Boundary controllability of a system modelling a partially immersed obstacle

¹ Institut de Mathématiques de Bordeaux, Université de Bordeaux, 351, cours de la Liberation 33405 Talence, France.
² Department of Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg Cauerstrasse 11, 91058 Erlangen, Germany.

^* Corresponding author: yue.wang@fau.de

Received: 25 February 2021
Accepted: 30 June 2021

Abstract

In this paper, we address the problem of boundary controllability for the one-dimensional nonlinear shallow water system, describing the free surface flow of water as well as the flow under a fixed gate structure. The system of differential equations considered can be interpreted as a simplified model of a particular type of wave energy device converter called oscillating water column. The physical requirements naturally lead to the problem of exact controllability in a prescribed region. In particular, we use the concept of nodal profile controllability in which at a given point (the node) time-dependent profiles for the states are required to be reachable by boundary controls. By rewriting the system into a hyperbolic system with nonlocal boundary conditions, we at first establish the semi-global classical solutions of the system, then get the local controllability and nodal profile using a constructive method. In addition, based on this constructive process, we provide an algorithmic concept to calculate the required boundary control function for generating a solution for solving these control problem.