Their names and symbols are as follows:. Base Physical Quantity Symbol for Quantity Name of SI Unit Symbol for SI Unit length l metre m mass m kilogram kg time t second s electric current I ampere A thermodynamic temperature T kelvin K amount of substance n mole mol luminous intensity I v candela cd The symbol for a physical quantity is a single letter of the Latin or Greek alphabet printed in italic sloping type. The symbol for a unit is printed in roman upright type.
Neither symbol should be followed by a full stop period. The physical quantity 'amount of substance' or 'chemical amount' is proportional to the number of elementary entities - specified by a chemical formula - of which the substance is composed. The proportionality factor is the reciprocal of the Avogadro constanct L 6. The amount of substance should no longer be called 'number of moles'.
Examples of relations between "amount of substance" and other physical quantities:. Prefixes to form the names and symbols of the decimal multiples and submultiples of SI units. Examples of SI derived units with special names and symbols. Physical Quantity for symbols see Sect. These units were used in older literature. They are given here for the purpose of identification and conversion to SI units.
Under suitable differentiabilty and commensurability assumptions we derive necessary optimality conditions in the form of a Pontryagin type Minimum Maximum Principle. For state-constrained control problems in full generality, a Maximum Principle in Pontryagin form was derived very recently by Richard Vinter, Imperial College London. We propose discretization and nonlinear programming methods to solve the time-delayed optimal control problem.
The discretization approach allows us to check the necessary conditions with high accuracy. The numerical approach is illustrated by the optimal control of a two-tank continuous stirred tank reactor CSTR , where the delay is caused by the transport of reactants between the two tanks.
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Then we present a further application in biomedicine, the optimal control of a tuberculosis model. The terminology "stabilization" is used more widely than "stablizability" when we discuss stability of a controlled system. The reason lies in using the so-called "Second Method of Lyapunov" excessively. As a structural character of the controlled system like controllability , stabilizability should be more important.
But the feedback set should be presented before we discuss stablizability. The larger the feedback set is, the more possible stabilizability becomes.
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In this talk, we will focus on the concept of feedback and discuss the stabilizability problem from the viewpoint of optimal control and game theory. In our framework, feedback is defined on information. How to value different information? Could stabilizability be improved when we enlarge the original feedback set? How to design a stabilizable controlled system as simply as possible?.
We present a new result on the controllability of closed quantum systems in infinite-dimensional Hilbert spaces, obtained in collaboration with M. We consider the case where the controlled Hamiltonian depends affinely on finetely many scalar inputs and the drift Hamiltonian has discrete spectrum. Under some mild regularity assumptions on the control operators and some generic conditions on the controllability of the Galerkin approximations, we show exact controllability in projection on an arbitrary number of eigenstates, provided that the required projection has a smaller norm than the initial condition.
Our method relies on a combination of Lie-algebraic control techniques applied to the Galerkin approximations and some topological arguments issuing from degree theory. We present a computational approach to the synthesis of controllers for a class of switched systems, based on the use of finite-state approximations called symbolic models. We first show how to compute symbolic models for switched systems enjoying an incremental stability property. Then, we show how these models can be used for controller synthesis for specifications given by a hybrid automaton. Finally, a computationally more efficient approach is presented based on the use of multi-scale symbolic models.
In this talk we will present some results on the control and stabilization of linear hyperbolic systems. First, we will investigate the stabilization of switched linear hyperbolic systems with a Lyapunov analysis.
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Then, we will consider the trajectory generation problem for 2 x 2 linear hyperbolic systems by backstepping and we will analyze how the tracking problem may be handled. Obtained results will be illustrated by academic examples and physically relevant dynamical systems as Shallow-Water equations and Aw-Rascle-Zhang equations.
We present structure-preserving numercical schemes for several fluid models used in oceanic and atmospheric circulations, such as the Boussinesq and anelastic equations. The resulting variational integrators allow for a discrete version of Kelvin circulation theorem, are applicable to irregular meshes and, being symplectic, exhibit excellent long term energy behavior. We then present a series of preliminary tests for rotating stratified flows such as hydrostatic and geostrophic adjustments, and inertial instability.
