Nodal Stabilization of the Flow in a Network with a Cycle

Abstract

In this paper we discuss an approach to the stability analysis for classical solutions of closed loop systems that is based upon the tracing of the evolution of the Riemann invariants along the characteristics. We consider a network where several edges are coupled through node conditions that govern the evolution of the Riemann invariants through the nodes of the network. The analysis of the decay of the Riemann invariants requires to follow backwards all the characteristics that enter such a node and contribute to the evolution. This means that with each nodal reflection/crossing the number of characteristics that contribute to the evolution increases. We show how for simple networks with a sufficient number of damping nodal controlers it is possible to keep track of this family of characteristics and use this approach to analyze the exponential stability of the system. The analysis is based on an adapted version of Gronwall’s lemma that allows us to take into account the possible increase of the Riemann invariants when the characteristic curves cross a node of the network. Our example is motivated by applications in the control of gas pipeline flow, where the graphs of the networks often contain many cycles.

Keywords

• nodal stabilization
• classical solutions
• networks
• gas pipeline
• characteristics
• cycle
• nodal control