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Abstract

This paper presents an inverse problem for the nonlinear 1-d Kuramoto-Sivashinsky (K-S) equation. More precisely, we study the nonlinear inverse problem of retrieving the anti-diffusion coefficient from the measurements of the solution on a part of the boundary and at some positive time everywhere. Uniqueness and Lipschitz stability for this inverse problem are proven with the Bukhgeim-Klibanov method. The proof is based on a global Carleman estimate for the linearized K-S equation.

Abstract

We consider the problem of active reduction of vibrations in a fluid-flexible structure system. In the aerospace domain, we are actually interested in the system that couples the deflection of a plane wing and the sloshing of the fuel inside the wing's tank. The control is performed using piezoelectric patches and the main difficulty comes from the complex coupling between the flexible modes of the wing and the sloshing modes of the fuel. We establish an infinite-dimensional model for the global system and then a finite-dimensional approximation calculated using the first modes and validated on the experimental setup. A feedback controller is used to show the effectiveness of the closed loop in attenuating vibrations. Index Terms--PDE experiment, fluid-flexible structure, pole

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Abstract

We apply robust control technics to an adaptive optics system including a dynamic model of the deformable mirror. The dynamic model of the mirror is a modification of the usual plate equation. We propose also a state-space approach to model the turbulent phase. A continuous time control of our model is suggested taking into account the frequential behavior of the turbulent phase. An H1 controller is designed in an infinite dimensional setting. Due to the multivariable nature of the control problem involved in adaptive optics systems, a significant improvement is obtained with respect to traditional single input single output methods.

Abstract

Often considered in numerical simulations related to the control of quantum systems, the so-called monotonic schemes have not been so far much studied from the functional analysis point of view. Yet, these procedures provide an efficient constructive method for solving a certain class of optimal control problems. This paper aims both at extending the results already available about these algorithms in the finite-dimensional case (i.e., the time-discretized case) and at completing those of the continuous case. This paper starts with some results about the regularity of a functional related to a wide class of models in quantum chemistry. These enable us to extend an inequality due to Aojasiewicz to the infinitedimensional case. Finally, some inequalities proving the Cauchy character of the monotonic sequence are obtained, followed by an estimation of the rate of convergence.

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Abstract

This paper concerns the inverse problem of retrieving a stationary potential for the Schrödinger evolution equation in a bounded domain of RN with Dirichlet data and discontinuous principal coefficient a(x) from a single time-dependent Neumann boundary measurement. We consider that the discontinuity of a is located on a simple closed hyper-surface called the interface, and a is constant in each one of the interior and exterior domains with respect to this interface. We prove uniqueness and lipschitz stability for this inverse problem under certain convexity hypothesis on the geometry of the interior domain and on the sign of the jump of a at the interface. The proof is based on a global Carleman inequality for the Schrödinger equation with discontinuous coefficients, result also interesting by itself.

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Abstract

This paper concerns the inverse problem of retrieving a stationary potential for the wave equation with Dirichlet data and discontinuous principal coefficient a(x) from a single time-dependent Neumann boundary measurement on the boundary. We consider that the discontinuity of a is located on a simple closed curve À1, called the interface, and a is constant in each one of the interior and exterior domains with respect to this interface. We prove uniqueness and lipschitz stability for this inverse problem under certain convexity hypothesis on the geometry of the interior domain and on the sign of the jump of a at the interface. The proof is based on a global Carleman inequality for the wave equation with discontinuous coefficients, result also interesting by itself.

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