Recent extensions of this geometric approach to compressible fluids willbe also presented. The structure of a collection of shapes such as a series of segmented anatomical structures is commonly studied through diffeomorphisms warping one shape onto another. These diffeomorphisms are obtained as flows of vector fields, and then a metric on the space of vector fields allows to build a metric on the space of shapes.
Different choices of vector fields and metrics lead to different metrics on the shape space, and various models have been developed to be adapted to various problems. However these non parametric frameworks do not allow to study the structure of data through a descriptive language for deformations as introduced by Ulf Grenander. In this talk we will present an attempt in that direction.
We constrain vector fields to be generated by a small dimension base of interpretable vector fields, which depend on the shape and evolve with it during the integration of the flow. Then, studying deformations transporting one shape onto another amounts to an optimal control problem in finite dimension, and enables to equip the shape space with a sub-Riemannian metric.
In order to ensure the coherence of this framework, we introduce the concept of deformation module which is a structure, stable under combination, generating vector fields of particular, user-defined type. The talk will aim to illuminate the control properties of quantum systems from a few different angles, including classical Fourier expansion and semiclassical methods.
Obstacles to controllability will be described, as well as attempts to identify subsets of controllable states. Sloppy models are statistical models that have numerous number of parameters, that usually depends on them nonlinearly and implicitely. Differential Equations are typical examples where sloppiness arises. Indeed, it is common that some paramaters influence weakly the trajectories generated for some regions of the state space.
As a consequence, the inverse problem of parameter estimation becomes ill-posed, as the Fisher Information can be become degenerated. The parameters are hard to estimate accurately by standard technics say Maximum Likelihood , and moreover they become sensitive to model misspecification. We focus on the case of Ordinary Differential Equations and we define an M-estimator based on a perturbed model.
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The parameter estimation becomes then the identification of the parameter that needs the smallest perturbation in the model in order to fit the data. Interestingly, this estimation problem can be turned into a deterministic optimal control. For linear ODEs, the standard Linear Quadratic theory gives an efficient framework where the deterministic Kalman Filter plays an unusual role.
The same approach is extended to the general case of nonlinear ODEs, where the Pontryagin Maximum Principle gives a tractable expression for our statistical criterion. We show that the corresponding criterion is smooth enough, so that we can derive root- n consistency and asymptotic normality. We discuss this approach for different sloppy models, as in well and mis-specified cases. The field of quantum information processing quantum computing and quantum communication has seen a tremendous progress during the past few decades. Many proof-of-principle experiments on small scale few physical degrees of freedom quantum systems have been realized within various physical frameworks such as NMR Nuclear Magnetic Resonance , trapped ions, cavity quantum electrodynamics, linear optics and superconducting circuits.
Despite all these achievements, and in order to make this a useful technology, a major scaling step is required towards many-qubit quantum bit protocols. The main obstacle here is the destruction of quantum coherence called decoherence due to the interaction of the quantum system with its environment.
The next critical stage in the development of quantum information processing is most certainly the active quantum error correction QEC. Through this stage one designs, possibly using many physical qubits, an encoded logical qubit which is protected against major decoherence channels and hence admits a significantly longer effective coherence time than a physical qubit.
While a theory of quantum error correction has existed and developed since mid s, the first experiments are being currently investigated in the physics laboratories around the world. I will review some of the theory behind this field and the role that feedback control plays in it. I will also review some of the most recent progress on the experimental side. In this talk, we present a result of local exact controllability to special trajectories of the micropolar fluid system in dimension 3. Previous results showed that this system is locally controllable when both the linear and the angular velocities are controlled.
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Here, we show that local controllability is still possible when acting only on the linear velocity equation or on the angular velocity equation. For both equations, saturating control laws will be designed. By taking into consideration the presence of amplitude-limited controls, we will apply nonlinear semigroup theory and Lyapunov techniques, among other methods.
