Virginie Régnier

Boundary controllability of a chain of serially connected Euler-Bernoulli beams with interior masses

The aim is to study the boundary controllability of a system modelling the vibrations of a network ofN Euler-Bernoulli beams serially connected by (N − 1) vibrating interior point masses. Using the classical Hilbert Uniqueness Method, the control problem is reduced to the obtention of an observability inequality. The solution is then expressed in terms of Fourier series so that one of the sufficient conditions for the observability inequality is that the distance between two consecutive large eigenvalues of the spatial operator involved in this evolution problem is superior to a minimal fixed value. This property called spectral gap holds. It is proved using the exterior matrix method due to W.H. Paulsen. Two more asymptotic estimates involving the eigenfunctions are required for the observability inequality to hold. They are established using an adequate basis.

Expansions in Generalized Eigenfunctions of the Weighted Laplacian on Star-shaped Networks

We are interested in evolution phenomena on star-shaped networks composed of n semi-infinite branches which are connected at their origins. Using spectral theory we construct the equivalent of the Fourier transform, which diagonalizes the weighted Laplacian on the n-star. It is designed for the construction of explicit solution formulas to various evolution equations such as the heat, wave or the Klein-Gordon equation with different leading coefficients on the branches.

Exponential stability of a network of serially connected Euler–Bernoulli beams

The aim is to prove the exponential stability of a system modelling the vibrations of a network of N Euler–Bernoulli beams serially connected. Using a result given by K. Ammari and M. Tucsnak, the problem is reduced to the estimate of a transfer function and the obtention of an observability inequality. The solution is then expressed in terms of Fourier series so that one of the sufficient conditions for both the estimate of the transfer function and the observability inequality is that the distance between two consecutive large eigenvalues of the spatial operator involved in this evolution problem is superior to a minimal fixed value. This property called spectral gap holds. It is proved using the exterior matrix method due to W. H. Paulsen. Two more asymptotic estimates involving the eigenfunctions are required. They are established using an adequate basis.

THE INFLUENCE OF THE TUNNEL EFFECT ON L ∞ -TIME DECAY

We consider the Klein-Gordon equation on a star-shaped network composed of n half-axes connected at their origins.W e add a potential that is constant but different on each branch.E xploiting a spectral theoretic solution formula from a previous paper, we study the L∞ -time decay via Hormander’s version of the stationary phase method.W e analyze the coefficient c of the leading term \(c\cdot t^{ - {1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace}2}}\) of the asymptotic expansion of the solution with respect to time.F or two branches we prove that for an initial condition in an energy band above the threshold of tunnel effect, this coefficient tends to zero on the branch with the higher potential, as the potential difference tends to infinity. At the same time the incline to the t-axis and the aperture of the cone of \( t^{ - {1 \mathord{\left/{\vphantom {1 2}} \right.\kern-\nulldelimiterspace} 2}}\) -decay in the (t, x)-plane tend to zero.

Energy Flow Above the Threshold of Tunnel Effect

We consider the Klein-Gordon equation on two half-axes connected at their origins. We add a potential that is constant but different on each branch. In a previous paper, we studied the L ∞-time decay via Hormander’s version of the stationary phase method. Here we apply these results to show that for initial conditions in an energy band above the threshold of the tunnel effect a fixed portion of the energy propagates between group lines. Further we consider the situation that the potential difference tends to infinity while the energy band of the initial condition is shifted upwards such that the particle stays above the threshold of the tunnel effect. We show that the total transmitted energy as well as the portion between the group lines tend to zero like \( {{a_{2}}^{-1/2}} \)in the branch with the higher potential a2 as a2 tends to infinity. At the same time the cone formed by the group lines inclines to the t-axis while its aperture tends to zero.