# Controllability of the semilinear wave equation governed by a multiplicative control

## TL;DR

This paper investigates the approximate and exact controllability of semilinear wave equations using a single multiplicative control, providing constructive methods and explicit controls, with applications to one-dimensional systems.

## Contribution

It introduces new controllability results for semilinear wave equations with multiplicative control, including explicit steering controls and uniform exact controllability in undamped cases.

## Key findings

- Approximate controllability achieved with a single multiplicative control.
- Explicit steering controls are constructed for the system.
- Exact controllability established for one-dimensional undamped wave equations.

## Abstract

In this paper we establish several results on approximate controllability of a semilinear wave equation by making use of a single multiplicative control. These results are then applied to discuss the exact controllability properties for the one dimensional version of the system at hand. The proof relies on linear semigroup theory and the results on the additive controllability of the semilinear wave equation. The approaches are constructive and provide explicit steering controls. Moreover, in the context of undamped wave equation, the exact controllability is established for a time which is uniform for all initial states.