\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