J. of Dyn. and Contr. Sys. - Vol. 3 (1997), No. 2

Abstracts

Title: Explicit Solution of some First Order PDE's

Author: E.N. Barron, R. Jensen, W. Liu

Mathematics subject classification: 49L25, 35F05, 35F10

Keywords: Hopf--Lax formula, Hamilton--Jacobi equation,
obstacle problem, minimax solutions

Abstract: 
The classical Hopf--Lax formula for an explicit solution of
a Hamilton Jacobi equation is extended to equations of the form
$u_t+H(u,Du)=0$ with terminal data $u(T,x)=g(x)$ assumed to be merely
quasiconvex, i.e., having convex level sets.  Using a new quasiconvex
conjugate $g^*(\g,p)$, the formula is given by
\[u(t,x)=\left(g^*(\g,p)-(T-t)H(\g,p)\right)^*.\] We give a direct
proof that $u$ is a Subbotin minimax solution of the problem.  The
first order obstacle problem associated with optimal control in
$L^{\i}$ is also studied and an explicit solution given.




Title: The generic local structure of time-optimal synthesis
with a target of codimension one in dimension greater than two

Author: G. Launay, M. Pelletier

Mathematics subject classification: 49J15, 57N80, 93B06, 93C10, 93C95

Keywords: Optimal trajectory, cut-locus, synthesis, exceptional,
stratification, generic

Abstract: 
In previous papers, we gave in dimension 2 and 3 a
classification of generic synthesis of analytic systems ${\dot v}
(t)=X (v(t))+u (t)Y(v(t))$ with a terminal submanifold $N$ of
codimension one when the trajectories are not tangent to $N$. We
complete here this classification in all generic cases in dimension 3,
giving a topological classification and a model in each case. We prove
also that in dimension $n \geq 3$, out of a subvariety of $N$ of
codimension three, we have described all the local synthesis.



Title: Homogeneous Tangent Vectors and High-Order Necessary
Conditions for Optimal Controls

Author: F. Ancona

Mathematics subject classification: 49K15, 93B03, 93C10

Keywords: Affine control system, homogeneous approximations,
graded structure, high-order tangent vectors

Abstract: 
An extension of the classical Pontryagin maximum principle
for Mayer problems without terminal constraints, subject to affine
control systems $\dot x = X_0 (x) + \sum_{j=1}^m u_j X_j (x)$, is
proved. In connection with a suitable dilation on the state space
$\Bbb R^n$, we introduce a class of ``homogeneous tangent vectors''
which provide a nonlinear, high-order approximation of the attainable
set in the case where the usual linear approximation reveals to be
inadequate.  By studying control variations which generate homogeneous
tangent vectors, we derive new necessary conditions for optimality
that are particularly effective for basically nonlinear optimal
control problems where other high-order tests provide no conclusive
information.




Title: Non-Abelian statistics for Calogero--Sutherland systems of particles

Authors: V.Golubeva, V. Leksin

Mathematics subject classification: 58F07

Keywords: anyons, non-Abelian statistics, Knizhnik--Zamolodchikov equations,
spin Calogero--Sutherland operators, root systems

Abstract: 
The mathematical aspects of the theory of non-Aabelian
statistics and anyons are discussed. The dynamics and kinematics of
these objects are defined by Hamiltonians which are the sums of terms
associated to the different roots of a given root system.  The
equations of the Fuchsian type, that is, the so-called generalized
Knizhnik--Zamolodchikov equations associated to the root systems
$A_n$, $B_n$, and $C_n$, are presented. Some results concerning the
generalized pure braid groups and their application for the definition
of the generalized anyon are discussed. The factorization of the spin
Calogero--Sutherland operators and their connection with the theory of
non-Abelian statistics and anyons are given.





Title: On mappings of bounded variation

Author: V.V. Chistyakov

Mathematics subject classification: 26A45, 54C60, 54C65, 49J45

Keywords: Mappings of bounded variation, metric space-valued mappings,
jumps, geodesic paths, Helly's selection principle, set-valued mappings,
regular selections

Abstract: 
We present the properties of mappings of bounded variation
defined on a subset of the real line with values in metric and normed
spaces and show that major aspects of the theory of real-valued
functions of bounded variation remains valid in this case. In
particular, we prove the structure theorem and obtain the continuity
properties of these mappings as well as jump formulas for the
variation.  We establish the existence of Lipschitz continuous
geodesic paths and prove an analog of the well-known Helly selection
principle. For normed space-valued smooth mappings we obtain the usual
integral formula for the variation without the completeness assumption
on the space of values. As an application of our theory we show that
compact set-valued mappings (= multifunctions) of bounded variation
admit regular selections of bounded variation.