Controllability of surface gravity waves and the sloshing problem

TL;DR

This paper investigates the controllability of surface gravity waves and sloshing in a 2D container using conformal maps, operator theory, and observability inequalities to establish conditions for controlling free surface motion.

Contribution

It introduces a novel formulation of the control problem using conformal maps and the Dirichlet-Neumann operator, and analyzes exact controllability via observability inequalities.

Findings

Formulated the control problem as a second order evolutionary equation.

Established Sobolev space framework for solutions.

Proved observability inequality for the adjoint system.

Abstract

We study the problem of controlling the free surface for a two dimensional solid container in the context of the gravity waves and the sloshing problem. By using conformal maps and the Dirichlet-Neumann operator, the problem is formulated as a second order evolutionary equation on the free surface involving a self-adjoint operator. We present then the appropriate Sobolev spaces where having solutions for the system and study the exact controllability through an observability inequality for the adjoint problem.