# Internal control of systems of semilinear coupled 1-D wave equations

## TL;DR

This paper establishes the internal controllability of certain 1-D coupled wave systems using the fictitious control method, addressing both non-degenerate and degenerate cubic couplings with novel trajectory constructions.

## Contribution

It extends controllability results to coupled wave equations with degenerate cubic coupling by adapting trajectory construction techniques.

## Key findings

- Proves controllability for systems with non-degenerate coupling.
- Establishes controllability for systems with degenerate cubic coupling.
- Develops new trajectory construction methods for degenerate cases.

## Abstract

We prove the internal controllability of some systems of two coupled wave equations in one space dimension, with one control, under certain conditions on the coupling. To do this we apply the "fictitious control method" in two cases: general systems with a "non-degenerate" coupling, and a particular case where the coupling is "degenerate", namely a cubic coupling. In the latter case, our proof requires to find nontrivial trajectories of the control system that go from $0$ to $0$. We build these trajectories by adapting (in $1$ space dimension) a construction developed by Jean-Michel Coron, Sergio Guerrero and Lionel Rosier for the study of coupled parabolic systems.