Keywords: Hamiltonian dynamics, Port-Hamiltonian systems
Abstract: This talk will focus on the concept of contraction in the context of port-Hamiltonian control systems. Motivated and driven by ten years of collaboration with Schneider-Toshiba on control of induction motors and power electronics, we will illustrate the relevance of this framework to rigorously address two pillars of industrial control: PI feedback and anti windup. After 40 years of dissipativity theory, physics keeps providing efficient encounters between control, analysis, and differential geometry.

Keywords: Quantum control, Nonlinear control
Abstract: A quantum harmonic oscillator (spring subsystem) is stabilized towards a target Fock state by reservoir engineering. This passive and open-loop stabilization works by consecutive and identical Hamiltonian interactions with auxiliary systems, here three-level atoms (the auxiliary ladder subsystem), followed by a partial trace over these auxiliary atoms. A scalar control input governs the interaction, defining which atomic transition in the ladder subsystem is in resonance with the spring subsystem. We use it to build a time-varying interaction with individual atoms, that combines three non-commuting steps. We show that the resulting reservoir robustly stabilizes any initial spring state distributed between 0 and 4n+3 quanta of vibrations towards a pure target Fock state of vibration number n. The convergence proof relies on the construction of a strict Lyapunov function for the Kraus map induced by this reservoir setting on the spring subsystem. Simulations with realistic parameters corresponding to the quantum electrodynamics setup at Ecole Normale Superieure further illustrate the robustness of the method.

Keywords: Quantum control, Hamiltonian dynamics, Conservation laws
Abstract: Pointer states are states of an open quantum system that are able to survive the constant monitoring of the system by an environment. It has been shown that open systems that are prepared in superpositions of such pointer states quickly decohere and evolve into classical statistical mixtures of (pure) pointer states. In this paper we demonstrate, using appropriate modeling assumptions for the system environment interaction, the following result: An individual trajectory of the system state involves towards a specific pointer state (and not just a statistical mixture of the same) if one monitors the environment state by measuring environmental observables even if only a fraction of these measurement outcomes are known to the observer. The central tool used to demonstrate this is the identification of conserved quantities that correspond to the eigenprojections of the system-environment Hamiltonian. We construct Lyapunov functions using this Hamiltonian to demonstrate the stability of the pointer states.

Keywords: Quantum control
Abstract: Determining whether an entangled state of interest can be asymptotically prepared by realistic open-system dynamics has important applications across quantum engineering. This problem has been recently solved for purely dissipative quasi-local dynamics described by a continuous-time Markovian semigroup. Here, we extend our previous analysis by addressing the role of internal Hamiltonian dynamics as well as of Hamiltonian control resources for achieving the same task. We show how Hamiltonians that are not frustration-free can genuinely extend the class of stabilizable states. In particular, we present stabilizing Hamiltonians, along with necessary and sufficient conditions for their existence, for maximally entangled GHZ-states and translationally invariant W-states, none of which are generally stabilizable by
dissipation alone.

Keywords: Quantum control, Distributed parameter systems, Optimal control
Abstract: In this note we investigate what is the best L^p-norm in order to describe the relation between the evolution of the state of a bilinear quantum system with the L^p-norm of the external field. Although L^2 has a structure more easy to handle, the L^1 norm is more suitable for this purpose. Indeed for every p>1 it is possible to steer, with arbitrary precision, a generic bilinear quantum system from any eigenstate of the free Hamiltonian to any other with a control of arbitrary small L^p norm. Explicit optimal costs for the L^1 norm are computed on an example.

Keywords: Optimal control
Abstract: In this work, we have studied the optimal cooling limit when cooling a quantum mechanical oscillator by coupling it to an auxiliary. Since this limit is necessarily confined by the energy transfer rate of the control, i.e., the magnitude of the spectrum of the control Hamiltonian, we search for the optimal control Hamiltonian that can achieve the maximal cooling fidelity, given a bounded energy transfer rate. First we analyze and present a class of Hamiltonian which generates the perfect cooling result in the absence of decoherence. Then with perturbation analysis, we further show that such Hamiltonian is also optimal at least to the first order of the dynamical dissipative terms. With the help of numerical optimization, we have studied cases with different damping rates of the system and the auxiliary, and derived the plot of the optimal cooling fidelity curve versus time as well as the optimal time point to achieve the maximal fidelity.

