A topological obstruction to the controllability of nonlinear wave equations with bilinear control term

TL;DR

This paper demonstrates that a known controllability obstruction for linear systems also applies to certain nonlinear wave equations with bilinear controls, using an abstract framework and specific applications.

Contribution

It extends the Ball-Marsden-Slemrod controllability obstruction to nonlinear wave equations with integrable bilinear controls, including Sine-Gordon and cubic wave/Klein-Gordon equations.

Findings

Obstruction applies to nonlinear wave equations with bilinear controls

Abstract result established for general nonlinear systems

Specific applications to Sine-Gordon and Klein-Gordon equations

Abstract

In this paper we prove that the Ball-Marsden-Slemrod controllability obstruction also holds for nonlinear equations, with integrable bilinear controls. We first show an abstract result and then we apply it to nonlinear wave equations. The first application to the Sine-Gordon equation directly follows from the abstract result, and the second application concerns the cubic wave/Klein-Gordon equation and needs some additional work.