Local exponential stabilization of nonlinear infinite-dimensional systems

Local exponential stabilization of nonlinear distributed parameter systems (DPS) by linear state feedbacks is adressed. The results rely on an adapted concept of Fréchet differentiability which is in general easier to deal with. As main contribution, it is first shown how to link the Fréchet differen-tiability of the nonlinear semigroup generated by the operator dynamics with the Fréchet differentiability of the closed-loop semigroup obtained by injecting a linear state feedback into the dynamics. As a second result, an appropriately stabilizing state feedback for the linearized system around any equilibrium is proved to be locally stabilizing for the nonlinear system, under some boundedness assumption on the control operator. A class of controlled systems satisfying the required assumptions is then identified. The theoretical results are illustrated for the state regulation of a diffusion equation perturbed by a nonlinear term.