Keywords: Quantum control, Optimal control, Hamiltonian dynamics
Abstract: The control of the dynamics of spin systems by magnetic fields has opened intriguing possibilities in quantum computing and in Nuclear Magnetic Resonance spectroscopy. In this framework, optimal control theory has been used to design control fields able to realize a given task while minimizing a prescribed cost such as the energy of the field or the duration of the process. However, some of the powerful tools of optimal control had not been used yet for NMR applications in medical imagery. Here, we show that the geometric control theory approach can be advantageously combined with NMR methods to crucially optimize the imaging contrast. This approach is applied to a benchmark problem but it gives a strong evidence for the possibility of using optimal control theory for enhancing the contrast and the resolution of medical images.

Keywords: Port-Hamiltonian systems, Modeling, Distributed parameter systems
Abstract:In this paper, a special class of damping model is introduced for second order dynamical systems. This class is built so as to leave the eigenfunctions invariant, while modifying the dynamics: for mechanical systems, well-known examples are the standard fluid and structural dampings. In the finite-dimensional case, the so-called Caughey series are a general extension of these standard damping models; the damping matrix can be expressed as a polynomial of a matrix, which depends on the mass and stiffness matrices. Damping is ensured whatever the eigenvalues of the conservative problem if and only if the polynomial is positive for positive scalar values. This can be recast in the port-Hamiltonian framework by introducing a port variable corresponding to internal energy dissipation (resistive element). Moreover, this formalism naturally allows to cope with systems including gyroscopic effects (gyrators). In the infinite-dimensional case, the previous polynomial class can be extended to rational functions and more general functions of operators (instead of matrices), once the appropriate functional framework has been defined. In this case, the resistive element is modelled by a given static operator, such as an elliptic PDE. These results are illustrated on several PDE examples: the Webster horn equation, the Bernoulli beam equation; the damping models under consideration are fluid, structural, rational and generalized fractional Laplacian or bi-Laplacian.

Keywords: Distributed parameter systems, Port-Hamiltonian systems, Irreversible thermodynamics
Abstract: Infinite dimensional Port Hamiltonian representation of non isothermal chemical reactors is proposed in the case of mass transport diffusion and chemical reaction without convection. The proposed approach uses thermodynamic variables. The presentation is given for one dimensional spatial domain by using the internal energy and the opposite of the entropy as Hamiltonian functions.

Keywords: Distributed parameter systems, Dirac structures, Boundary control
Abstract: The aim of this paper is to study a conservative wave equation coupled to a diusion equation : this coupled system naturally arises in musical acoustics when viscous and thermal eects at the wall of the duct of a wind instrument are taken into account. The resulting equation, known as Webster-Lokshin model, has variable coecients in space, and a fractional derivative in time. The port-Hamiltonian formalism proves adequate to reformulate this coupled system, and could enable another well-posedness analysis, using classical results from port-Hamiltonian systems theory. First, an equivalent formulation of fractional derivatives is obtained thanks to so-called diusive representations: this is the reason why we rest concentrate on rewriting these diusive representations into the port-Hamiltonian formalism; two cases must be studied separately, the fractional integral operator as a low-pass lter, and the fractional derivative operator as a high-pass lter. Second, a standard finite-dimensional mechanical oscillator coupled to both types of dampings, either low-pass or high-pass, is studied as a coupled pHs. The more general PDE system of a wave equation coupled with the diffusion equation is then found to have the same structure as before, but in an appropriate innite-dimensional setting, which is fully detailed.

Keywords: Irreversible thermodynamics, Mechatronics
Abstract: This paper presents a thermodynamically consistent model of MSMA (Magnetic Shape Memory Alloys) under port Hamiltonian framework. It is based on previous works on MSMA proposed in (Gauthier et al., 2008; Calchand et al., 2011). The main difference lies in the choice of the state variables and manipulated thermodynamic forces. Furthermore in (Gauthier et al., 2008), subsequent experiments revealed a highly hysteretic behavior of these materials. Here, the simplified hysteretic behavior is incorporated into the port-Hamiltonian model to obtain a finer and more precise model. Such modeling will allow the use of a wide range of energy based methods to design the associated control system. The paper ends with some extensions to more complex hysteresc phenomena by using Preisach like model. First ideas are proposed to extend the previous physical model to systems with internal hysteretic loops.

Keywords: Port-Hamiltonian systems, Conservation laws, Modeling
Abstract: In this paper we consider the port-Hamiltonian formulation of systems of two conservation laws defined on two complementary intervals of some interval of the real line and coupled by some moving interface. We recall first how two port Hamiltonian systems coupled by an interface may be expressed as an port Hamiltonian systems augmented with two variables being the characteristic functions of two spatial domains. Then we consider the case of a moving interface and show that it may be expressedas the preceeding port Hamiltonian system augmented with an input, being the velocity of the interface and define a conjugated output variable. We conclude by giving some definition of the passivity of interface relations coupling the external variables associated with the interface.