Null controllability of some degenerate wave equations

TL;DR

This paper investigates the null controllability of one-dimensional linear degenerate wave equations using boundary control, establishing conditions under which controllability is possible and providing explicit controllability times.

Contribution

It demonstrates null controllability when control acts on the non-degenerate boundary, contrasting with previous results, and offers explicit controllability time expressions.

Findings

Controllability achieved for all initial states with boundary control on non-degenerate boundary.

Explicit formula for controllability time is provided.

Counterexamples show limitations for certain degenerate wave equations.

Abstract

This paper is devoted to a study of the null controllability problems for one-dimensional linear degenerate wave equations through a boundary controller. First, the well-posedness of linear degenerate wave equations is discussed. Then the null controllability of some degenerate wave equations is established, when a control acts on the non-degenerate boundary. Different from the known controllability results in the case that a control acts on the degenerate boundary, any initial value in state space is controllable in this case. Also, an explicit expression for the controllability time is given. Furthermore, a counterexample on the controllability is given for some other degenerate wave equations.