It will allow us to derive well-posedness results and asymptotic stability properties of the closed-loop systems. Some numerical simulations illustrate the convergence property of the solutions to the closed-loop nonlinear partial differential equations. The intracellular metabolism of living cells is usually represented by a metabolic network under the form of a directed hypergraph that encodes the elementary biochemical reactions taking place within the cell. In this hypergraph, the nodes represent the involved metabolites and the edges represent the metabolic fluxes.
Then it is discussed how these two fundamental techniques can be exploited to design minimal bioreaction models by using a systematic model reduction approach that automatically produces a family of equivalent minimal models which are fully compatible with the underlying metabolism and consistent with the available experimental data. The theory is illustrated with an experimental case-study on CHO cells. The problem of maximizing the contrast in Nuclear Magnetic Resonance spectroscopy is an important issue in Medicine.
It depends on the biological matter blood, cerebral matter, water Too low a contrast must be enhanced by providing contrast accelerators to the patient, with toxicity issues. The aim of the project is to give mathematical methods to get the optimal contrast just by controlling the spectroscope. We present a model consisting of two coupled vector fields controlled by magnetization parameters, and we address the contrast problem in terms of algebraic questions in the parameter space.
Knowing precisely the underlying geometry of the problem is the first step in constructing effective control strategies. This is a joint project involving B. Chyba and J. Martinon Inria Saclay , O. I will present a definition of curvature for affine optimal control problems with a Tonelli Lagrangian. This extends the well known concept of Riemannian sectional curvature to a large class of geometric structures, including sub-Riemannian, Finsler and sub-Finsler ones.
This is a joint work with A. Agrachev and D. Barilari, to appear in Memoirs of the AMS. Consider a three-dimensional fluid in a rectangular tank, bounded by a flat bottom, vertical walls and a free surface evolving under the influence of gravity. Applications of the above-mentioned observability inequalities are also presented, such as the null controllability over a measurable set, time or norm optimal control, bang-bang control and so on.
These results can also be established for more general parabolic systems with real-analytic coefficients. In this talk we review some recent developments on stabilization and state estimation of reaction-diffusion equation on n-balls which in 2-D reduce to a disk and in 3-D reduce to a sphere with constant coefficients. The backstepping method is used to design both the control law and the observer, by reducing the system to an infinite sequence of 1-D systems using spherical harmonics. Very remarkably, in the constant coefficient case the resulting control law can be found explicitly.
It can be written as a multiple integral whose kernel is the product of the classical backstepping kernel used in control of one-dimensional reaction-diffusion equations and a function closely related to the Poisson kernel in the n-ball. An application to multi-agent deployment will be discussed. In addition, a number of open problems will be presented. We will supplement the previous works on how to design multi-boundary feedback controllers to stabilize the general inhomogeneous hyperbolic PDEs by carefully choosing the target control systems.
Some perspectives will be finally shown. We focus on finite perimeter sets, i. Under an assumption on the nilpotent approximation, we prove a blowup theorem, providing a first step towards a generalization of De Giorgi rectifiability theorem. This is a joint work with L. Ambrosio and V. In this talk, we apply the concept of Stackelberg-Nash strategies to the control of parabolic equations.
We assume that we can act on the system through a hierarchy of controls. A first control the leader is assumed to choose the policy. The main novelty in this work is that, in this way, we can obtain the exact controllability to a prescribed but arbitrary trajectory. We study linear and semilinear problems and also problems with pointwise constraints on the followers.
For systems modelled by partial differential equations, there is generally choice in the type of actuator and sensors used and also their locations. Since it is often difficult to move hardware, and trial-and-error may not be effective when there are multiple sensors and actuators, analysis is crucial. Integrating controller design with actuator location can lead to better performance without increased controller cost.
Similarly, better estimation can be obtained with carefully placed sensors.
