Seminar on Geometric Control Theory and its Applications
Academic year 2001 - 2002
- Wednesday 5 June, 2002:
V.M.Zakalyukin, Moscow State University
Boundary singularities and Envelopes of Wave fronts
In many problems of the thery of differential equations and optimal control
the stadard A,D,E singularities of wave fronts and there metamorphoses
with the course of time arise. However the equivalence group used here
is the group of diffeomorphisms of the space-time fibered over
the time axis. Such diffeomorphisms do not preserve the envelopes
of fronts.
A natural question of the study of the singularities of an evolving front with respec
t to the group of diffeomorphisms of the space-time fibered over
the space was essentially investigated in the singularity theory in other
contexts (for example, they are related to the singularities of mappings of
the space, containing the discriminant of a simple singularity,
to the singularities of bicaustics).
We describes the classfication of the envelopes of families
of wavefronts in an explicit form. The list up to the dimension three
coincides with the list of discriminants of simple projections
of curves. We describe various connections with the other
classification problems in geometry.
For example with the singularities of families of chords in
affine geometry.
We discuss a simple example of time-optimal control problem
where a singularity from the list appears.
- Wednesday 8 May, 2002:
Ayse Kara (ICTP)
Observability of General Linear Pairs
- Tuesday 23 April, 2002 at 17:30 in room D:
V. Capasso, Milano
Collective systems with spatial structure (ant systems)
- Wednesday 11 and 18 April, 2002:
U. Boscain, SISSA
K+P Problem for a Three-level Quantum System
We apply techniques of subriemannian geometry on Lie groups to
laser-induced population transfer in a three-level quantum system.
The aim is to induce transitions by two laser pulses, of arbitrary shape
and frequency, minimizing the pulse energy. We prove that the
Hamiltonian system given by the Pontryagin Maximum Principle is completely
integrable, since this problem can be stated as a ``K+P problem''
on a semi-simple Lie group. Optimal trajectories and controls are
exhausted. The main result is that optimal controls correspond to lasers
that are ``in resonance''. The ``K+P problem'' is discussed in detail.
- Wednesday 20 March, 2002:
A. Marigo, CNR
Normal forms for Free nilpotent Lie Algebras.
Local properties of a control system
\[
\dot x=f_0(x)+\sum_{j=1}^d u_jf_j(x)
\]
are completely determined by the structure of the Lie Algebra
$\Ggi$ generated by the vector fields $f_0,f_1,\ldots, f_d$.
Of particular importance, from applications point of view, are nilpotent system.
This fact motivates the effort of the research in approximating a general system with
a nilpotent one ("nilpotent approximation"). Moreover it is fundamental for an
approximating nilpotent system to have a "simple" and "compact" form, i.e. a canonical
form.
In this work we consider the class of abstract nilpotent Lie Algebras freely generated
by a set of $d$ elements.
We determine those nilpotent Lie Algebras stable under tranformation of the generators
and build the relative multiplication tables.
Finally we go back to nilpotent Lie Algebras of vector fields and, for them we provide
a canonical coordinate representation.
- Wensday March 6, 2002:
I. Zelenko, Technion
Fundamental form of rank 2 vector distribution
Rank k vector distribution D on the manifold M is by definition a
k-dimensional subbundle of the tangent bundle TM.
In other words, for each point q in M a
k-dimensional subspace D(q) of the tangent space T_qM is chosen and
D(q) depends smoothly on q. Two vector distributions D_1 and D_2 are
called locally equivalent at some point q_0 in M , if there exists a
diffeomorphism F of some neighbourhood U of q_0, which transforms
distribution D_1 to D_2, i.e., F_*D_1(q)=D_2 (F(q)) for all q in U. The
question is when two distributions are locally equivalent?
In the present talk we will restrict ourselves to the case k=2.
If dim M=3 or 4, all generic germs of rank 2 distributions are equivalent
(Darboux's and Engel's theorems). If dim M >= 5, the normal forms of these
germs contain (dim M-4) functional parameters. Using the general theory
of curves in the Lagrange Grassmannian developed in [1], we will describe
the construction of
basic invariant of the distribution, fundamental form, which is
the obstacle to local equivalence. The case dim M=5 will be discussed
in greater details.
References:
[1] A. Agrachev, I. Zelenko, Geometry of Jacobi
curves.I, J. Dynamical and Control Systems, 8(1):93-140, 2002
- Thursday January 31, 2002:
M. Sigalotti, SISSA
On the local structure of control functions corresponding to time-optimal
trajectories in R^3
In the talk it is analized the structure of a control function u(t)
corresponding to an optimal trajectory for the system \dot q =f(q)+u g(q)
in R^3 nearby a point where some nondegenericity conditions are satisfied.
As we will see, the control turns out to be the concatenation of some
bang and some singular arcs. Studying the index of the second variation of
the switching times, the number of such arcs is bounded by six.
- Thursday January 24, 2002:
C. Altafini, SISSA
A few issues in the control of quantum mechanical systems
In this survey talk I will review the setting commonly used for control of closed qua
ntum mechanical systems driven by external fields (i.e. bilinear control systems on sph
eres and compact semisimple Lie groups) and present a few of the methods currently in u
se for control design in this setting.
- December 12, 2001:
F. Aicardi
Classification of singularities of convex envelopes of apparent contours
The apparent contour of a surface S in R^3, is the set of critical
values%/1iso8859-15 of the projection of S on a plane.
The apparent contour is a collection of plane 'fronts', i.e. closed curves
with double points and cusps. The convex envelope%/1iso8859-15 of an apparent contour is
the boundary of its convex hull.
We classify all singularities up to codimension 3, i.e. all singularities in
generic 3-parametric families of convex envelopes of apparent contours.
