Fifth SIAM Conference on Control and Its Applications
July 11 – 14, 2001
Town & Country Hotel, San Diego, CA

The SIAM Activity Group on Control and Systems Theory joined
forces with the SIAM Annual Meeting in hosting the Fifth
SIAM Conference on Control and Its Applications in San
Diego. Short courses on Flow Control by Max Gunzburger and
on Shape Optimization by Michel Delfour preceded the
conference, setting the stage for a program that included 84
minisymposia and 13 invited presentations. The conference
literally kicked off with a plenary talk by Raffaello
D’Andrea on cooperative control, a field dealing with the
control of entities that exchange information while working
towards an overall group objective. Highlights of the talk
were video clips of Cornell’s world championship robotics
soccer victories in 1999 (Stockholm) and 2000 (Melbourne).
D’Andrea attributed their RoboCup victories to the
incorporation of optimal control techniques in their team
strategies. In the first joint session with the annual
meeting, Jerry Marsden explained the role of celestial
dynamics in trajectory design for space missions, focusing
on the role of three and four body problems. The theory was
illustrated in the context of the upcoming Genesis mission
which collects solar wind samples and brings them to earth.
Invariant manifolds, periodic orbits, and the equilibrium
points of the three body problem were introduced, and saddle
point controllers for remaining on the so-called halo orbit
were described. In the second joint session, P. Kumar
examined wireless communication networks. In the wireless
environment, packets can be relayed from node to node. When
a node broadcasts its own information, it can also interfere
with the broadcasts of other nodes. One of many observations
of Kumar might be paraphrased in the following way: the
most information is passed through the network when the
broadcast range of each node is as small as possible without
disconnecting the network. Gilbert Strang, in the address
of the outgoing president, also sought to unify the annual
meeting with the control conference by providing a linear
algebra view of the well-known Kalman filter, observing
that Kalman’s filter was essentially a statement about
tridiagonal matrices and their factorization.

Two prizes were given at the conference. First, the W. T.
and Idalia Reid Prize in Mathematics was given to Eduardo
Sontag in honor of his contributions to nonlinear control
theory. In his talk following the prize presentation,
Sontag examined “Feedback Control Theory and the Challenges
of Postgenomic Molecular Biology.” Now that the genomes of
many species are substantially mapped, and the structure of
the encoded proteins are being solved for, the next step is
to develop a “systems molecular biology” to characterize
the behavior of complete cellular signal pathways. The
scientific and medical payoff of such knowledge will
literally change our understanding of life and result in
revolutionary therapies. A major effort involving the
biology, computer science, chemistry, and physics
communities is under way in this direction, and Sontag
discussed in his lecture the potential role to be played by
concepts and techniques from feedback control theory.

The second prize, awarded by the control activity group to a
younger researcher, was given to Vincent Blondel for his
research at the interface between system theory and
computational complexity theory. His work on the
computational complexity of problems ranging from the robust
stability of time-varying linear systems to controllability
of hybrid systems to stability of saturated linear systems
addressed fundamental problems in systems and control theory
from a novel point of view, delineating the limitations of
mathematical analysis and computation in certain contexts.
As an illustration in his talk, he observed that easy to
state problems concerning the growth of products of matrices
or the decay of solutions of dynamical systems were
undecidable. That is, there is no algorithm that always
halts with the right answer.

Although optimal control emerged initially in the context of
differential equations, applications soon advanced to
partial differential equations including the pioneering work
of J. L. Lions on a variety of topics ranging from impulse
control to decomposition methods. Lions, who passed away
recently, was remembered by Tom Banks before the Reid Prize
lecture was given. The short courses on flow control and
shape optimization at the start of the conference were
followed by a series of related minisymposia. As one might
expect, flow control involves the application of optimal and
feedback control in the design of optimal flows around
objects. The minisymposia approached this field via
numerical methods (reduction of these high dimensional
complex problems to lower dimensions using reduced order
models or proper orthogonal decompositions) and a rich
variety of applications from chemical vapor deposition to
air flow around a wing.

The plenary talk of Olivier Pironneau provided a survey of
the huge, growing, and difficult field of shape
optimization. Applications are pervasive, from structural
design problems (design a bridge in order to maximize the
minimal wind velocity that causes oscillation), to the
design of ship hulls to minimize drag, to the design of low
weight solar collectors for satellites.

Another research direction in PDE control showcased at the
conference concerns the control of interactive structures.
These structures, surveyed in the plenary lecture of Irena
Lasiecka, are described by systems of PDEs with coupling
terms on interfaces. For example, the equations of
elasticity might be coupled at the boundary (interface) with
the plate or shell equation, a typical configuration in
structural acoustic control problems with smart actuators
and sensors. One may also consider additional coupling with
the heat equation in describing a thermoelastic system whose
elastic properties are modified by heating effects. These
couplings lead to a new class of control models with a
strong mixture of hyperbolic and parabolic characteristics.
New theory exploiting propagation of hyperbolicity and
smoothing effects due to parabolicity were used to describe
stabilization and controllability properties.

John Betts, an aerospace engineer from Boeing, provided an
industrial facet to the conference. Betts discussed
practical issues that arise when trying to solve control
problems with “black box” dynamics. That is, in
industrial applications, the equations that model a physical
process may be quite complicated. The engineer is given, in
essence, a black box describing the system, which must be
controlled or optimized. Numerical issues in optimal
control were highlighted in the talks of Assen Dontchev and
Fredi Troeltzsch from both the ODE and PDE viewpoint.
Dontchev focused on the role of solution regularity and
Lipschitzian stability in the derivation of error estimates
for discretizations, using splines and state constrained
problems for illustrations. Troeltzsch noted that certain
embeddings, valid in 1 dimension, break down in higher
dimensions, leading to a more complicated analysis involving
bootstrapping arguments. Troeltzsch’s talk concluded with a
movie (the North American premier) of an optimally cooled
steel beam.

A talk with a strong applied component, given by Ralph
Smith, explained how hysteresis and inherent nonlinearities
can be accommodated in the design of systems which utilize
smart materials. The atomic force microscope, an instrument
requiring nanoscale precision, was used as a motivating
application. Smith pointed out the crucial importance of
collaboration between mathematicians and engineers in the
development of effective control strategies that are backed
up by rigorous mathematical theory.

Another facet of the conference addressed algebra and
operator theoretical issues, culminating in the invited
address of William Helton on matrix inequalities and
computer assisted analysis. In his talk he showed how
matrix inequalities arise and he sketched recent
developments of symbolic methods for automatically
determining if a given function of matrices is “matrix
convex.” Helton’s talk concluded with the following short
summary of the conference: “it was fun.”