–16] ∗ Corresponding author. E-mail address: jalzabut@psu.edu.sa (J.O. Alzabut). 0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.04.026