Shallow water waves generated by a floating object: a control theoretical perspective

TL;DR

This paper models and analyzes the control of shallow water waves generated by a floating object, demonstrating conditions for controllability and stabilizability of the system through linearized equations.

Contribution

It derives and studies the linearized shallow water equations with a floating body, establishing controllability and stabilizability results under symmetry assumptions.

Findings

Symmetric waves can be generated from equilibrium via control force.

The system is approximately controllable and strongly stabilizable when projected onto symmetric states.

In general, the system is not controllable without symmetry assumptions.

Abstract

We consider a control system describing the interaction of water waves with a partially immersed rigid body constraint to move only in the vertical direction. The fluid is modeled by the shallow water equations. The control signal is a vertical force acting on the floating body. We first derive the full governing equations of the fluid-body system in a water tank and reformulate them as an initial boundary value problem of a first-order evolution system. We then linearize the equations around the equilibrium state and we study its well-posedness. Finally we focus on the reachability and stabilizability of the linear system. Our main result asserts that, provided that the floating body is situated in the middle of the tank, any symmetric waves with appropriate regularity can be obtained from the equilibrium state by an appropriate control force. This implies, in particular, that we can project this system on the subspace of states with appropriate symmetry properties to obtain a reduced system which is approximately controllable and strongly stabilizable. Note that, in general, this system is not controllable (even approximately).