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.
Highlights
Free surface interactions with fixed or floating structures have been intensively studied by the mathematical community in the last years with respect to modelling, well-posedness, numerical simulations, etc
In his work Lannes implemented a method for the full water wave equations and for reduced asymptotic models, such as the Boussinesq and the nonlinear shallow-water equations, where the pressure exerted by the fluid on the partially immersed structure appears as a Lagrange multiplier associated with the constraint that under the floating structure, the surface of the fluid coincides with the bottom of the structure
In a similar way and motivated for mathematical modeling and simulations of a specific type of wave energy converting device, the so-called oscillating water column (OWC) device, Bocchi, He and Vergara-Hermosilla discuss in [1] about the fluid-structure interaction of the partially submerged fixed wall structure of the OWC device, considering the nonlinear shallow water equations to describe the fluid movement, and obtain explicit transmission conditions for the system and respective reduced transmission problems. Bocchi, He and Vergara-Hermosilla propose in [2] a new and general approach on the mathematical modelling of the OWC, where include the presence of the timedependent air pressure in the device and prove a local well-posedness result in a Sobolev setting
Summary
Bocchi, He and Vergara-Hermosilla propose in [2] a new and general approach on the mathematical modelling of the OWC, where include the presence of the timedependent air pressure in the device and prove a local well-posedness result in a Sobolev setting By considering their results, in this work we deep on a particular kind of boundary controllability on the transmission problems studied in [1] on an equivalent physical configuration. We deal with the exact boundary controllability of nodal profile on a system modelling a structure partially immersed in a fluid governed by the nonlinear shallow water equations [7], considering a discontinuity in the height of the fluid bottom and the transmission conditions developed in [1]
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