A Lyapunov-Razumikhin approach for stability analysis of logistics networks with time-delays

This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, redistribution , reselling , loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. Logistics network represents a complex system where different elements that are logistic locations interact with each other. This interaction contains delays caused by time needed for delivery of the material. Complexity of the system, time-delays and perturbations in a customer demand may cause unstable behaviour of the network. This leads to the loss of the customers and high inventory costs. Thus the investigation of the network on stability is desired during its design. In this article we consider local input-to-state stability of such logistics networks. Their behaviour is described by a functional differential equation with a constant time-delay. We are looking for verifiable conditions that guarantee stability of the network under consideration. Lyapunov–Razumikhin functions and the local small gain condition are utilised to obtain such conditions. Our stability conditions for the logistics network are based on the information about the interconnection properties between logistic locations and their production rates. Finally, numerical results are provided to demonstrate the proposed approach. 1. Introduction Real-world logistics networks are geographically distributed and can be considered as large-scale interconnected systems with a nonlinear behaviour. Moreover, they are subject to internal and external perturbations like changes in customer behaviour or transport congestions that can destabilise the network. This leads to high inventory costs, large throughput times and thus to the decrease in performance and finally to the loss of customers. Therefore, such basic properties as stability and robustness against disturbances must be assured for a reasonable behaviour of the logistics network. Mathematical models provide an opportunity to investigate the properties of logistics systems like performance, stability, robustness and to design appropriate controls to improve these properties modelling of logistics networks is of high interest for many logistic companies, requiring new techniques …