Juliette Leblond

Shortest paths of bounded curvature in the plane

Given two oriented points in the plane, we determine and compute the shortest paths of bounded curvature joining them. This problem has been solved recently by Dubins in the no-cusp case, and by Reeds and Shepp otherwise. We propose a new solution based on the minimum principle of Pontryagin. Our approach simplifies the proofs and makes clear the global or local nature of the results.

Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems

We establish some global stability results together with logarithmic estimates in Sobolev norms for the inverse problem of recovering a Robin coefficient on part of the boundary of a smooth 2D domain from overdetermined measurements on the complementary part of a solution to the Laplace equation in the domain, using tools from analytic function theory.

Recovery of pointwise sources or small inclusions in 2D domains and rational approximation

We consider the inverse problems of locating pointwise or small size conductivity defaults in a plane domain, from overdetermined boundary measurements of solutions to the Laplace equation. We express these issues in terms of best rational or meromorphic approximation problems on the boundary, with poles constrained to belong to the domain. This approach furnishes efficient and original resolution schemes.

How can the meromorphic approximation help to solve some 2D inverse problems for the Laplacian

We exhibit new links between approximation theory in the complex domain and a family of inverse problems for the 2D Laplacian related to non-destructive testing.

Line segment crack recovery from incomplete boundary data

We are concerned with non-destructive control issues, namely detection and recovery of cracks in a planar (2D) isotropic conductor from partial boundary measurements of a solution to the Laplace–Neumann problem. We first build an extension of that solution to the whole boundary, using constructive approximation techniques in classes of analytic and meromorphic functions, and then use localization algorithms based on boundary computations of the reciprocity gap.

Analytic extensions and Cauchy-type inverse problems on annular domains: stability results

We consider the Cauchy issue of recovering boundary values on the inner circle of a two-dimensional annulus from available overdetermined data on the outer circle, for solutions to the Laplace equation. Using tools from complex analysis and Hardy classes, we establish stability properties and error estimates.

Robust identification from band-limited data

Consider the problem of identifying a scalar bounded-input/bounded-output stable transfer function from pointwise measurements at frequencies within a bandwidth. We propose an algorithm which consists of building a sequence of maps from data to models converging uniformly to the transfer function on the bandwidth when the number of measurements goes to infinity, the noise level to zero, and asymptotically meeting some gauge constraint outside. Error bounds are derived, and the procedure is illustrated by numerical experiments.

Hardy approximation toL ∞ functions on subsets of the circle

We consider approximation ofL∞ functions byH∞ functions on proper substs of the circle. We derive some properties of traces of Hardy classes on such subsets, and then turn to a generalization of classical extremal problems involving norm constraints on the complementary subset.

Hardy spaces of the conjugate Beltrami equation

We study Hardy spaces of solutions to the conjugate Beltrami equation with Lipschitz coefficient on Dini-smooth simply connected planar domains, in the range of exponents

Weighted H 2 approximation of transfer functions

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