Automatica 51 (2015) 192–199 Contents lists available at ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Brief paper Boundary feedback stabilization of the Schlögl system ✩ Martin Gugat a , Fredi Tröltzsch b a Friedrich-Alexander-Universität Erlangen-Nürnberg, Department Mathematik, Cauerstr. 11, 91058 Erlangen, Germany b Technische Universität Berlin, Institut für Mathematik, Sekretariat MA 4-5, Str. des 17. Juni 136, 10623 Berlin, Germany article info Article history: Received 3 December 2012 Received in revised form 15 January 2014 Accepted 29 September 2014 Keywords: Lyapunov function Boundary feedback Robin feedback Parabolic partial differential equation Exponential stability Stabilization of periodic orbits Periodic operation Stabilization of desired orbits Poincaré–Friedrichs inequality abstract The Schlögl system is governed by a nonlinear reaction–diffusion partial differential equation with a cubic nonlinearity that determines three constant equilibrium states. It is a classical example of a chemical reaction system that is bistable. The constant equilibrium that is enclosed by the other two constant equilibrium points is unstable. In this paper, Robin boundary feedback laws are presented that stabilize the system in a given stationary state or more generally in a given time-dependent desired system orbit. The exponential stability of the closed loop system with respect to the L 2 -norm is proved. In particular, it is shown that with the boundary feedback law the unstable constant equilibrium point can be stabilized. © 2014 Elsevier Ltd. All rights reserved. 1. Introduction The Schlögl system has been introduced in Schlögl (1972)as a model for chemical reactions for non-equilibrium phase transi- tions. It describes the concentration of a substance in 1-d. In neu- rology, the same nonlinear reaction–diffusion system is known under the name Nagumo equation and models an active pulse transmission through an axon (Chen & Guo, 1992; Nagumo, 1962). It is also known as Newell–Whitehead–Segel equation (see Newell & Whitehead, 1969 and Segel, 1969). This system is governed by a parabolic partial differential equation with a cubic nonlinearity that determines three constant equilibrium states u 1 < u 2 < u 3 , where u 2 is unstable. In view of its simplicity, the Schlögl system may serve as a test case for the stabilization of an unstable equi- librium for reaction–diffusion equations that generate traveling waves. While this task might appear a little bit academic for the Schögl model, it is of paramount importance for more complicated ✩ This work was supported by DFG in the framework of the Collaborative Research Center SFB 910, project B6. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Nicolas Petit under the direction of Editor Miroslav Krstic. E-mail addresses: martin.gugat@fau.de (M. Gugat), troeltz@math.tu-berlin.de (F. Tröltzsch). equations such as the bidomain system in heart medicine, cf. Ku- nisch and Wagner (2013). Here, the goal of stabilization is to ex- tinguish undesired spiral waves as fast as possible and hereafter to control the system to a desired state. However, there are similar- ities between these models and it is therefore reasonable to con- sider the same problem for the Schögl system. The control functions can act in the domain (distributed control) or on its boundary. In this paper, the problem of boundary feedback stabilization is studied. Example 1 illustrates that, without the influence of the boundary conditions, the system state approaches exponentially fast a stable equilibrium, even if the initial state is arbitrarily close to the unstable equilibrium. Also, the more general case of boundary stabilization of time- dependent states of the system is considered in this paper. This includes the stabilization of periodic states that is interesting as a tool to stabilize the periodic operation of reactors, see Silverston and Hudgins (2013). This case also includes the stabilization of traveling waves. In this paper, linear Robin-feedback laws are presented that yield exponential stability with respect to the L 2 -norm for desired orbits of the system. The term desired orbit is used to describe a possibly time-dependent solution of the partial differential equation that defines the system. The exponential stabilization is particularly interesting since the boundary feedback allows to stabilize the system in the unstable equilibrium that is enclosed by the other two constant equilibrium points. http://dx.doi.org/10.1016/j.automatica.2014.10.106 0005-1098/© 2014 Elsevier Ltd. All rights reserved.