\magnification = 1250
\bigskip
\noindent Abstracts  of the talks at the Session \bf Optimization and Control,
\rm AMS - UMI joint meeting; Pisa, June 2002.
\bigskip
\noindent Organizers: R. Triggiani (Charlottesville) - T. Zolezzi (Genova).
\bigskip
\bigskip
\noindent \bf A. L. DONTCHEV

\noindent \rm Mathematical Reviews, Ann Arbor.
\bigskip
\it The many faces of the condition number theorem.
\bigskip
\noindent \rm  For a number of problems,
the distance to the manifold of ill-posed problems is equal to
the reciprocal of the condition number.
Results of this type are called condition number (distance) theorems.
\smallskip
\noindent I will talk about condition numbers theorems for
regularity properties of mappings associated with inequalities,
variational inequalities, and optimization problems.
\bigskip
\bigskip

  \noindent \bf B. PICCOLI

\noindent \rm Istituto per le Applicazioni del Calcolo, Roma.
\bigskip
\it Synthesis theory in optimal control.
\bigskip
\noindent \rm We consider optimal control problems of Bolza type for 
finite dimensional
systems. For these problems, we show that the concept of synthesis,
roughly speaking a collection of trajectories one for each initial point,
is the correct concept of solutions.
\smallskip
\noindent We show applications to low
dimensional systems. The problems of defining a solution to the
discontinuous ODE linked to optimal feedbacks is also discussed.
\bigskip
\bigskip

\noindent \bf I. LASIECKA

\noindent \rm Department of Applied Mathematics,  University of 
Virginia, Charlottesville.
\bigskip
\it Control Problems for PDE's with an interface.

\bigskip
\noindent \rm We shall consider PDE control systems defined on two different
but connected spatial regions with an interface. The dynamics 
governing the model consists of
coupled hyperbolic and parabolic equations.
The control action takes place on the  interface between two regions.
\smallskip
\noindent The aim of this talk is to discuss controllability and 
stability properties
of the coupled model.
Particular emphasis will be given to propagation of hyperbolicity and 
analyticity from one component of the system onto another. As easily 
predicted, this depends on the strength of the coupling. We shall 
witness various configurations where controllability and /or 
stability properties of one component can be propagated onto the 
entire system.
The geometry of the spatial region will play critical role for the 
results obtained.
\bigskip
\bigskip

\noindent \bf G. STEFANI

\noindent \rm Dipartimento di Matematica Applicata, Firenze.
\bigskip
\it Sufficient conditions for the bang-bang and singular case.
\bigskip
\noindent \rm Hamiltonian methods are used to prove sufficient 
conditions for the
optimality of a bang-bang or singular extremal. The notion of invariant
second variation and its relations with the Hamiltonian flow is used.
\bigskip
\bigskip
\noindent \bf H. FATTORINI

\noindent \rm Department of Mathematics, University of California, Los Angeles.
\bigskip
\it Optimal control of diffusions.

\bigskip
\noindent \rm We consider the controlled diffusion equation
$y'(t) = Ay(t) + \mu(t),$ where $A$ is a uniformly elliptic operator
in a $m$-dimensional bounded domain $\Omega$ with boundary $\Gamma,$ 
$A$ restricted by
a boundary condition $\beta$ on $\Gamma$ of Dirichlet or variational
type. The state space is
$L^1(\Omega)$ and the controls $\mu(t)$ belong to the space
$L_w^\infty(0, T; \Sigma(\overline \Omega))$ of all $C(\overline 
\Omega)$-weakly
measurable, essentially bounded $\Sigma(\overline \Omega)$-valued 
functions in $0 \le t
\le T$
$(C(\overline \Omega)$ is the space of all continuous functions in 
$\overline \Omega,$
$\Sigma(\overline \Omega)$ is the space of all bounded Borel measures in
$\overline \Omega).$ The
formulation includes both the time optimal and norm optimal problems
with exact targets.
\smallskip
\noindent The results are on
existence, uniqueness and a version of Pontryagin's maximum principle that
implies a {\it concentration principle:} for almost all $t$ optimal 
measure-valued
controls $\mu(t)$ are supported by ``small sets" (intersections of 
hypersurfaces in
dimension $m > 2,$ finite sets in dimension $m = 1).$ The conditions 
coming out the
maximum principle are both necessary and sufficient.
The problem  changes considerably if only positive controls
$(\mu(t)$ a positive measure a.e.) are allowed; in this case there 
are optimal controls
that don't satisfy the maximum principle.
\bigskip
\bigskip
\noindent \bf A. BACCIOTTI

\noindent \rm Dipartimento di Matematica, Politecnico di Torino.
\bigskip
  \it On the relationship between optimal regulation and nonlinear 
stabilization.

