On local exponential stability of equilibrium profiles of nonlinear distributed parameter systems

Abstract

Local exponential (exp.) stability of nonlinear distributed parameter, i.e. infinite-dimensional state space, systems is considered. A weakened concept of Fréchet differentiability (( Y,X )-Fréchet differentiability) for nonlinear operators defined on Banach spaces is proposed, including the introduction of an alternative space (Y) in the analysis. This allows more freedom in the manipulation of norm-inequalities leading to adapted Fréchet differentiability conditions that are easier to check. Then, provided that the nonlinear semigroup generated by the nonlinear dynamics is Fréchet-differentiable in the new sense, appropriate local exp. stability of the equilibria for the nonlinear system is established. In particular, the nonlinear semigroup has to be Fréchet differentiable on Y and (Y,X)-Fréchet differentiable in order to go back to the original state space X. This approach may be called ”perturbation-based” since exp. stability is also deduced from exp. stability of a linearized version of the nonlinear semigroup. Under adapted Fréchet differentiability assumptions, the main result establishes that local exp. stability of an equilibrium for the nonlinear system is guaranteed as long as the exp. stability holds for the linearized semigroup. The same conclusion holds regarding instability. The theoretical results are illustrated on a convection-diffusion-reaction system.