Controllability under positivity constraints of multi-d wave equations

We consider both the internal and boundary controllability problems for wave equations under non-negativity constraints on the controls. First, we prove the steady state controllability property with nonnegative controls for a general class of wave equations with time-independent coefficients. According to it, the system can be driven from a steady state generated by a strictly positive control to another, by means of nonnegative controls, when the time of control is long enough. Secondly, under the added assumption of conservation and coercivity of the energy, controllability is proved between states lying on two distinct trajectories. Our methods are described and developed in an abstract setting, to be applicable to a wide variety of control systems

Nonlinear Neumann boundary stabilization of the wave equation using rotated multipliers

The rotated multipliers method is performed in the case of the boundary stabilization by means of a(linear or non-linear) Neumann feedback. this method leads to new geometrical cases concerning the "active" part of the boundary where the feedback is apllied. Due to mixed boundary conditions, these cases generate singularities. Under a simple geometrical conditon concerning the orientation of boundary, we obtain a stabilization result in both cases.Comment: 17 pages, 9 figure

A LAGRANGE-MULTIPLIER APPROACH FOR THE NUMERICAL SIMULATION OF AN INEXTENSIBLE MEMBRANE OR THREAD IMMERSED IN A FLUID ∗

Abstract. The inextensibility constraint is encountered in many physical problems involving thin solids interacting with a fluid. It is generally imposed in numerical simulations by means of a penalty method. Here, we propose a novel saddle-point approach allowing to impose it through a Lagrange multiplier defined on the thin structure, the tension. The functional analysis of the problem allows to determine which boundary conditions are needed for this problem. The forces originating from the structure appear as a boundary condition for the fluid problem, defined on a moving boundary which represents the structure. The problem is discretised with mixed finite elements. The mesh of the thin solid is included in the mesh of the bulk, and is advected by its velocity in the course of time iterations. The appropriate choice of the finite element spaces for this mixed approach is discussed and it is shown that boundary conditions on the thin structure edges impact on this choice. Numerical tests are performed which demonstrate the convergence and robustness of the method. These examples include the simulation of a closed, membrane bound object (a vesicle) and of a filament or flag in a large Reynolds-number flow. Key words. Fluid-structure interactions, surface divergence, saddle-point problem, boundary Lagrange multiplier. AMS subject classifications. 65M60, 74K05, 74K15, 74F10, 76D05, 76D0

Stokes–Leibenson problem for Hele-Shaw flow: a critical set in the space of contours

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