DIFFERENTIAL GEOMETRIC METHODS IN THE CONTROL OF PARTIAL
DIFFERENTIAL EQUATIONS
Lectures and Working Sessions were held in
Engineering Room 1 B 40
at the University of Colorado, Boulder.
Registration was in Engineering Room
133, and breaks were in the Engineering Lower Patio.
The infusion of Riemann geometric methods to obtain
Carleman inequalities and sharp continuous observability
inequalities for variable-coefficient PDEs
Abstract:
This talk shall attempt to motivate and justify the
exploratory character of the conference: that is, the
injection of differential-geometric methods into the
control of PDEs, traditionally analysis-dominated. Two main
areas may be singled out to make this point: (i) the
estabishment of certain a-priori trace inequalities, which
arise in the control of PDEs; (ii) the modeling and
analysis of such complicated objects as shell equations.
Continuous observability inequalities (C.O.I.) are
the dual version of exact controllability results.
By now classical (mid-80s) energy methods succeeded in
obtaining C.O.I. for canonical PDEs (waves, Schrödinger,
plate equations, etc.) with constant coefficients in the
principal part, and lower order terms only BELOW energy
level. We shall see how Riemann geometric methods allow one
to overcome both these two limitations and obtain sharp and
checkable C.O.I. One additional positive feature of these
Riemann geometric methods is that they may be viewed as a
far-reaching generalization of classical methods. Second
order hyperbolic equations and Schrödinger equations will
be discussed in this talk. [The subsequent talk by Yao will
focus on platelike equations and shell equations.]
Comparison with the literature will be made. Open questions
which involve geometry and PDE theories will be presented,
to stimulate interaction between these two groups.
How Riemannian and subanalytic geometry can help in
solving certain overdetermined systems of linear PDE
Abstract:
We will give an outline of the proof of the local solvability of a
class of overdetermined systems of first-order linear PDE with analytic
complex coefficients using new bounds on the lowest eigenvalue of the
Hodge-Laplacian and basic facts about subanalytic sets. The condition is
necessary and sufficient for the local solvability of the system. The speaker
will not assume any prior knowledge of the subject and leave aside the
technical aspects of the proof.
This is joint work with S. Chanillo.
An isoperimetric inequality and
the first Steklov Eigenvalue
Abstract:
Let (M, g) be a compact Riemannian manifold with boundary and
dimension n >= 2.
In this talk we discuss the first non-zero Steklov eigenvalue
problem: phi is harmonic, with the Neumann-type boundary condition
that the normal derivative is a constant nu_1 times phi. The positive scalar
nu_1 is called
the first Steklov eigenvalue. First we discuss the importance of this
problem in several areas of mathematics and then
we discuss upper and lower estimates of the eigenvalues nu_1 in terms of the
geometry of the manifold (M, g).
Some of the estimates I will discuss are:
(1) A sharp estimate for surfaces with non-negative Gaussian curvature
which says that nu_1 >= k_0 where
k_0 is the minimum of the geodesic curvature;
(2) An upper estimate for a convex manifold with non-negative Ricci
curvature which is given in terms of the first non-zero eigenvalue
for the Laplacian on the boundary;
(3) An estimate from below for a starshaped domain on a manifold
whose Ricci curvature is bounded from below; and
(4) A comparison theorem
for simply-connected domains in a simply-connected manifold. We exhibit
annular domains for which the comparison theorem fails to be true.
We introduced the
isoperimetric constant I(M) defined as
the infimum over open subsets U of M of the volume
of the boundary of U in the interior of M, divided by the smaller
of the volume of the boundary of U on the boundary of M and the
volume of its complement in the boundary of M.
I will discuss
a Cheeger-type inequality that involves the isoperimetric
constant I(M).
Finally, we will discuss upper and lower estimates for the constant I(M)
in terms of isoperimetric constants of the boundary of M.
Chord Uniqueness and Controllability:
the View from the Boundary
Abstract:
A chord is a geodesic of minimum length joining two points
of the boundary of a Riemannian manifold with boundary. We assume
that the boundary is locally strictly convex and that the manifold,
including its boundary, is compact. The existence of a chord joining
any two boundary points follows from well-known geometric arguments.
If each chord is unique, we show that every geodesic arising from an
interior point leaves the manifold after a distance which is at most
the maximum distance between boundary points. This has consequences
for boundary control of a natural hyperbolic problem with coefficients
independent of time.
This is joint work with Walter Littman.
Optimization, free boundary and symmetry
breaking problems for the Laplace operator
11:30 -- 11:45 am: Guangcao Ji,
Texas Tech University, Lubbock
Uniform Feedback Stabilization via Boundary Moments
of a 3-Dimensional Structural Acoustics Model
11:45 am -- 12:00 pm: George Avalos,
Texas Tech University, Lubbock
Point Observations of a Structural Acoustics
Model
The Bochner technique and observability
inequalities
Abstract:
Exact controllability leads to some observability inequalities.
