Reconstructing initial data using iterative observers for wave type systems. A numerical analysis

A new iterative algorithm for solving initial data inverse problems from partial observations has been recently Proposed in Ramdani et al. (Automatica 46(10), 1616–1625, 2010). Based on the concept of observers (also called Luenberger observers), this algorithm covers a large class of abstract evolution PDE’s. In this paper, we are concerned with the convergence analysis of this algorithm. More precisely, we provide a complete numerical analysis for semi-discrete (in space) and fully discrete approximations derived using finite elements in space and an implicit Euler method in time. The analysis is carried out for abstract Schrödinger and wave conservative systems with bounded observation (locally distributed)

Reconstructing initial data using observers : error analysis of the semi-discrete and fully discrete approximations

A new iterative algorithm for solving initial data inverse problems from partial observations has been recently proposed in Ramdani, Tucsnak and Weiss [15]. Based on the concept of observers (also called Luenberger observers), this algorithm covers a large class of abstract evolution PDE's. In this paper, we are concerned with the convergence analysis of this algorithm. More precisely, we provide a complete numerical analysis for semi-discrete (in space) and fully discrete approximations derived using finite elements in space and finite differences in time. The analysis is carried out for abstract Schr\"odinger and wave conservative systems with bounded observation (locally distributed).Comment: 38 pages, 1 figure

Solving Thermoacoustic Tomography with an observer-based algorithm

In thermoacoustic tomography, the problem is to recover from surface measurements the initial state of a wave equation. We propose the use of the iterative observer-based algorithm of Ramdani et al. (Automatica 2010) to solve this inverse problem

Closed-loop perturbations of well-posed linear systems

We are concerned with the perturbation of a rather general class of linear time-invariant systems, namely well-posed linear system (WPLS), under additive linear perturbations seen as feedback laws. Let Σ be a WPLS with (A, B, C) as generating triple. For all control operator E, and all observation operator F such that (A, E, F ) is the generating triple of a WPLS, we prove that, if (A, B, F ) and (A, E, C) are also the generating triples of some WPLS, for all admissible feedback operator K for (A, E, F ), we can construct a WPLS Σ K whose generating triple is (A K , B K , C K ), where A K is the infinitisemal generator of the closed-loop of(A, E, F ) by the feedback operator K. Furthermore, we give necessary and sufficient condition such that exact controllability persists from Σ to Σ K . In particular, we show that this is the case for all sufficiently small bounded operator K

An observer-based approach for thermoacoustic tomography

We propose to use the observer-based algorithm of Ramdani, Tucsnak and Weiss (Automatica, 2010) for the initial state recovery of the wave equation involved in thermoacoustic tomography. We proved the rate of convergence of the iterative algorithm to the observable part of the initial state. We performed 3D numerical test in the relevant case where the measurement is performed on a grid of transducers on a half-sphere

Reconstructing initial data using iterative observers for wave type systems

An iterative algorithm for solving initial data inverse problems from partial observations has been proposed in 2010 by Ramdani, Tucsnak and Weiss (Automatica, 2010). In this work, we are concerned with the convergence of this algorithm when the inverse problem is ill-posed, i.e. when the observations are not sufficient to reconstruct any initial data. We prove that the state space can be decomposed as a direct sum, stable by the algorithm, corresponding to the observable and unobservable part of the initial data. We show that this result holds for both locally distributed and boundary observation

Recovering the initial state of dynamical systems using observers

The aim of this work is to show that the observer based algorithm proposed in (Ramdani et al., Automatica 2010) for solving initial data inverse problem allows to reconstruct the observable part of the initial data when observability assumption fails. This poster is an overview of the results obtained in (Haine, MCSS, In Revision)

Asymptotic stability of the multidimensional wave equation coupled with classes of positive-real impedance boundary conditions

This paper proves the asymptotic stability of the multidimensional wave equation posed on a bounded open Lipschitz set, coupled with various classes of positive-real impedance boundary conditions, chosen for their physical relevance: time-delayed, standard diffusive (which includes the Riemann-Liouville fractional integral) and extended diffusive (which includes the Caputo fractional derivative). The method of proof consists in formulating an abstract Cauchy problem on an extended state space using a dissipative realization of the impedance operator, be it finite or infinite-dimensional. The asymptotic stability of the corresponding strongly continuous semigroup is then obtained by verifying the sufficient spectral conditions derived by Arendt and Batty (Trans. Amer. Math. Soc., 306 (1988)) as well as Lyubich and V\~u (Studia Math., 88 (1988))

Observateurs itératifs en horizon fini. Application à la reconstruction de données initiales pour des EDP d'évolution

A new iterative algorithm for solving initial data inverse problems from partial observations has been recently proposed in Ramdani, Tucsnak and Weiss (Ramdani et al., 2010). The algorithm, which can be used for finite-dimensional and infinite-dimensional linear systems, is based on the use of two observers (also called Luenberger observers) used iteratively back and forth in time. In this paper, we first present the algorithm in a finite-dimensional context and then generalize it to Partial Differential Equations (PDE). To conclude, the method is applied to the 1D wave equation and numerical results will be given

Systèmes linéaires invariants en temps de dimension infinie : un aperçu

Un système linéaire invariant en temps (continu) $\Sigma$ décrit l'évolution d'un système dynamique linéaire, et dont la dynamique est statique. À une donnée initiale $z_0$ et une entrée $u(\cdot)_{|[0,t]}$, elle fait correspondre un état courant $z(t)$ et une sortie $y(\cdot)_{|[0,t]}$. C'est une modélisation bien connue en automatique, c'est-à-dire quand les espaces vectoriels sous-jacents (où ``vivent'' $u(t)$, $z_0$, $z(t)$, et $y(t)$ pour tout $t\ge 0$) sont de dimension finie. En particulier, un résultat classique nous dit qu'il existe quatre matrices $A$, $B$, $C$, et $D$, de dimensions appropriées, telles que l'on ait la représentation d'état \left\lbrace\begin{array}{l}\dot z(t) = A z(t) + B u(t), \Forall t\ge 0,\\y(t) = C z(t) + D u(t), \Forall t\ge 0,\end{array}\right.où $\dot z$ désigne la dérivée temporelle de $z$. Équations que l'on complète par la donnée initiale $z(0) = z_0$.\\Dans cette présentation, nous parlerons de la généralisation de ces systèmes et des concepts de contrôlabilité/stabilisabilité et d'observabilité/détectabilité qui sous-tendent l'existence d'observateurs (en horizon infini) lorsque l'on s'intéresse à des espaces de Hilbert en toute généralité. En effet, il s'agit du cadre classique de la réécriture d'un grand nombre d'équations aux dérivées partielles (linéaires) sous la forme qui nous intéresse