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

Abstract

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.

Keywords:

• input-to-state stability
• interconnected systems
• time-delay systems
• Lyapunov methods
• small-gain theorems

Acknowledgements

S. Dashkovskiy is partially supported by the DFG as a part of Collaborative Research Center 637 ‘Autonomous Cooperating Logistic Processes – A Paradigm Shift and its Limitations’. M. Kosmykov is supported by the Volkswagen Foundation (Project Nr. I/82684 ‘Dynamic Large-Scale Logistics Networks’).