论文标题
沉浸在2D完美不可压缩的液体中的刚体的远程轨迹跟踪
Remote trajectory tracking of rigid bodies immersed in a 2D perfect incompressible fluid
论文作者
论文摘要
我们考虑浸入二维不可压缩的完美流体中的几个刚体的运动。刚体的运动是由牛顿定律以液压而产生的，流体运动由不可压缩的Euler方程描述。我们的分析涵盖了身体周围流体速度的循环是非零的，并且流体涡度的界限。整个系统占据了一个边界的简单连接域与外部固定边界，除非在开放的非空部分上，否则可以通过控制正常的速度和进入涡流，从而允许某些流体进入和流出域。我们证明，可以通过受控正常速度在外部边界上的远程动作来准确地实现刚体的任何非coll缩平滑运动，该速度采用了状态反馈的形式，而零进入涡旋。 This extends the result of (Glass, O., Kolumb{á}n, J. J., Sueur, F. (2017). External boundary control of the motion of a rigid body immersed in a perfect two-dimensional fluid. Analysis \& PDE) where the exact controllability of a single rigid body immersed in a 2D irrotational perfect incompressible fluid from an initial position and velocity to a final position and velocity was调查。证明依赖于一种非线性方法来求解与具有常规非平凡零的二次操作员相关的非线性方程的线性扰动。在这里，此方法应用于一类边界控制所满足的二次方程，该方程是通过扩展在不受控制的情况下进行的牛顿方程的重新印度（Glass，O.，o.，lacave，C.，Munnier，A.，Sueur，A. 3561-3577）对控制在外部边界上作用。
We consider the motion of several rigid bodies immersed in a two-dimensional incompressible perfect fluid. The motion of the rigid bodies is given by the Newton laws with forces due to the fluid pressure and the fluid motion is described by the incompressible Euler equations. Our analysis covers the case where the circulations of the fluid velocity around the bodies are nonzero and where the fluid vorticity is bounded. The whole system occupies a bounded simply connected domain with an external fixed boundary which is impermeable except on an open non-empty part where one allows some fluid to go in and out the domain by controlling the normal velocity and the entering vorticity. We prove that it is possible to exactly achieve any non-colliding smooth motion of the rigid bodies by the remote action of a controlled normal velocity on the outer boundary which takes the form of state-feedback, with zero entering vorticity. This extends the result of (Glass, O., Kolumb{á}n, J. J., Sueur, F. (2017). External boundary control of the motion of a rigid body immersed in a perfect two-dimensional fluid. Analysis \& PDE) where the exact controllability of a single rigid body immersed in a 2D irrotational perfect incompressible fluid from an initial position and velocity to a final position and velocity was investigated. The proof relies on a nonlinear method to solve linear perturbations of nonlinear equations associated with a quadratic operator having a regular non-trivial zero. Here this method is applied to a quadratic equation satisfied by a class of boundary controls, which is obtained by extending the reformulation of the Newton equations performed in the uncontrolled case in (Glass, O., Lacave, C., Munnier, A., Sueur, F. (2019). Dynamics of rigid bodies in a two dimensional incompressible perfect fluid. Journal of Differential Equations, 267(6), 3561-3577) to the case where a control acts on the external boundary.
