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Convex Analysisの二,三の進展について

Optimal transport (OT) theory can be informally described using the words of the French mathematician Gaspard Monge (1746-1818): A worker with a shovel in hand has to move a large pile of sand lying on a construction site. The goal of the worker is to erect with all that sand a target pile with a prescribed shape (for example, that of a giant sand castle). Naturally, the worker wishes to minimize her total effort, quantified for instance as the total distance or time spent carrying shovelfuls of sand. Mathematicians interested in OT cast that problem as that of comparing two probability distributions, two different piles of sand of the same volume. They consider all of the many possible ways to morph, transport or reshape the first pile into the second, and associate a "global" cost to every such transport, using the "local" consideration of how much it costs to move a grain of sand from one place to another. Recent years have witnessed the spread of OT in several fields, thanks to the emergence of approximate solvers that can scale to sizes and dimensions that are relevant to data sciences. Thanks to this newfound scalability, OT is being increasingly used to unlock various problems in imaging sciences (such as color or texture processing), computer vision and graphics (for shape manipulation) or machine learning (for regression, classification and density fitting). This short book reviews OT with a bias toward numerical methods and their applications in data sciences, and sheds lights on the theoretical properties of OT that make it particularly useful for some of these applications.

Computational Optimal Transport

The evolution of the state of many systems modeled by linear partial diﬁerentialequations (PDEs) or linear delay-diﬁerential equations can be described by operatorsemigroups. The state of such a system is an element in an inﬂnite-dimensionalnormed space, whence the name \inﬂnite-dimensional linear system".The study of operator semigroups is a mature area of functional analysis, which isstill very active. The study of observation and control operators for such semigroupsis relatively more recent. These operators are needed to model the interactionof a system with the surrounding world via outputs or inputs. The main topicsof interest about observation and control operators are admissibility, observability,controllability, stabilizability and detectability. Observation and control operatorsare an essential ingredient of well-posed linear systems (or more generally systemnodes). Inthisbookwedealonlywithadmissibility, observabilityandcontrollability.We deal only with operator semigroups acting on Hilbert spaces.This book is meant to be an elementary introduction into the topics mentionedabove. By \elementary" we mean that we assume no prior knowledge of ﬂnite-dimensional control theory, and no prior knowledge of operator semigroups or ofunbounded operators. We introduce everything needed from these areas. We doassume that the reader has a basic understanding of bounded operators on Hilbertspaces, diﬁerential equations, Fourier and Laplace transforms, distributions andSobolev spaces on

Observation and Control for Operator Semigroups

The past few decades have seen substantial growth in Additive Manufacturing (AM) technologies. However, this growth has mainly been process-driven. The evolution of engineering design to take advantage of the possibilities afforded by AM and to manage the constraints associated with the technology has lagged behind. This paper presents the major opportunities, constraints, and economic considerations for Design for Additive Manufacturing. It explores issues related to design and redesign for direct and indirect AM production. It also highlights key industrial applications, outlines future challenges, and identifies promising directions for research and the exploitation of AM's full potential in industry.

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Design for additive manufacturing: trends, opportunities, considerations, and constraints

This book presents methods to study the controllability and the stabilization of nonlinear control systems in finite and infinite dimensions. The emphasis is put on specific phenomena due to nonlinearities. In particular, many examples are given where nonlinearities turn out to be essential to get controllability or stabilization. Various methods are presented to study the controllability or to construct stabilizing feedback laws. The power of these methods is illustrated by numerous examples coming from such areas as celestial mechanics, fluid mechanics, and quantum mechanics. The book is addressed to graduate students in mathematics or control theory, and to mathematicians or engineers with an interest in nonlinear control systems governed by ordinary or partial differential equations.

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Control and Nonlinearity

In this chapter we will introduce the basic concepts of optimal control for linear elliptic partial differential equations. At first we present the classical theory in functional spaces “à la J.L.Lions”, see [Lio71, Lio72]; then we will address the methodology based on the use of the Lagrangian functional (see, e.g., [Mau81, BKR00, Jam88]). Finally, we will show two different numerical approaches for control problems, based on the Galerkin finite element method.