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Proper placement when there are disturbances present is in general different from that appropriate for reducing the response to an initial condition, and both are different from locations based on optimizing controllability or observability. The dependence of optimal location on the selected criterion is illustrated with several applications. Numerical algorithms and issues will be briefly discussed.. This talk will give a quick overview of what has been be done in this direction through the use of action of groups of diffeomorphisms and mainly right invariant distances.
We will show that simple ideas coming from Riemannian geometry can be recast in the shape setting providing a range of practical tools as well as challenging questions in geometry, probability and statistics. Stanley Durrleman Statistical anatomical models: how to compute with phenotypes? Group studies in neuroimaging raise the need for statistical methods to find in variants in large data sets of anatomical structures. In this talk, we will present a generic statistical framework that can deal with 3D structural images as well as shapes segmented from images such as white matter fiber tracts, meshes of the cortical structures, sulcal ribbons, etc.
The mean is given as a typical anatomical configuration that captures the geometric invariants within the studied population. The variance is described by typical deformations of the mean configuration. The framework relies on the metric of large diffeomorphic deformations in an adaptive finite-dimensional setting. Extension of this framework for the analysis of longitudinal shape data sets will be also presented.
The talk will be illustrated by various examples taken from neuroanatomical studies. Video Slides. Enfin, nous proposons une extension permettant la prise en compte de la polarisation des ondes. You need to install a Flash Player to watch this video! In the first part of the talk, we consider two approaches of quaternion geometric weighting Lp averaging working on the exponential and logarithm maps of full quaternions.
The first formulation is based on computing the Euclidean weighted Lp center of mass in the tangent space of quaternions.
Besides giving explicit forms of these algorithms, their application for quaternion image processing is shown in the second part of the talk, by introducing the notion of quaternion bilateral filtering. The third part of the talk deals with the extension of mathematical morphology operators to quaternion images. The lack of natural ordering in non-Euclidean spaces present an inherent problem when defining morphological operators extended to quaternion-valued images.
We analyze how a robust estimate of the center of mass can be used to obtain a notion of quaternion local origin which can be used to compute rank based operators in the quaternion tangent space. Hence, the notions of local supremum and infimum are introduced, which allow to define the quaternion dilation and erosion, and other derived morphological operators. It is a 4-hour tutorial based mainly on the coming new edition of our Encyclopedia of Distances Springer, Besides mathematics and computer science, most intense search for useful distances occurs now in computational biology, image analysis, speech recognition, and information retrieval.
So, this tutorial consists of 3 parts:. The brief introductions will be followed by nets of definitions almost without results or any proofs. Such a large scale approach does not permit also to present history and applications, but each of the multitude of presented definitions will be traceable and usually we can say more on it upon request. Distances in pattern recognition, audio and image. Distances in biology. Information geometric optimization algorithms: A unifying picture via invariance principles.
In this talk I will review the use of optimal transport methods to tackle various imaging problems such as texture synthesis and mixing, color transfer, and shape retrieval. Representing texture variations as well as shapes geometry can be achieved by recording histograms of high dimensional feature distributions. I will present a fast approximate Wasserstein distance to achieve fast optimal transport manipulations of these high dimensional histograms. The resulting approximate distance can be optimized using standard first order optimization schemes to perform color equalization and texture synthesis.
It is also possible to use this optimal transport as a data fidelity term in standard inverse problems regularization. One can try online several ideas related to Wasserstein imaging as many other imaging methods by visiting www. Optimal transport in imaging sciences. In the first part of the talk, after recalling some basic facts from optimal transport theory, we will explain how some matching problems arising in mathematical economics are intimately related to optimal transport problems.
In the second part of the talk, focusing on the quadratic case, we will relate the problem to a notion of barycenters that generalizes the McCann interpolation to the case of more than two marginals. We will give existence, characterization, uniqueness and regularity results for these barycenters and will consider some examples.
This talk will be based on joint works with Martial Agueh and Ivar Ekeland. Matching, multi-marginals problems and barycenters in the Wasserstein space.