On local stability of equilibrium profiles of nonisothermal axial dispersion tubular reactors

Exponential (exp.) stability of equilibrium profiles for a nonisothermal axial dispersion tubular reactor is considered. This model is described by nonlinear partial differential equations (PDEs) whose state components are the temperature, the reactant and the product concentrations inside of the reactor. It is shown how to get appropriate local exponential stability of the equilibria for the nonlinear model, on the basis of stability properties of its linearized version and some relaxed Fréchet differentiability conditions of the nonlinear semigroup generated by the dynamics. In the case where the reactor can exhibit only one equilibrium profile, the latter is always locally exponentially stable for the nonlinear system. When three equilibria are highlighted, local bistability is established, i.e. the pattern (locally)”(exp.) stable - unstable - (exp.) stable” holds. The results are illustrated by some numerical simulations. As perspectives, the concept of state feedback is also used in order to show a manner to stabilize exponentially a nonlinear system on the basis of its capacity to stabilize exponentially a linearized version of the nonlinear dynamics and some Fréchet differentiability conditions of the corresponding closed-loop nonlinear semigroup.