This paper studies the exact controllability of a finite dimensional system obtained by discretizing in space and time the linear 1-D wave system with a boundary control at one extreme. It is known that usual schemes obtained with finite difference or finite element methods are not uniformly controllable with respect to the discretization parameters and . We introduce an implicit finite difference scheme which differs from the usual centered one by additional terms of order and . Using a discrete...
We consider a linear damped wave equation defined on a two-dimensional domain Ω, with a dissipative term localized in a subset ω. We address the shape design problem which consists in optimizing the shape of ω in order to minimize the energy of the system at a given time T . By introducing an adjoint problem, we first obtain explicitly the (shape) derivative of the energy at time T with respect to the variation in ω. Expressed as a boundary integral on ∂ω, this derivative is then used as an advection...
This paper studies the
exact controllability of a finite dimensional system obtained by
discretizing in space and time the linear 1-D wave system with a
boundary control at one extreme. It is known that usual schemes
obtained with finite difference or finite element methods are not
uniformly controllable with respect to the discretization
parameters and Δ. We introduce an implicit finite
difference scheme which differs from the usual centered one by
additional terms of order
...
In the framework of the linear fracture theory, a commonly-used tool
to describe the smooth evolution of a crack embedded in a bounded domain Ω is the so-called
energy release rate defined as the variation of the mechanical
energy with respect to the crack dimension. Precisely, the
well-known Griffith's criterion postulates the evolution of the
crack if this rate reaches a critical value. In this work, in the anti-plane scalar case, we
consider the shape design problem which consists in optimizing...
This work is devoted to the
analysis of a viscous finite-difference space semi-discretization
of a locally damped wave equation in a regular 2-D domain. The
damping term is supported in a suitable subset of the domain, so
that the energy of solutions of the damped continuous wave
equation decays exponentially to zero as time goes to infinity.
Using discrete multiplier techniques, we prove that adding a
suitable vanishing numerical viscosity term leads to a uniform
(with respect to the mesh size)...
This paper deals with the numerical computation of boundary null controls for the 1D wave equation with a potential. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. We do not apply in this work the usual duality arguments but explore instead a direct approach in the framework of global Carleman estimates. More precisely, we consider the control that minimizes over the class of admissible null...
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