\bigskip

\noindent \rm Classical studies in the theory of linear systems 
demonstrate that the
relationship between the problem of minimizing a quadratic cost functional
over the infinite horizon and the problem of feedback stabilization are
strictly related, in the sense that if we are able to solve one problem,
that we can also deduce a solution for the other.
Similar relationship can be established in the case of nonlinear (affine)
systems, provided that the value function of the optimization problem is
assumed to be of class C^1.
\smallskip
\noindent We investigate here the possibility of further extensions, 
for the case
where the value function is just locally Lipschitz continuous.
The results are based on nonsmooth analysis methods, and the stabilizing
feedback involved in our approach are in general discontinuous.
\bigskip
\bigskip
\noindent \bf H. J. SUSSMANN


\noindent \rm  Department of Mathematics, Rutgers University.
\bigskip
     \it Open mapping theorems for generalized differentials
\bigskip
\noindent \rm Open mapping theorems assert, roughly, that if the differential
$DF(x)$ of a map $F$ at a point $x$ is nonsingular, then $F$ maps
neighborhoods of $x$ to neighborhoods of $F(x)$. The simplest such theorem
is the one for maps of class $C^1$. If the target space is
finite-dimensional, then a similar theorem is true for maps that are just
differentiable at a point and continuous on a neighborhood.
\smallskip
\noindent These
well-known facts have been generalized by Clarke, Warga, Halkin and others
to maps that are not classically differentiable but admit "differentials"
in a generalized sense. To define such notions of differential, two main
approaches---one based on uniform approximations by $C^1$ maps, another one
involving factorizations $F(x)=G(x)x$---have been pursued, leading to two
very general theories that happen not to be comparable with each other. We
propose a new theory, of "path-integral generalized differentials," that
has all the good properties of a generalized differential---such as the
chain rule---and in addition satisfies a directional open mapping theorem
and contains all the other theories.
\bigskip
\bigskip
\noindent \bf A. AGRACHEV

\noindent \rm SISSA (Trieste) and Steklov Mathematical Institute (Moscow).
\bigskip
\it Generalized "Ricci Curvature" of Optimal Control Problems
  and Hamiltonian Systems.
\bigskip
\noindent \rm The talk is devoted to the construction and properties of a
fundamental feedback (gauge) invariant of smooth optimal control problems;
namely, to a far going generalization of the classical Ricci tensor in
Riemannian Geometry. We start from the space of extremals treated, in the
spirit of the Pontryagin maximum principle, as curves in the cotangent
bundle $T^*M$, where $M$ is the state space of the control system.
\smallskip
\noindent Generalized Ricci curvature is a real function defined on 
the domain in $T^*M$ filled by
the extremals, both regular and singular. In the regular case, the
extremals are just trajectories of a Hamiltonian system on $T^*M$. In fact,
the generalised Ricci curvature can be defined directly for the
Hamiltonian system on $T^*M$ even if the system doesn't arise from an
optimal control problem. This curvature is actually a basic invariant of
the space of trajectories of the system projected in $M$.
\bigskip
\bigskip
\noindent \bf W. LITTMAN

\noindent \rm School of Mathematics, University of Minnesota, Minneapolis.
\bigskip


\it Control from the boundary on two dimensional Riemannian manifolds

\bigskip
\noindent \rm Let $M$ be a compact two dimensional smooth Riemannian 
manifold with convex
boundary.  Assume there are no closed geodesics on $M$. Then there exists a
pseudo convex function on the closure of $M$.
\smallskip
\noindent Corollary:
The wave equation on $M$ can be controlled from the boundary in finite time.
Other results and counterexamples are given.  This is joint work with
Santiago Betelu and Robert Gulliver.
\bigskip
\bigskip
\noindent \bf L. PANDOLFI

