Linearized Controllability Analysis of Semilinear Partial Differential Equations

TL;DR

This paper demonstrates that for semilinear infinite-dimensional systems, the controllability of the linearized system guarantees the controllability of the nonlinear system, extending classical finite-dimensional results.

Contribution

It establishes a controllability equivalence between linearized and nonlinear semilinear infinite-dimensional systems under certain conditions.

Findings

Linearized controllability implies nonlinear controllability for semilinear systems.

Results extend finite-dimensional controllability theory to infinite-dimensional cases.

Conditions on nonlinear operators are similar to those in stability analysis literature.

Abstract

It is well-known that the controllability of finite-dimensional nonlinear systems can be established by showing the controllability of the linearized system. However, this classical result does not generalize to infinite-dimensional nonlinear systems. In this paper, we restrict ourselves to semilinear infinite-dimensional systems, and show that the exact controllability of the linearized system implies exact controllability of the nonlinear system. The restrictions concerning the nonlinear operator are similar to those that can be found in the literature about the linearized stability analysis of semilinear systems.