Controllability under positivity constraints of multi-d wave equations

TL;DR

This paper investigates the controllability of multi-dimensional wave equations with positivity constraints on controls, establishing conditions under which states can be driven between steady states or trajectories using nonnegative controls.

Contribution

It introduces new controllability results for wave equations with positivity constraints, applicable to a broad class of control systems, including steady states and trajectories.

Findings

Proves steady state controllability with nonnegative controls for wave equations.

Establishes controllability between states on different trajectories under energy conservation.

Develops an abstract framework applicable to various control systems.

Abstract

We consider both the internal and boundary controllability problems for wave equations under non-negativity constraints on the controls. First, we prove the steady state controllability property with nonnegative controls for a general class of wave equations with time-independent coefficients. According to it, the system can be driven from a steady state generated by a strictly positive control to another, by means of nonnegative controls, when the time of control is long enough. Secondly, under the added assumption of conservation and coercivity of the energy, controllability is proved between states lying on two distinct trajectories. Our methods are described and developed in an abstract setting, to be applicable to a wide variety of control systems.