


SS8  Équations différentielles et commande / SS8  Differential Equations and Control Org: F. Clarke (Lyon) et/and R. Stern (Concordia)
 HEDY ATTOUCH, Université Montpellier II
Finite time stabilization results and dry friction

We show some finite time stabilization results concerning nonlinear
oscillators subject to dry friction. Applications to physical sciences
and decision sciences are given.
 JEANPIERRE BOURGUIGNON, IHES

 PIERMARCO CANNARSA, Università di Roma "Tor Vergata", Dipartimento di
Matematica, Via della Ricerca Scientifica 1, 00133 Roma,
Italy
Interior sphere property of attainable sets and time optimal
control problems

The main object of the talk is the geometric analysis of attainable
sets in time T > 0 for nonlinear control systems y¢(t) = f (y(t),u(t) ), obtaining sufficient conditions for these sets to
satisfy a uniform interior sphere condition. This result is then used
to recover the semiconcavity of the value function of time optimal
control problems with a general target.
 THIERRY CHAMPION, Université du Sud ToulonVar
Coupling penalty schemes and the steepest descent method in
convex programming

We study the asymptotic convergence of the flow associated with a
dynamical system obtained by coupling the continuous steepest descent
method with a penalty scheme for a convex program. The penalty
parameter involved in this dynamical system may be seen as a control
variable. We establish the convergence of the primal trajectories
towards an optimal solution whose characterization depends on the
behaviour of the penalty parameter, and we also discuss the
convergence of the associates dual paths.
This is joint work with M. Courdurier (Univ. Washington, Seattle).
 MONICA COJOCARU, Dept. of Mathematics, University of Guelph, Guelph, ON, Canada
Projected Differential Equations. Projected Dynamical
Systems. Overview and New Developments

We present here a type of constrained dynamics, given by an ODE with
discontinuous and nonlinear righthand side. Differential inclusions
related to this type of equation, which we call projected differential
equation (PrDE), appeared in the mathematical literature as early as
1973, then continued to be studied in the '80s. However, the
definition in the form we use today was introduced in the early '90s,
on Euclidean space.
Although with discontinuities, a PrDE has solutions in the class of
absolutely continuous functions. We then associate a dynamics, given by
these solutions, and obtain the now widely used projected dynamical
system (PDS). A projected dynamics is essentially a positional control
problem, since the solutions of any PrDE are constrained to evolve
within and on the boundary of a closed, convex subset of the
underlying space.
A crucial trait of a PrDE is that its critical points (the zeros of
the righthand side) coincide with the solutions to a variational
inequality problem (VI), thus making the associated projected dynamics
extremely useful in applications, for example: spatial price
equilibria, financial equilibria and transportation.
In this talk we present a brief history of this topic, highlighting
the researchers who contributed to its advance, as well as the
contribution of the author, together with collaborators, to the very
recent developments in this area.
 CYRIL IMBERT, Université Montpellier 2, CC 051, Place E. Bataillon, 34095
Montpellier Cedex 5
Effet régularisant d'un opérateur non local sur les
équations de HamiltonJacobi du premier ordre

Nous étudions la régularité des solutions d'équations
intégrodifférentielles. Ces équations peuvent être
interprétées comme des équations de BellmanIsaacs de certains
problèmes de contrôle optimal.
Les équations que nous étudions sont des équations
d'HamiltonJacobi d'ordre 1 perturbées par un opérateur non local,
le laplacien fractionnaire. Il existe déjà une littérature
importante à ce sujet et la théorie des solutions de viscosité
permet de donner un sens faible, non seulement aux dérivées mais
aussi à l'opérateur.
Nous prouvons que sous des hypothèses naturelles sur l'Hamiltonien
(celles qui assure l'unicité de la solution de l'équation non
perturbée), la solution de viscosité est deux fois
différentiable en espace et une fois en temps. Pour ce faire, nous
construisons tout d'abord une solution de viscosité bornée et
Lipschitzienne, puis nous utilisons une représentation intégrale
de l'équation et des méthodes de point fixe pour prouver que la
solution est en fait C^{2}.
Nous estimons enfin la différence entre la solution de l'équation
non perturbée et la solution de l'équation perturbée par un
opérateur non local évanescent.
 OLIVIER LEY, Université de Tours, France
BellmanIsaacs equations under quadratic growth assumptions
and applications to control