The latter is sometimes easy and sometimes very difficult. The
Bochner technique describes a method initiated by S. Bochner about
fifty years ago for proving some identities of geometric interest.
This talk will report some recent progress in estimating observability
inequalities by this technique for plates and thin shells.
Exact observability of a hyperbolic equation of
2nd order in the Neumann case and its applications to inverse problems
Abstract:
We will prove the exact observability inequality (Lipschitz
stability in the lateral Cauchy problem) for a hyperbolic equation
of the second order with the zero Neumann boundary condition.
For the observability, we take Dirichlet data on a suitable subboundary.
The key is the multiplier method and a Carleman estimate which was
derived by Shishatskii (on the basis of integration by parts).
Furthermore we apply the observability to an inverse problem of
determining coefficients or right hand sides from lateral Cauchy data
on the suitable subboundary.
Boundary control theory and partial differential
equations
Abstract:
In this talk we discuss a strategy, proposed by the
speaker, to achieve exact controllability for evolution
equations. Unlike much of the literature, the
proposed strategy is direct, rather than being focused on
proving continuous observability inequalities, which then
by standard duality are equivalent to exact controllability
results. The talk will emphasize the two-way
interrelationship between PDE theories and boundary
control theories; and how each side contributes to enrich
the other.
11:30 -- 11:45 am: John Cagnol,
École des Mines, Paris
Intrinsic Geometric Model for the Vibration of a
Constrained Shell (joint work with J.-P. Zolesio)
11:45 am -- 12:00 pm: Thierry Horsin,
Université de Versailles, Saint-Quentin
Chess in Controlling the Burgers Equations
On the controllabillity and the stabilizability of
incompressible fluids
Abstract:
We survey recent results on the controllability and on the
stabilizability of the equations of incompressible fluids. Roughly speaking,
the problem of controllability we are considering is the following one:
given two divergence-free velocity vector fields on the domain occupied
by the incompressible fluid, can one pass from the first velocity vector
field to the second one by acting on the fluid in a suitable way on a
given part of the domain or of the boundary of the domain?
The situation is now rather well understood for the case of inviscid
incompressible fluids. We have partial results in the case of viscous
incompressible fluids. Concerning the stabilizability, for suitable
boundary controls, one has null global asymptotic stabilizability by
means of explicit feedback laws for the 2-D inviscid incompressible fluids,
even if the linearized control system around the null solution is not
asymptotically stabilizable.
Global Existence of Nonlinear Wave Equations
Outside Convex Obstacles
Abstract:
We prove global existence for certain nonlinear wave equations outside of
smooth convex obstacles in three space dimensions. The analysis combines
the conformal method of [1], [12] with relatively new
estimates for the linear wave equation, developed in [2],
[14], and [17]. These linear estimates are
generalizations of the null-form estimates of Klainerman and Machedon
[11] for the linear wave equation on Minkowski space.
This is joint work with H. Smith and C. Sogge. References:
[1] D. Christodoulou: Global solutions of nonlinear hyperbolic eq
uations for small initial data, Comm. Pure Appl. Math.39
(1986),
267--282.
[2] V. Georgiev and P. P. Schirmer: Global existence of low regularity
solutions of non-linear wave equations, Math. Z. 219
(1995), 1--19.
[11] M. Machedon and S. Klainerman: Space-time estimates for null forms
and the local existence theorem, Comm. Pure Appl. Math. 46
(1993),
1221--1268.
[12] R. Penrose: Conformal treatment of infinity, Relativity, Groupes
et Topologie (Lectures, Les Houches, 1963 Summer School of Theoret. Phys.,
Univ. Grenoble), pp. 565--584.
[14] H. Smith and C. D. Sogge: Null form estimates for (1/2, 1/2) symbols
and local existence for a quasilinear Dirichlet-wave equation, preprint.
[17] C. D. Sogge: On local existence for nonlinear wave equations
satisfying variable coefficient null conditions, Comm. Partial Differential
Equations 18 (1993), 1795--1821.
11:30 -- 11:45 am: Mary Ann Horn,
Vanderbilt University, Nashville, Tennessee
Implications of Sharp Trace Regularity Results on Boundary
Stabilization of the System of Linear Elasticity
11:45 am -- 12:00 noon: Matthias Eller,
Tennessee Tech University, Cookesville
A Uniqueness Theorem for a Thermoelastic System
Abstract:
We discuss the following problems:
1) existence and geometric properties of the envelope of holomorphy of a
real submanifold of C^2;
2) evolution of compact subsets of C^2; and
3) recent existence and regularity theorems for the the Levi equation.
