In this poster, we prove exponential decay for a system of two Schrodinger equations in a wave guide, with coupling and damping at the boundary. This relies on the spectral analysis of the corresponding coupled Schrodinger operator on the one-dimensional cross section. We show in particular that we have a spectral gap and that the corresponding generalized eigenfunctions form a Riesz basis.

References

[1] R. Ayechi, I. Boukhris, and J. Royer, A system of Schrodinger equations in a wave guide, Journal of Mathematical Physics, 64(11), 2023.

[2] J. Royer, Exponential decay for the Schrodinger equation on a dissipative wave guide, Ann. Henri Poincar´e, 16(8):1807–1836, 2015.

[3] K. J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Springer, 2000.

Key words : Schrodinger equations, Stabilization, Wave guide.

Field : Stabilization Problem

Sedimentary rocks contain traces of the Earth's ancient magnetic field, forming a valuable archive for reconstructing the geodynamic history of the Earth: continental drift, magnetic reversals, or internal movements of the core. To read this fossil memory, geophysicists rely on measurement devices, such as superconducting quantum interference device (SQUID) or quantum diamond microscopes (QDM), which locally measure components of the magnetic field at a fixed distance above the sample. However, the measured field strength is particularly weak, and extremely sensitive to noise, which leads to an ill-posed inverse problem for reconstructing the internal magnetization of the sample.

The magnetic field measurements recorded by these microscopes are modeled as the projection, along a direction $\vec{l} \in \mathbb{R}^3$, of the field generated by an unknown magnetization, supported in a planar domain $S \in \mathbb{R}^2$ containing the rock sample. This magnetization is a vector-valued function $m$ assumed to belong $[L^2(S)]^3$ , while the measurements are performed on a set $Q \in \mathbb{R}^2$ located at a height $h > 0$ above the sample (see Figure 1).

The link between the magnetization $\vec{m}$ and the measured field is based on an integral operator, derived from Maxwell's equations in the static regime, and relies on the Poisson−Laplace equation $\Delta \phi = \nabla \cdot \vec{m}$, satisfied by the scalar magnetic potential $\phi$. This potential is then expressed using the Poisson kernel and the Riesz transforms. Within this framework, we define the measurement operator $b_{\vec{l}}$.

The objective of this work is to develop a mathematically rigorous and numerically stable approach to estimate the magnetic moment $<\vec{m}> = \displaystyle\int \int_S \vec{m}(t)\, dt \in \mathbb{R}^3$ of a sample from directional magnetic field measurements $b_{\vec{l}}$ on $Q$.

References

[1] L. Baratchart, D. P. Hardin, E. A. Lima, E. B. Saff, and B. P. Weiss, Characterizing Kernels of Operators Related to Thin-Plate Magnetizations via Generalizations of Hodge Decompositions, Inverse Problems, vol. 29, no. 1, 2013.

[2] CL. Baratchart, S. Chevillard, D. P. Hardin, J. Leblond, and E. Andrade Lima, Magnetic Moment Estimation and Bounded Extremal Problems, Inverse Problems and Imaging, vol. 13, no. 1, 2019.

Key words : Inverse source problems, elliptic partial differential equations, harmonic functions, geosciences and planetary sciences, paleomagnetism.

Field : Inverse Problems

The Kohn-Vogelius method is an effective approach for solving inverse parameter identification problems. We present an abstract framework that generalizes the work of [1, 2, 3] to non-symmetric, second-order, linear PDE operators. This framework allows us to solve inverse problems involving the identification of singular parameters from boundary measurements. The method transforms the inverse problem into an optimization problem by constructing an energy-like cost functional $J$ defined over an admissible set $Q_{(ad)}$ , such that the inverse problem’s solution q is the unique minimizer of $J$. The difficulty for non symmetric PDE is that the functional $J$ cannot coincide with the bilinear form associated with the forward problem. We study the derivatives of $J$ with respect to various perturbations of $q$, either in terms of $L^\infty$ perturbations or in terms of perturbations of $q$’s discontinuity points, to enable the use of gradient descent algorithms. Finally, we present several applications related to reconstructing the discontinuity points of $q$, especially in impedance or generalized-impedance models, accompanied by two-dimensional numerical illustrations.

References

[1] S. Chaabane, I. Feki, N. Mars, Numerical reconstruction of a piecewise constant Robin parameter in the two-or three-dimensional case, Inverse Problems, 28 (6), (2012).

[2] S. Chaabane, B. Charfi, H. Haddar. Reconstruction of discontinuous parameters in a second order impedance boundary operator, Inverse Problems, 32 (10), (2016).

[3] B. Charfi, Identification du support de singularit´e d’une imp´edance g´en´eralis´ee, PHD Thesis, Facult´e des Sciences de Sfax, (2019).

Key words : Identification of discontinuity sets, Kohn-Vogelius method.

Field : Inverse Problems

In this work, we examine the asymptotic behavior of a Timoshenko beam system incorporating fractional delays and nonlinear external sources. Under suitable conditions on the damping, delay, and initial data, we prove that the solution exhibits an exponential decay rate.