\noindent \rm Dipartimento di Matematica, Politecnico di Torino.
\bigskip
\it Approximate identification of inputs to distributed parameter systems
\bigskip
\noindent \rm Input identification is a particular deconvolution 
problem, the (approximate)
identification of $u$ from sample of the function $y(t)$ given by
$$ y(t) =C\int_0^t e^{A(t-s)} Bu(s) {\em d\/}\, s\,, \qquad 0\leq t\leq T $$
where $A $ generates a $C_0$--semigroup on a Hilbert space $X$ while 
$B$ and $C$ are suitable operators.
The output $y$ is sampled at discrete time instants $kT/n$. The 
identification problem is ill posed. Hence we
shall relay on a penalization technique. In this way we construct a 
``sequence" $\{v_{n,\alpha}\}$ (which depends
on the sampling step $T/n$, the penalization parameter $\alpha$ and 
also on the tolerance of the observation
error).
\smallskip
\noindent The results that we expect are as follows: in the case of 
regular inputs $u$, and/or ``regular" enough
input/output operators, we expect convergence of $\{v_{n,\alpha}\}$ 
to $u$ in the uniform norm. We expect that
time domain techniques can be used for this. Otherwise we expect 
$L^2$ convergence and we expect that the
simplest approach to this problem in such generality is via frequency 
domain techniques.
\bigskip
\bigskip
\noindent \bf R. TRIGGIANI

\noindent \rm Mathematics Department, University of Virginia, 
Charlottesville, VA, USA
\bigskip
  \it Boundary stabilization of dynamic shallow shells by non-linear 
dissipation in
physical  boundary conditions
\bigskip

\noindent \rm  We consider a dynamic shallow shell model, which is 
mathematically expressed in
a Riemann  geometric setting, as recently proposed by P.Y.Yao. The 
shell (calotta) may be
viewed as  a portion of a curved surface. It consists of two 
hyperbolic-like PDEs, defined on
the  curved surface: a dynamic system of elasticity in the 2-d, 
in-plane dispalcement, and a
Kirchhof-like equation in the scalar normal displacement. Boundary 
conditions are clamped
for the first equation and involve moments and stresses for the 
second equation.
Under homogeneous BC, the (free) shell is conservative in a natural 
energy space.
We then select suitable, non linear dissipative terms in the physical 
BC, so that two
desirable properties are achieved:
(i) the new boundary feedback problem is well posed in the non-linear 
semigroup sense in
the energy space;
(ii) its solutions decay uniformly with sharp rates.
The solution of the stabilization problem relies, among other things, 
on the combination
of Riemann geometric methods with microlocal analysis methods. The 
latter are needed to
estimate the first order (boundary) traces of the system of 
elasticity component (a critical
task for the very solution of the problem) as well as the second 
order traces of the Kirhhof-type
component (to eliminate unnecessary geometrical conditions).

\noindent This is joint work with I. Lasiecka to appear in JMAA.


\bigskip
\bigskip
\noindent \bf F. RAMPAZZO

\noindent \rm Dipartimento di Matematica Pura ed Applicata, Padova.

\bigskip

     \it Lie brackets of locally Lipschitz continuous vector fields.
\bigskip
  \noindent \rm This is a joint work with  Hector Sussmann, Rutgers University.
\smallskip


  \noindent The Lie bracket $[f,g]$ of vector fields $f$ and $g$
  is a fundamental tool in  Differential Geometric  Control Theory.
  If $f$ and $g$
are differentiable  on  $\Omega\subseteq {\Bbb R}^n$, then
$[f,g]$
is a  vector field on $\Omega$
  defined, at each $x\in\Omega$, by
$$
[f,g](x) = Dg(x)\cdot f(x)-Df(x)\cdot g(x).
$$
It is well known that the {\it commutativity} (on
sufficiently small
  time intervals) of the
flows of $f$ and $g$ is equivalent to the fact that $[f,g](x)$ is
equal to zero everywhere. On the opposite side, one has {\it
controllability} results, the most classic of which being Chow's theorem.
\smallskip
\noindent We have investigated such basic issues in some cases when 
the involved
vector fields do not possess all the derivatives needed in the classic
results. For instance, a natural question could
be the following one: if $f$ and $g$ are locally Lipschitz continuous
and $[f,g](x)=0$
at almost every  $x\in\Omega$, is it true that the associated
flows commute? Analogous questions can be posed for the
controllability issue.
\smallskip
We are able to give some positive answers to such questions by
introducing a  notion of  set-valued Lie bracket for {\it locally
Lipschitz continuous vecor fields} $f$ and $g$. In particular,
at any point of $\Omega$ the Lie bracket of $f$ and $g$ is a nonempty,
compact
convex set, which reduces to the singleton $\{[f,g](x)\}$
as soon as $f$ and $g$  are differentiable.




\end

