Abstracts

Abstract: TBA

Abstract: TBA

Abstract: After presenting a few astrochemical motivations, I will introduce the recent quantum chemical methods, that we have recently proposed to solve the molecular, electronic Schrödinger equation. These methods are based on the classical concept of the Lewis structure of a molecule, and make use of geminals, which are wave functions for electron pairs.

Abstract: We consider a stochastic dynamics describing the evolution of a qubit controlled by an external field and subject to continuous-time measurements. Motivated by stabilization techniques recently developed in, e.g., [10], [11], we investigate the support of the corresponding solution, which is a random

variable taking values on the space of two-by-two density matrices. By making use of the Strook-Varadhan support theorem and by classical geometric control arguments, we compute the support for two possible choices of the measurement and Hamiltonian operators. This is a joint work with Paolo Mason.

Abstract: We provide sufficient conditions for the approximate controllability of infinite-dimensional quantum control systems corresponding to form perturbations of the drift Hamiltonian modulated by a control function. To this extent, we rely on previous results on controllability of quantum bilinear control systems and obtain a priori L1-bounds of the controls for generic initial and target states; we then apply a stability theorem for the non-autonomous Schrödinger equation to extend such results to systems defined by form perturbations, including singular ones. As an application, we prove approximate controllability of a quantum particle in a one-dimensional box with a point (Dirac) interaction at the center of the box with tuneable strength. Based on arXiv:2402.02955. Joint work with A. Balmaseda and J. M. Pérez-Pardo.

Abstract: Dans cet exposé, nous parlerons de la stabilisation rapide locale de l'équation de Schrödinger bilinéaire. Nous verrons les limitations de l'approche par backstepping de Fredholm dans ce cadre, les récents avancés d'Hoai-Minh Nguyen sur la question grâce au Grammien de stabilisation et les liens possibles entre le backstepping de Fredholm et le Grammien de stabilisation. 

Abstract: The goal of this presentation is to give an introduction to the relations between the exact controllability of nonlinear PDEs and the control theory for ODEs based on Lie brackets, through a study of the Schrödinger PDE with bilinear control. We focus on the small-time local controllability (STLC) around an equilibrium, when the linearized system is not controllable. We study the second order order term in the Taylor expansion of the state, with respect to the control. For ODEs, this result is a consequence of Sussman’s sufficient condition S(θ) (when focused on quadratic terms), but we propose a new proof, designed to prepare an easier transfer to PDEs. This proof relies on a new representation formula of the state (inspired by the Magnus formula). By adapting it, we prove a new STLC result for the multi-input bilinear Schrödinger PDE. This presentation is based in a joint work with Karine Beauchard and Frédéric Marbach.

Abstract: In this talk, I will discuss how a general bilinear finite-dimensional closed quantum system with dispersed parameters can be steered between eigenstates. We show that, under suitable conditions on the separation of spectral gaps and the boundedness of parameter dispersion, rotating wave and adiabatic approximations can be employed in cascade to achieve population inversion between arbitrary eigenstates. 

Abstract: We present some considerations on when linearly independent eigenfunctions of the Schrödinger operator may have the same modulus. In particular, in the one-dimensional case, we show that this may happen only when the ambient manifold is a circle. The study is motivated by its application to the bilinear control of the Schrödinger equation, where it has been shown in the works of  Chambrion, Duca, Nersesyan, and Pozzoli that, under suitable saturation assumptions on the potentials of interaction, the system can be steered arbitrarily fast between wave functions having the same modulus. 

Abstract: Despite the importance of control systems governed by bilinear controls for the description of phenomena that cannot be realistically modeled by additive controls, the action of multiplicative controls is generally not so widely studied as it happens for boundary and locally distributed controls. The main reasons of this fact might be found in the intrinsic nonlinear nature of such problems and furthermore, for controls that are scalar functions of time, in an ineluctable obstruction for proving results of exact controllability, which is contained in the celebrated work of Ball, Marsden and Slemrod, 1982. In this talk, I will present results on the stabilization and controllability of parabolic-type evolution equations using a scalar input bilinear control. Specifically, I will focus on steering the system toward particular target trajectories known as eigensolutions. Finally, I will introduce a recent extension of this work that enables us to control the Fokker-Planck equation by the drift term.

References:

• F. Alabau-Boussouira, P. Cannarsa, C. Urbani "Superexponential stabilizability of evolution equations of parabolic type via bilinear control'', Journal of Evolution Equations, vol. 21, pages 941-967 Springer (2021)

• P. Cannarsa, C. Urbani "Superexponential stabilizability of degenerate parabolic equations via bilinear control'', Inverse Problems and Related Topics, vol. 310, pages 31-45, Springer Singapore (2020)

• F. Alabau-Boussouira, P. Cannarsa, C. Urbani "Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control", Nonlinear Differ. Equ. Appl., vol. 29, pages 38 (2022)

• P. Cannarsa, A. Duca, C. Urbani "Exact controllability to eigensolutions of the bilinear heat equation on compact networks", Discrete and Continuous Dynamical Systems - Series S, vol. 15, No. 6, pages 1377-1401 (2022)

• F. Alabau-Boussouira, P. Cannarsa, C. Urbani "Bilinear control of evolution equations with unbounded lower order terms. Application to the Fokker-Planck equation'', Comptes Rendus Mathématiques, vol. 362, pp. 511-545 (2024)