Carleman estimates in the uniqueness of the
continuation, inverse problems, and optimal control
Abstract:
We consider Carleman type estimates (with an additional large
parameter) for classical second order operators and relate them to the
uniqueness of the continuation for these operators and also for Maxwell's,
elasticity, and thermoelasticity systems. We describe applications to
inverse problems and optimal control (exact and approximate controllability).
The results are due to Belishev, Bukhgeim, Klibanov, Nakamura, Tataru,
Yamamoto, and the speaker.
4:00 pm: Working Session
5:45 pm: Buses leave from Kittredge Commons to the top of
Flagstaff Mountain
Topological derivative for optimal control
problems
Abstract:
Assume that Omega is an open set of
R^N and that there is given a shape functional J, which defines
a real number J(Omega \ K)
for any compact subset K of the closure of Omega. We denote by
B(x), x in Omega, the ball of radius rho > 0:
B(x) = {y in R^N | |y-x| < rho};
I(rho) = J(Omega\closure of B(x)); and
assume that there exists the following limit, as rho tends to zero:
T_Omega(J)(y) = lim [d I(rho) / d( | B(y) | )] .
The topological derivative T_Omega(J)(y)
of the shape functional J(Omega)
is introduced in [sz97] in order to characterize the
infinitesimal variation of J(Omega) with respect to the
infinitesimal variation of the topology of the domain Omega.
In general, the topological derivative allows us to derive the new optimality
condition for the shape optimization problem:
J(Omega*) = inf {J(Omega\K) | K compact subset of Omega}.
The optimal domain Omega* is characterized by the first order condition
defined on the boundary of the optimal domain Omega*,
d J(Omega* ;V) >= 0, for all admissible vector fields V,
and by the following optimality condition defined in the interior
of the domain Omega*:
T_Omega*(J)(x) >= 0 for all x in Omega* .
We present the results [sz98a] on the application of the topological
derivative to the shape sensitivity analysis of optimal control problems.
The form of the topological derivative is derived for the optimal
value of the cost functional and
the results of computations for a model problem are provided. References:
[sz97] Sokolowski, J., Zochowski, A.
On topological derivative in shape optimization,
SIAM Journal on Control and Optimization.
Volume 37, Number 4, 1999, pp. 1251-1272.
[sz98a] Sokolowski, J., Zochowski, A.
Topological derivative for optimal control problems,
Proceedings of Fifth International Symposium on Methods and Models in
Automation and Robotics, Micedzyzdroje,
Poland, August, 1998, pp. 111-116; paper to appear in Control and
Cybernetics.
Uniform stabilizability of systems of PDE's
coupled at the interface of two regions
Abstract:
We consider problems related to uniform stabilization of
systems described by coupled PDE's of different type (hyperbolic-parabolic
like). The coupling takes place on the boundary (interface) of two
spatial regions, where the interface represents a vibrating flat
wall.
A prototype for this problem is a structural acoustic model. This is
described by the wave equation in a region (acoustic chamber), coupled
at the interface (boundary of the three-dimensional chamber) with a
plate equation. The wave equation describes the acoustic pressure in a
three-dimensional chamber, whereas the plate equation describes the
oscillations of a vibrating flat wall, which along with hard walls,
surrounds the chamber. Depending on the plate model considered, the
dynamics representing the vibrating wall may be described by an analytic
semigroup, in which case the coupling in the structure is of
hyperbolic-parabolic type.
The main goal is to establish uniform decay rates for the energy of the
overall system, subject to a minimal amount of a nonlinear damping placed
on the hard walls (boundary damping).
Two special cases (configurations) will be considered: 1. The (nonlinear) damping is placed only on the interface
(vibrating wall). 2. The (nonlinear) damping affects the "hard walls" only.
In the first case, the proof for uniform decay rates requires
convexity type conditions to be imposed on the "hard walls".
In the second case, instead, there is no need for any geometric
constraints, but the area subjected to damping is larger than in the
first case. This result is obtained by using recently developed "sharp"
trace estimates (in turn, achieved by microlocal analysis methods) for
solutions to wave and plate equations.
The problem described above leads to several open questions, in the theme
of the conference.
Is it possible to relax the convexity condition in problem 1, and obtain
the same result under less severe geometric constraints? This is a
differential geometry problem.
How can one generalize this result in order to account for more realistic
"curved" vibrating walls? This particular question lends itself directly
into shell theory, as the appropriate models for vibrating walls are
governed by shell equations. In this case, the corresponding analysis is
more subtle and several questionsrelated to geometry and microlocal analysis
need to be addressed for this new configuration.
11:30 am -- 12:00 noon:
Discussion Session: Future Directions
Presentation of open problems and directions for inquiry, over the coming
five years, by: Jean-Michel Coron,
University of Paris-Sud Markus Keel,
CalTech and Roberto Triggiani,
University of Virginia.