Local Exponential Stabilization of a Coupled Burgers’ PDE-ODE System † Agus Hasan and Shu-Xia Tang Abstract— This paper concerns the boundary stabilization problem of a coupled system consisting of the Burgers’ equation and a linear ordinary differential equation (ODE). The Burgers’ equation is a widely considered nonlinear partial differential equation (PDE), partially due to its low order and partially due to its structure analogous to the Navier-Stokes equation which describes fluid dynamics. The controller we employ for stabi- lizing this nonlinear coupled system was firstly developed from the boundary control problem of the corresponding linearized system, based on an infinite-dimensional backstepping trans- formation. By construction of a strict Lyapunov functional, the closed-loop nonlinear system with the backstepping controller is proved to be locally exponentially stable. Index Terms— coupled PDE-ODE systems, nonlinear sys- tems, backstepping, boundary control. I. I NTRODUCTION Systems modelled by a coupled PDE and ODE can be found in many engineering problems, e.g., in flexible cable of an overhead crane [1], battery management systems ([2], [3], [4], [5]), and automated drilling systems ([6], [7]). The objective of this paper is to stabilize a coupled system of a (viscous) Burgers’ PDE and a linear ODE, through one boundary controller. Burgers’ equation is usually considered as a simplified form of the one-dimensional Navier-Stokes equation, and it takes the following form [8] u t (x, t) = ǫu xx (x, t) - u(x, t)u x (x, t) (1) The equation contains a nonlinear term and it is the simplest equation for which the solutions can develop shock waves. Using the Hopf-Cole transformation [9], one can change the equation into a linear parabolic equation. Therefore, the solution do not exhibit chaotic features like sensitivity with respect to initial condition. Stabilization of PDE systems with boundary control was considered as a challenging topic until around two decades ago, when the backstepping technique which is a systematic method for nonlinear ODE control problems was introduced into PDE control design and estimation problems, see [10] for a tutorial overview of this approach. Since then this method has been used to design stabilizing control laws for many PDEs. † This work was supported by the Free the Drones (FreeD) project and the National Natural Science Foundation of China (61773112). A. Hasan is with the Center for Unmanned Aerial Vehicles, The Maersk McKinney Møller Institute, University of Southern Denmark, 5230 Odense, Denmark. Email: agha@mmmi.sdu.dk. S.-X. Tang is with the Department of Applied Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada. Email: s74tang@uwaterloo.ca. Corresponding author: S.-X. Tang. Most of the early efforts focus on the backstepping control of linear PDE and linear coupled PDE-ODE systems. For example, the control problem of a linear coupled hyperbolic PDE-ODE system is studied in [11]. While considering the ODE state as a disturbance source of the hyperbolic PDE, it was shown that the designed backstepping control law, acting on the PDE boundary, is able to attenuate the disturbance. Other works include control with Neumann interconnections [12], coupled ODE-Schrodinger equation [13], and adaptive control for a class of PDE-ODE cascade systems with uncertain harmonic disturbances [14]. It is worth mentioning that the backstepping control technique has also been applied to a coupled system of a linearized Burgers’ PDE and a linear ODE, see, [15]. In engineering problems, the backstepping method has found several applications. For examples, the backstepping controller is used in [16] and [17] to find an optimal oil rate under gas coning conditions. Furthermore, the backstepping controller has been used for slugging con- trol [18] and lost circulation and kick control in oil well drilling ([19], [20]). Although most physical systems are nonlinear, only a few results are available for stabilization of nonlinear PDE systems, such as the Korteweg-de Vries equation ([21], [22]), Benjamin-Bona-Mahony equation [23] and Ginzburg-Landau equation [24]. Even less results exist for stabilizing the coupled systems involving PDEs. For example, feedback control design of nonlinear coupled system of two hetero- directional hyperbolic PDEs, called the 2×2 quasilinear hyperbolic system, is presented in [25], which uses the backstepping technique and achieves local stabilization. To the best knowledge of the authors, the only existing result for stabilizing coupled systems of a linear ODE and a nonlinear PDE is [26], which studies the observer design for a coupled system consisting of a linear ODE and a nonlinear PDE. In this paper, our control design for the nonlinear coupled Burger’s PDE-ODE system follows a similar idea as [25]. In particular, the backstepping feedback controller, which exponentially stabilizes the linearized Burger’s PDE-ODE system in the sense of the H 1 norm, is proved to locally stabilize the nonlinear Burgers’ PDE-ODE system in the sense of the H 2 norm with an exponential decay rate. The rest of this paper begins with a problem formulation in Section II, which is followed by a presentation of some preliminary results from the boundary control problem of the corresponding linearized coupled PDE-ODE system in Section III. The main result is presented in Section IV, with a proof presented in Section V. Finally, conclusions and some possible future works are presented in section VI. 2017 IEEE 56th Annual Conference on Decision and Control (CDC) December 12-15, 2017, Melbourne, Australia 978-1-5090-2873-3/17/$31.00 ©2017 IEEE 2479