We are concerned with secondorder degenerate parabolic
HamiltonJacobiBellman and Isaacs equations. We prove a comparison
principle between semicontinuous viscosity sub and supersolutions
growing at most quadratically.
As an application, we consider a finite horizon stochastic control
problem with unbounded controls and we prove that the value function
is the unique viscosity solution of the corresponding dynamic
programming equation.
This is a joint work with Francesca Da Lio (Universita di Torino,
Italy).
 JEANPIERRE RAYMOND, Université Paul Sabatier, Laboratoire MIP, 31062 Toulouse
Cedex 4, France
Feedback stabilization of parabolic systems

We are interested in the local stabilization of parabolic equations or
systems around an unstable stationary solution. The feedback control
is an internal or a boundary control. We are mainly concerned with the
case where the nonlinear term of the state equation is unbounded in
the space of initial data. We prove different local stabilization
results by linear feedback laws (obtained by solving some algebraic
Riccati equations), or by nonlinear ones (by considering
HamiltonJacobiBellman equations).
Applications to the NavierStokes equations and to other nonlinear
parabolic equations will be given.
 LUDOVIC RIFFORD, Lyon

 CARLO SINESTRARI, Università di Roma "Tor Vergata", Dipartimento di
Matematica, Via della Ricerca Scientifica 1, 00133 Roma
Semiconcavity of the value function for exit time problems
with nonsmooth target

We consider a class of nonlinear optimal control problems where the
cost functional depends on the arrival time of the trajectory on a
given target set. A well known example is the minimum time problem.
We study the conditions under which the value function of the problem
is semiconcave. In contrast to previous works (e.g.CannarsaSinestrari [1995]) which required an interior sphere
condition on the target, in the result we present here the target can
be completely general. On the other hand, the dynamics of the system
is assumed to satisfy suitable regularity properties. In particular,
the set of admissible velocities for the system must have a smooth
boundary at every point in a neighborhood of the target.
 EMMANUEL TRELAT, Université ParisSud (Orsay), Mathématiques, Labo. ANEDP,
UMR 8628, Bat. 425, 91405 Orsay Cedex
Global subanalytic solutions of HamiltonJacobi type equations

In the '80s Crandall and Lions introduced the concept of viscosity
solution in order to get existence and/or uniqueness results for
HamiltonJacobi equations. We first investigate the Dirichlet and
CauchyDirichlet problems for such equations, where the Hamiltonian is
associated to a problem of calculus of variations, and prove that if
the data are analytic then the viscosity solution is moreover
subanalytic. We then extend this result to HamiltonJacobi equations
stemming from optimal control problems, in particular from
subRiemannian geometry, which are generalized eikonal equations.
 PETER WOLENSKI, Louisiana State University, Baton Rouge, Louisiana 70803
Invariance results for impulsive differential inclusions

We consider the measuredriven differential inclusion

ì ï í
ï î

dx Î F 
æ è

x(t) 
ö ø

dt + G 
æ è

x(t) 
ö ø

dm(t) 
 
 

\leqno(DI) 

where F : R^{n} \rightrightarrows R^{n} and G :R^{n} \rightrightarrows M_{n×m} are given
multifunctions with nonempty, compact, and convex values, and m is
a Borel vectorvalued measure with values in a closed convex cone
K Í R^{m}. We will discuss the nature of a
"solution" to (DI) and how it can be obtained via a
graphical limit of Eulerpolygonal arcs. We shall also introduce and
provide local characterizations of weak and strong invariance.

