/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

Convex bodies: the brunn–minkowski theory

“Multivalued Analysis” is the theory of set-valued maps (called multifonctions) and has important applications in many different areas. Multivalued analysis is a remarkable mixture of many different parts of mathematics such as point-set topology, measure theory and nonlinear functional analysis. It is also closely related to “Nonsmooth Analysis” (Chapter 5) and in fact one of the main motivations behind the development of the theory, was in order to provide necessary analytical tools for the study of problems in nonsmooth analysis. It is not a coincidence that the development of the two fields coincide chronologically and follow parallel paths. Today multivalued analysis is a mature mathematical field with its own methods, techniques and applications that range from social and economic sciences to biological sciences and engineering. There is no doubt that a modern treatise on “Nonlinear Functional Analysis” can not afford the luxury of ignoring multivalued analysis. The omission of the theory of multifunctions will drastically limit the possible applications.

SET-Valued Analysis

Convex Analysis: (pms-28)

Deterministic and Stochastic Optimal Control

In this paper we formulate a time-optimal control problem in the space of probability measures The main motivation is to face situations in finite-dimensional control systems evolving deterministically where the initial position of the controlled particle is not exactly known, but can be expressed by a probability measure on $\mathbb {R}^{d}$ We propose for this problem a generalized version of some concepts from classical control theory in finite dimensional systems (namely, target set, dynamic, minimum time function) and formulate an Hamilton-Jacobi-Bellman equation in the space of probability measures solved by the generalized minimum time function, by extending a concept of approximate viscosity sub/superdifferential in the space of probability measures, originally introduced by Cardaliaguet-Quincampoix in Cardaliaguet and Quincampoix (Int Game Theor Rev 10, 1–16, 2008) We prove also some representation results linking the classical concept to the corresponding generalized ones The main tool used is a superposition principle, proved by Ambrosio, Gigli and Savare in Ambrosio et al [3], which provides a probabilistic representation of the solution of the continuity equation as a weighted superposition of absolutely continuous solutions of the characteristic system

Generalized control systems in the space of probability measures

Let $M$ be a Riemannian manifold and let $\varOmega $ be a bounded open subset of $M$. It is well known that significant information about the geometry of $\varOmega $ is encoded into the properties of the distance, $d_{\partial \varOmega }$, from the boundary of $\varOmega $. Here, we show that the generalized gradient flow associated with the distance preserves singularities, that is, if $x_0$ is a singular point of $d_{\partial \varOmega }$ then the generalized characteristic starting at $x_0$ stays singular for all times. As an application, we deduce that the singular set of $d_{\partial \varOmega }$ has the same homotopy type as $\varOmega $.

Singular gradient flow of the distance function and homotopy equivalence

The paper develops a model of traffic flow near an intersection,
where drivers seeking to enter a congested road wait in a
buffer of limited capacity. Initial data comprise the vehicle density on each road,
together with the percentage of drivers approaching the intersection
who wish to turn into each of the outgoing roads.
 &nbsp 
If the queue sizes
within the buffer are known, then the initial-boundary value
problems become decoupled and can be independently solved
along each incoming road.
Three variational problems are introduced, related to different kind of
boundary conditions. From the
value functions, one recovers the traffic density
along each incoming or outgoing road by a Lax type formula.
 &nbsp 
Conversely,
if these value functions are known, then the queue sizes can be determined
by balancing the boundary fluxes of all incoming and outgoing roads.
In this way one obtains a contractive transformation, whose fixed point
 yields the unique solution of the Cauchy problem
for traffic flow in an neighborhood of the intersection.
 &nbsp 
The present model accounts for
 backward propagation of queues along roads leading to a crowded intersection,
it achieves well-posedness for general $L^\infty $ data, and continuity w.r.t. weak convergence of the initial densities.

Conservation law models for traffic flow on a network of roads

This paper establishes the global existence of weak solutions to the Burgers--Hilbert equation, for general initial data in ${\bf L}^2(\mathbb{R})$. For positive times, the solution lies in ${\bf L}^2\cap{\bf L}^\infty$. A partial uniqueness result is proved for spatially periodic solutions, as long as the total variation remains locally bounded.

Global Existence of Weak Solutions for the Burgers--Hilbert Equation

A minimum time problem with a nonlinear smooth dynamics and a target satisfying an internal sphere condition is considered. Under the assumption that the minimum time $T$ be continuous and the normal cone to the hypograph of $T$, $N_{hypo(T)}$, be pointed, we show that $hypo(T)$ is $\varphi$-convex, i.e., satisfies a strong external sphere condition. Consequently, $T$ is a.e. twice differentiable and satisfies some further regularity properties. Our results are based on a representation of Clarke generalized gradient of $T$. An example is provided, showing that if $N_{hypo(T)}$ is not pointed, then the result may fail.

/pdf/on-the-structure-of-the-minimum-time-function-2vw8dctft3.pdf

Khai T. Nguyen

Papers

Generalized control systems in the space of probability measures

Singular gradient flow of the distance function and homotopy equivalence

Conservation law models for traffic flow on a network of roads

Global Existence of Weak Solutions for the Burgers--Hilbert Equation

On the Structure of the Minimum Time Function