Optimal boundary control for steady motions of a self-propelled body in a Navier-Stokes liquid

Consider a rigid body ${\mathcal S} \subset {\mathbb R}^3$ immersed in an infinitely extended Navier-Stokes liquid and the motion of the body-fluid interaction system described from a reference frame attached to ${\mathcal S}$. We are interested in steady motions of this coupled system, where the region occupied by the fluid is the exterior domain $\Omega = {\mathbb R}^3 \setminus {\mathcal S}$. This paper deals with the problem of using boundary controls $v_*$, acting on the whole $\partial\Omega$ or just on a portion $\Gamma$ of $\partial\Omega$, to generate a self-propelled motion of ${\mathcal S}$ with a target velocity $V(x):=\xi+\omega \times x$ and to minimize the drag about ${\mathcal S}$. Firstly, an appropriate drag functional is derived from the energy equation of the fluid and the problem is formulated as an optimal boundary control problem. Then the minimization problem is solved for localized controls, such that supp $v_*\subset \Gamma$, and for tangential controls, i.e, $v_*\cdot n|_{\partial \Omega}=0$, where $n$ is the outward unit normal to $\partial \Omega$. We prove the existence of optimal solutions, justify the G\^ateaux derivative of the control-to-state map, establish the well-posedness of the corresponding adjoint equations and, finally, derive the first order optimality conditions. The results are obtained under smallness restrictions on the objectives $|\xi|$ and $|\omega|$ and on the boundary controls

The motion of a fluid-rigid ball system at the zero limit of the rigid ball radius

International audienceWe study the limiting motion of a system of a rigid ball moving in a Navier-Stokes fluid flow in ${\mathbb R}^3$ as the radius of the ball goes to zero. Recently, Dashti and Robinson solved this problem in the 2D case, in the absence of rotation of the ball. This restriction was caused by the difficulty in obtaining appropriate uniform bounds on the second order derivatives of the fluid velocity when the rigid body can rotate. In this paper, we show how to obtain the required uniform bounds on the velocity fields in the 3D case. These estimates then allow to pass to the zero limit of the ball radius and show that the solution of the coupled system converges to the solution of the Navier-Stokes equations describing the motion of only fluid in the whole space. The trajectory of the centre of the ball converges to a fluid particle trajectory, which justifies the use of rigid tracers for finding Lagrangian paths of fluid flow

A boundary control problem for the steady self-propelled motion of a rigid body in a Navier-Stokes fluid

International audienceWe study the self-propelled motions of a rigid body immersed in a viscous incompressible fluid which fills the exterior domain of the rigid body.The mechanism used by the body to reach the desired motion is modeled through a distribution of velocities at its boundary. The fluid motion is modeled by the stationary Navier-Stokes system. These equations are coupled with two relations for the balance of forces and torques.We prove that there exists a control allowing the rigid body to move with a prescribed rigid velocity provided the velocity is small enough. We also show that since the net force exerted by the fluid to the rigid body vanishes, we have a better summability of the fluid velocity than the classical summability result for the solutions of the stationary Navier-Stokes system in exterior domains

Solving inverse source problems using observability. Applications to the Euler-Bernoulli plate equation

International audienceThe aim of this paper is to provide a general framework allowing to use exact observability of infinite dimensional systems to solve a class of inverse source problems. More precisely, we show that if a system is exactly observable, then we can identify a source term in this system by knowing the corresponding intensity and appropriate observations which often correspond to the measure of some boundary traces. This abstract theory is then applied to a system governed by the Euler-Bernoulli plate equation. Using a different methodology, we show that exact observability can be used to identify both the locations and the intensities of combinations of point sources in the plate equation

Search for dark matter produced in association with bottom or top quarks in √s = 13 TeV pp collisions with the ATLAS detector

A search for weakly interacting massive particle dark matter produced in association with bottom or top quarks is presented. Final states containing third-generation quarks and miss- ing transverse momentum are considered. The analysis uses 36.1 fb−1 of proton–proton collision data recorded by the ATLAS experiment at √s = 13 TeV in 2015 and 2016. No significant excess of events above the estimated backgrounds is observed. The results are in- terpreted in the framework of simplified models of spin-0 dark-matter mediators. For colour- neutral spin-0 mediators produced in association with top quarks and decaying into a pair of dark-matter particles, mediator masses below 50 GeV are excluded assuming a dark-matter candidate mass of 1 GeV and unitary couplings. For scalar and pseudoscalar mediators produced in association with bottom quarks, the search sets limits on the production cross- section of 300 times the predicted rate for mediators with masses between 10 and 50 GeV and assuming a dark-matter mass of 1 GeV and unitary coupling. Constraints on colour- charged scalar simplified models are also presented. Assuming a dark-matter particle mass of 35 GeV, mediator particles with mass below 1.1 TeV are excluded for couplings yielding a dark-matter relic density consistent with measurements