References

[1] A. Adnane, A. Benaissa, K. Benomar, “Uniform stabilization for a Timoshenko beam system with delays in fractional order internal damping”, SeMA Journal, pp. 1–20, 2022.

[2] R. Aounallah, A. Benaissa, A. Zara¨ı, “Blow-up and asymptotic behavior for a wave equation with a time delay condition of fractional type”, Rendiconti del Circolo Matematico di Palermo, Series 2, 70(2):1061–1081, 2021.

[3] J. Ferreira, E. Piskin, C. Raposo, M. Shahrouzi, H. Y¨uksekkaya, “Stability Result for a Kirchhoff Beam Equation with Variable Exponent and Time Delay”, Universal Journal of Mathematics and Applications, 5(1):1–9, 2022

Key words : Timoshenko system; energy decay; nonlinear systems; fractional delay

Field : TBA

We consider a nonlinear beam equation of Kirchhoff type with dissipative boundary conditions and source term. The dissipative terms is considered by memory term with weakly singular kernel and frictional term.We establish a sufficient condition of the initial data with arbitrarily high initial energy such that the solution blow up in finite time.

References

[1] F.Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, In Annales de l’Institut Henri Poincare (C) Non Linear Analysis, volume 23, pages 185-207, 2006.

[2] H. A. Levine. Instability and nonexistence of global solutions to nonlinear wave equations of the form, Transactions of the American mathematical society, 192:1-21,1974.

[3] H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form, Transactions of the American mathematical society, 192:1- 21,1974.

Key words : Blow up, Beam of Kirchhoff type, Dissipative term

Field : Control

In this paper, we present an optimal control approach subjected to a nonlocal monodomain model with discrete and stochastic aspects. We consider the control as an electrical shock (defibrillation) applied to a region of the heart boundary to eliminate reentrant waves (arrhythmias) and ensure tissue recovery. We establish the well-posedness of the martingale solutions of the stochastic nonlocal model. Furthermore, we prove the existence of the optimal control and its first order conditions for both cases. Lastly, we present the optimization problem’s numerical simulations.

Key words : TBA

Field : Inverse Problems

We consider a classical underdamped harmonic oscillator. By controlling its frequency over time subject to pointwise constraints, we seek to amplify the amplitude of motion of the oscillator as quickly as possible.

Our analysis reveals the structure of optimal controls and in particular that periodicity is a necessary property for a control of the frequency to be timeoptimal [4]. Together with Pontryagin’s maximum principle and Floquet theory, this observation enables us to bridge optimal control theory and stability analysis of dynamical systems [3, 2]. Most notably we find that the frequency of the optimal control is twice as large as the natural frequency of the oscillator. This is a wellknown condition for parametric resonance, a type of instability arising in response to periodic variations of a system’s parameter [1].

References

[1] L.D. Landau and E.M. Lifshits, 1960. Mechanics. CUP Archive.

[2] C. Chicone, 2006. Ordinary differential equations with applications.

[3] V. Hardel, P-A. Hervieux, K. Lutz, G. Manfredi and Y. Privat, 2025. Parametric Resonance: Bridging Optimal Control Theory and Dynamical System Stability. Preprint.

[4] K. Lutz and Y. Privat, 2025. Minimal Time Control of Underdamped Parametric Oscillators. Preprint.

Key words : Underdamped oscillator, time minimal, parametric resonance

Field : Control

In this work, we prove a stability result by the measurement of one component for a phase-field system on a bounded domain $\Omega \subset \mathbb{R}^{n}$ for the dimension $n \leq 3$. More precisely, we deal with the identification of two parameters for a phase-field system using a modified Carleman estimate by one observation. The key idea is to adapt the technique introduced in \cite{Cristofol2011} in order to derive a suitable Carleman estimate with a single observation on a subdomain $\omega \Subset \Omega$, thereby establishing a Lipschitz stability inequality for the simultaneous recovery of two coefficients.

References

[1] G. Caginalp, An analysis of a phase field model of a free boundary, Archive for Rational Mechanics and Analysis, Vol. 92, 1986, pp. 205{245.

[2] F. Ammar Khodja, A. Benabdallah, C. Dupak, I. Kostin, Controllability to the trajectories of phase field models by one control force, SIAM Journal on Control and Optimization, Vol. 42, 2003, pp. 1661{1680.

[3] N. Baranibalan, K. Sakthivel, K. Balachandran, J.-H. Kim, Inverse problems for the phase field system with one observation, Applicable Analysis, Vol. 88, No. 4, 2009, pp. 529{545.

[4] N. Baranibalan, K. Sakthivel, K. Balachandran, J.-H. Kim, Reconstruction of two time independent coecients in an inverse problem for a phase eld system, Nonlinear Analysis, Vol. 73, No. 4, 2010, pp. 1003{1012.

Key words : Inverse problem, phase-field system, Carleman estimate

Field : Inverse Problems