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.
Highlights
The boundary stabilization of quasilinear systems has been the subject of numerous investigations, see for example [13], [5]
For the initial boundary value problems that describe the evolution of the state on the network, we obtain semi-global classical solutions on a given time interval [0, T ], provided that the initial data and the boundary data is sufficiently small in the C1-sense and the initial data are compatible with the boundary conditions and the node conditions
We show that for a networked quasi-linear system that is defined on a graph with a cycle, the C1−norm of the state decays exponentially fast if there is sufficient nodal control action located in the cycle
Summary
The boundary stabilization of quasilinear systems has been the subject of numerous investigations, see for example [13], [5]. In this paper we consider problems of flow control with linear Riemann dampers that have the property that the maximum norm of the Riemann invariants is decreased as the characteristic curves cross the damping location. Note that while stabilization problems for quasilinear hyperbolic problems have been studied in depth (see for example [6], [18], [19]) in these contributions networked systems with cycles have not been considered. In this paper we follow a different approach that is based upon the tracing of the characteristic curves along which the values of the Riemann invariants evolve during the process This type analysis is related to the tracing of optical rays in the analysis of the boundary controllability properties of the wave equation, see [4].
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