On stability of nonlinear hyperbolic systems with reaction and switching Hao Yang, Bin Jiang, Vincent Cocquempot, and Abdel Aitouche Abstract— This paper investigates the exponential stability in L 2 norm of scalar nonlinear hyperbolic systems of balance laws with the reaction that may be accumulative or dissipative. Two Lyapunov-based stability criteria that depend on the system parameters and boundary data are proposed with fully considering the reactions’ characteristics. The new results can help to construct a common Lyaunov function to stabilize the switched nonlinear hyperbolic systems under arbitrary switching. Several traffic system examples are taken to illustrate the theoretical results. Index Terms— Nonlinear hyperbolic partial differential equa- tion; Switched systems; Stability; Boundary control. I. I NTRODUCTION Hyperbolic partial differential equations (PDE) of balance laws can model the fundamental dynamics of many physical systems that are represented by the flow in networks and are controlled at the boundary nodes. Examples include transportation systems [1], production systems [2], open channel systems [3], etc. Various stability results of such hy- perbolic systems have been proposed by means of boundary conditions. These boundary conditions only rely on system parameters and the boundary data. The stability results can be traced back to two main methods: One method relies on the direct estimates of the solutions and their derivatives along the characteristic curves [4], [5], [6]; The other one utilizes the Lyapunov method [3], [7], [8]. The reaction term in the hyperbolic PDE introduces a source or a sink effect that may increase or decrease the state value along the characteristic curves. Most of existing results assume that the reactions are small enough or dissipative, and design the boundary control independently from the effect of reactions with only a few exceptions. Ref. [9] considers a special nonlinear reaction with Lotka-Volterra structure. In [10], backstepping controllers that use both boundary data and interior data are designed for first-order hyperbolic PDEs with reactions in integral form. Ref [11] shows that in some cases the basic quadratic Lyapunov function may not exist for the hyperbolic system with reaction terms. Indeed, the existing boundary conditions can be further relaxed in the This work is supported by National Natural Science Foundation of China (61104116,61273171), Doctoral Fund of Ministry of Education of China (20113218110011), A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), and NUAA Research Funding (NS2011016). H. Yang and B. Jiang are with College of Automation Engineering (CAE), Nanjing University of Aeronautics and Astronautics, China. Email: haoyang@nuaa.edu.cn (H. Yang), binjiang@nuaa.edu.cn (B. Jiang). V. Cocquempot and A. Aitouche are with LAGIS, UMR CNRS 8219, Universit´ e Lille 1 : Sciences et Technologies, France. Email: vincent.cocquempot@univ-lille1.fr (V. Cocquempot), abdel.aitouche@hei.fr (A. Aitouche). presence of the significantly dissipative reactions, and are not available if the reactions are severely accumulative. On the other hand, many engineering applications can be modeled by switched systems due to the existence of various jumping parameters [12]. Fruitful results have been reported on stability and stabilization of switched systems in which modes are governed by ordinary differential equations (ODE), e.g., [13]-[15]. Recently, switched system with each mode driven by hyperbolic PDE has attracted some attentions since it can effectively model the hyperbolic PDE with the switching of dynamic parameters, e.g., the transportation system [16], the bioremediation model with biomass that can either be active or dormant [17], the switching delay control for hyperbolic PDE [18], etc. However, until now few works are devoted to the stability of switched hyperbolic PDEs of balance laws. In [19], a common quadratic Lyapunov function is provided when the semigroup generators commute. Ref. [20] uses the estimates of the solutions along the characteristic curves and proves that the switched linear hyperbolic system is exponentially stable under arbitrary switching if the boundary gains satisfy some commutativity conditions. In [21], Lyapunov-based method is utilized to establish boundary conditions subject to switching signals. All above works assume that the reactions of each mode are small enough or none, and all modes are stable. In this work, we use a Lyapunov-based approach to ana- lyze the stability of switched nonlinear hyperbolic systems of balance laws with reactions that may be drastically accu- mulative or dissipative. The main contributions are twofold: 1. For the non-switched scalar nonlinear hyperbolic sys- tem, we fully consider the reaction characteristics and propose two novel boundary conditions respectively for two cases where the reaction is accumulative or dissi- pative. A key technique is to construct a comparative system which can help to link boundary behaviors and interior behaviors. The obtained new results signifi- cantly extend existing boundary conditions to the cases of various reactions. 2. For switched nonlinear hyperbolic system with reaction, we design individual boundary conditions for each mode to establish a common Lyapunov function, which guarantees that the switched system is exponentially sta- ble under arbitrary switching. This idea is quite different from that in [20]. The proposed methods guarantee the existence of solution of class C 0 over the switching process rather than the solution in broad sense as in [20], [21]. In the rest of the paper, the non-switched hyperbolic