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Register a new userShallow water waves generated by a floating object: a control theoretical perspective
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Authors
Pei Su
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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).
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It has been proved by Zuazua in the nineties that the internally controlled semilinear 1D wave equation $partial_{tt}y-partial_{xx}y + g(y)=f 1_{omega}$, with Dirichlet boundary conditions, is exactly controllable in $H^1_0(0,1)cap L^2(0,1)$ with controls $fin L^2((0,1)times(0,T))$, for any $T>0$ and any nonempty open subset $omega$ of $(0,1)$, assuming that $gin mathcal{C}^1(R)$ does not grow faster than $betavert xvert ln^{2}vert xvert$ at infinity for some $beta>0$ small enough. The proof, based on the Leray-Schauder fixed point theorem, is however not constructive. In this article, we design a constructive proof and algorithm for the exact controllability of semilinear 1D wave equations. Assuming that $g^prime$ does not grow faster than $beta ln^{2}vert xvert$ at infinity for some $beta>0$ small enough and that $g^prime$ is uniformly Holder continuous on $R$ with exponent $sin[0,1]$, we design a least-squares algorithm yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order $1+s$ after a finite number of iterations.
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This article treats two problems dealing with control of linear systems in the presence of a jammer that can sporadically turn off the control signal. The first problem treats the standard reachability problem, and the second treats the standard linear quadratic regulator problem under the above class of jamming signals. We provide necessary and sufficient conditions for optimality based on a nonsmooth Pontryagin maximum principle.
Numerical simulations of flows are required for numerous applications, and are usually carried out using shallow water equations. We describe the FullSWOF software which is based on up-to-date finite volume methods and well-balanced schemes to solve this kind of equations. It consists of a set of open source C++ codes, freely available to the community, easy to use, and open for further development. Several features make FullSWOF particularly suitable for applications in hydrology: small water heights and wet-dry transitions are robustly handled, rainfall and infiltration are incorporated, and data from grid-based digital topographies can be used directly. A detailed mathematical description is given here, and the capabilities of FullSWOF are illustrated based on analytic solutions and datasets of real cases. The codes, available in 1D and
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