Raja Dziri

Dynamical shape control in non-cylindrical hydrodynamics

We are interested in a dynamical shape control problem in hydrodynamics. The considered cost functional is the kinetic energy of a perfect and incompressible fluid. We establish the existence of a non-autonomous vector field which constructs the tube followed by the fluid in such a way as to minimize its kinetic energy. By introducing a specific adjoint state closely related to the shape of the tube, we derive the necessary optimality condition.

Drag reduction for non-cylindrical Navier-Stokes flows

We address the minimal drag problem for a viscous and incompressible fluid contained in a moving domain Ω t whose boundary has a speed V(t) which turns to be the control parameter. The flow velocity u(t) verifies u(t)=V(t) on the boundary as the fluid sticks to the boundary. Then, the control acts on the non-cylindrical Navier–Stokes solution through two aspects. Directly on the Dirichlet non-homogeneous boundary condition and also through its flow mapping T t (V) which builds the moving domain starting from the initial one. The drag functional J is given in terms of the dissipated viscous energy and depends on the control V. We give a sense to the gradient with respect to V by making use of the transverse field technique which enables us to obtain an explicit expression for the gradient. We give a feedback loop and an associated existence result in order to generate the best admissible boundary displacement. In this problem, we have no fluid–structure coupling nor free boundary. The moving boundary is an active (strongly) nonlinear control.

SHAPE CONTROL OF NOISE IN A STRUCTURAL ACOUSTIC SYSTEM

Abstract The considered problem is of varying the thickness of a beam along its length in order to produce an optimal design which reduces the structural noise generated into an air filled cavity by the beam vibration. The design parameter is the thickness of that beam. We consider two models which differ only in taking or not into account the back-pressure effects of the air on the vibration. We provide necessary optimality conditions which guide the optimal beam design.