Optimal Control of Partial Differential Equations

Part 1: general overview on the contents of the book. Part 2 Modelling of networks of elastic strings: modelling of nonlinear elastic strings networks of nonlinear elastic strings linearization wellposedness of the network equations controllability of networks of elastic strings - exact controllability of tree networks, lack of controllability for networks with closed circuits stabilizability of string networks string networks with masses at the nodes. Part 3 Networks of thermoelastic beams modelling of a thin thermoelastic curved beam the equations of motion - some remarks on warping and torsion rotating beams - dynamic stiffening nonlinear nonshearable 3-d beams - approximation-generalizations linear shearable 3-d beams nonlinear shearable 2-d beams - approximation-generalizations a list of beam models - damping networks of beams - geometric joint conditions, rigid joints pinned joints, dynamic joint conditions, rigid joints, pinned joints rotating two-link nonlinear shearable beams. Part 4 A general hyperbolic model for networks: the general model some special cases - string networks, networks of planar Timoshenko beams, networks of linear shearable beams, networks of initially curved bresse beams, beams and strings existence and regularity of solutions energy estimates for hyperbolic systems exact controllability of the network model stabilizability of the network model. Part 5 Spectral analysis and numerical simulation: eigenvalue problems for networks - notation networks of strings setworks of Timoshenko beams Euler-Bernoulli beams the eigenvalue problem for mechanical networks - the string case, homogeneous network of strings, examples, the homogenous Timoshenko network numerical simulations of controlled networks, introductory remarks, absorbing controls, cirecting controls finite element approximations of networks - dry friction at joints. Part 6 Interconnected membranes: modelling of dynamic nonlinear elastic membranes - equations of motion, edge conditions, Hamilton's Principle systems of interconnected elastic membranes - geometric junction conditions, dynamic conditions, linearization, well-posedness of the linear model controllability of linked isotropic membranes - observability estimates for the homogeneous problem, a priori estimates for serially connected membranes, a priori estimates for single jointed membrane systems, the reachable states, serially connected membranes, membrane transmission problems. Part 7 Systems of linked plates: modelling of dynamic nonlinear elastic plates (Part contents)

Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures

The design of auxetic structures is traditionally based on engineering experience, see, for instance, refs. [1,2]. The goal of this article is to demonstrate how an existing auxetic structure can be improved by structural optimization techniques and manufactured by a selective electron beam melting (SEBM) system. Auxetic materials possess negative Poisson’s ratios ν which is in contrast to almost all engineering materials (e. g. metals ν ≈ 0.3). The negative Poisson’s ratio translates into the unusual geometric effect that these materials expand when stretched and contract when compressed. This behavior is of interest for example for fastener systems and fi lters. [ 2 , 3 ] It also results in interesting mechanical behavior like increased penetration resistance for large negative Poisson’s ratios [ 4 ] and in the case of isotropy for higher shear strength relative to the Young’s modulus [ 2 ] of the structure. The auxetic behavior can always be traced back to complex mesoor microscopic geometric mechanisms, e. g. unfolding of inverted elements or rotation of star shaped rigid units relative to each other. [ 5 – 7 ] These geometries can be realized on vastly different scales from specifi cally designed polymers to reactor cores. [ 2 , 8 ]

Design of Auxetic Structures via Mathematical Optimization

We consider traffic flow models for road networks where the flow is controlled at the nodes of the network. For the analytical and numerical optimization of the control, the knowledge of the gradient of the objective functional is useful. The adjoint calculus introduced below determines the gradient in two ways. We derive the adjoint equations for the continuous traffic flow network model and derive also the adjoint equations for a discretized model. Numerical examples for the solution of problems of optimal control for traffic flow networks are presented.

Günter Leugering

Papers

Optimal Control of Partial Differential Equations

Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures

Design of Auxetic Structures via Mathematical Optimization

Optimal control for traffic flow networks

Uniform stabilization of a nonlinear beam by nonlinear boundary feedback