Modelling Analysis And Control Of Some Water Waves Rigid Body Interactions PDF Download

This PhD dissertation deals with the study of some fluid-structure interaction models in shallow water regimes. Particularly, the manuscript brings contributions on interactions between a fluid with fixed and floating rigid bodies. In a first part, we start by studying the mathematical modelling and simulations of an oscillating water column. In this device, the waves come from shore, find a bottom step, and then reach a chamber to change the air volume to drive a turbine to produce electricity. In this first approach, constant atmospheric pressure is assumed on the free surface of the fluid inside the chamber. Waves governed by the one-dimensional shallow water equations in the presence of this device are essentially reformulated as two transmission problems: the first one is associated with a step in front of the structure, and the second one is related to the wave structure interaction. On the other hand, by considering the notion of Riemann invariants to obtain the discretized version of the transmission conditions, we implement the Lax-Friedrichs scheme to get numerical solutions of the model. Furthermore, by using the concept of nodal profile controllability, we address the problem of boundary controllability for the introduced model. Finally, we close the first part of the thesis by proposing a second nonlinear model of an oscillating water column device involving a transmission condition that describes a time-dependent air pressure flow inside the chamber, which is obtained by considering the free surface Bernoulli's equation and properties relative to the shallow water regime. In a second part of the manuscript, we study some models describing the vertical motion of a solid floating at the free surface of a viscous shallow fluid. The rigid structure involved is assumed to be controlled by a vertical force exerted via an actuator. We start by proving the well-posedness of the model, which is obtained by considering adequate function spaces and convenient operators between them. Furthermore, by obtaining an explicit form of the transfer function associated, we prove some results on the input-output stability of the system. Finally, we study a Cummins type integro-differential equation describing the motion of a partially immersed structure floating in a viscous fluid in an unbounded domain. Relying on the stability results of Matignon for fractional systems, the explicit solutions of the system are established, leading to an explicit knowledge of the long-time behaviour of them.