Stability of limit regimes in general reaction-diffusion type systems

Abstract

In this paper, we consider the stability of limit regimes for a general class of nonlinear distributed mathematical models named Reaction-Diffusion models. RD systems naturally arise in many applications. For instance, in the biological mathematical modeling and in the signal transmission theory the FitzHugh–Nagumo model, whose distributed variant is a particular case of general RD system, is widely used. We investigate the problem of stability of attracting sets for an infinite-dimensional RD system with respect to bounded external signals (disturbances). The interaction functions as well as nonlinear perturbations do not assume to be Lipschitz continuous. Therefore, we cannot expect the uniqueness of solution for the corresponding initial-value problem and we have to use a multi-valued semigroup approach. An undisturbed system is considered to have a global attractor, i.e., the minimal compact uniformly attracting set. The main purpose is to estimate the deviation of the trajectory of the disturbed system from the global attractor of the undisturbed one as a function of the magnitude of external signals. Such an estimate can be obtained in the framework of the theory of input-to-state stability (ISS). The paper proposes a new approach to obtaining estimates of robust stability of the attractor in the case of a multivalued evolutionary operator. In particular, it is proved that the multivalued semigroup generated by weak solutions of a nonlinear reaction-diffusion system has the property of local ISS with respect to the attractor of the undisturbed system.

Keywords

• reaction-diffusion system
• system without uniqueness
• input-to-state stability
• robust stability
• global attractor