The study was motivated by phase transitions in thermodynamics but the
problem is also intimately related to plane nonlinear optimal control
problems.
- December 5, 2001:
Andei Agrachev, SISSA,
On the curvature of Hamiltonian systems and second order ODEs
The goal of the talk is to present a construction of the canonical
connection and the curvature (generalizing Levi Civita connection and
Riemannian sectional curvature) for a wide class of Hamiltonian systems
and other vector fields on cotangent and tangent bundles. The construction
is purely "dynamical": it involves only the flow generated by the vector
field and the fiber bundle structure. The correspondent curvature enjoys
properties similar to the Riemannian case, i.e. the case of the geodesic
flow on the tangent bundle. In particular, it allows to state very general
"Comparizon theorems".
No preliminary knowledge in Riemannian geometry is required.
- November 28, 2001:
Andrey Sarychev, University of Aveiro, Portugal
Extremal and Optimal Controls for Chained Systems
We consider optimal control problems for 2-chained systems
- a class of systems which arise naturally in applications
(e.g. in robotics). Applying Pontryagin Maximum Principle
and some other tools of optimal control theory we study
structural properties of extremal and optimal trajectories
for these problems.
- November 14, 2001 at 16:30:
Natalia Chtcherbakova, SISSA,
On the configuration study of the "falling-cat" problem
In this talk there will be discussed some aspects of so-called "falling cat" problem.
Let us consider a configurational space of the mechanical system subject to the
condition that the angular momentum of the system is zero.The problem is to reach the d
esired position of the system by changing its inner configuration without violating the
condition for the angular momentum.
The first part of the talk is concerned to the purely kinematic study of the co
nfiguration space of the system consisted of n free particles in R^d. The result will b
e presented is due to A. Guichardet. In his article [1] he has shown that for any syste
m such that the number of the
particles is strictly bigger than the dimension if the space the result of any pure rot
ation of the initial
configuration can be achieved by performing the inner configuration of the particles.
For the case n <= d the particles will always remain in some less-dimensional constant
subspace of R^d.
In the second part of the talk there will be considered the system consisted of four eq
ual
masses such that they form a break-line of three equal segments with the fixed length.
As before, the total angular momentum of the system is supposed to be zero.
There will be discussed the study of the configurational space of this system in R^3.
References
[1] Guichardet,A. 1984. On rotation and vibration motions of molecules.
Ann. Inst. H. Poincare Phys. Theor. 40(3): 329-342.
- November 7, 2001:
F. Ancona, University of Bologna,
Stabilization by Patchy Feedbacks and Robustness Properties
This talk is concerned with the problem of
constructing discontinuous stabilizing
feedbacks for nonlinear
control system, which enjoy robusteness properties
with respect to external and internal perturbations.
We first consider ``patchy'' vector fields,
a class of discontinuous, piecewise smooth
vector fields introduced in [1],
and we prove the stability of the corresponding
solution set with respect to impulsive perturbations.
A linear estimate of the effect produced by such perturbations
is also established for a generic class of patchy vector fields
in the plane, that admit discontinuities across polygonal lines.
Next, we apply these results to derive robusteness properties
with respect to
both (internal) measurement errors and persistent external disturbances
for ``patchy feedbacks'':
a class of feedback laws that generate patchy vector fields.
References
[1] F. Ancona and A. Bressan,
Patchy vector fields and asymptotic stabilization.
ESAIM - Control, Optimiz. Calc. Var.
Vol. 4, (1999), pp. 445-471.
[2] F.H. Clarke, Yu.S. Ledyaev, L. Rifford,
R.J. Stern,
Feedback stabilization and Lyapunov functions,
SIAM J. Control Optim., 39, (2000), no. 1, pp. 25-48.
[3] E.D. Sontag,
Stability and stabilization: discontinuities and the
effect of disturbances, in Proc. NATO Advanced Study Institute -
Nonlinear Analysis, Differential Equations, and Control,
(Montreal, Jul/Aug 1998), F.H. Clarke and R.J. Stern eds.,
Kluwer, (1999), pp. 551-598.
- October 24, 2001:
B. Piccoli, University of Salerno and SISSA,
Cooperative and safety controls for Dubin's car
We illustrate the use of cooperative controls to produce a
safety control for a Dubin's car with non expert driver.
The mathematical formulation of the problem leads to consider
discontinuous ODEs and optimal synthesis. The concept of solution plays a
key role in solving the problem and in numerical simulations.
-
October 16, 2001:
A. A. Davydov,
Vladimir State University, Russia
Local controllability of generic control systems near singular points
For a control system a point $P$ of the boundary $B$ of set of points with
small
time local transitivity property is called $s$-{\it singular}, with $s \in N$,
if there exists a support plane to the velocity indicatrix at this point,
passing trough the zero velocity, having exactly $s$ points in common with
$I(P)$ and lying in the tangent space to the boundary $B$ at the point $P$.
At a singular point a generic control system sometimes have and sometimes have
not (small time) local transitivity property. Results provided some necessary
and some sufficient conditions for (small time) local transitivity property of
a
generic system at its singular point are presented
-
October 10, 2001:
U. Serres,
SISSA-ISAS
Curvature of 2-dimensional Control Systems and Zermelo's Navigation Problem
In Riemannian geometry the Gaussian curvature of a manifold reflects intrinsic properti
es of the geodesic flow.
For example, the geodesics of the surface have no conjugate points if the curvature is
non-positive.
Indeed, these geodesics are just extremals of a the time-optimal control problem which
dynamics are given by q'(t) = u_1X_1(q)+u_2X_2(q), with u_1^2+u_2^2=1, where {X_1,X_2
} form an orthonormal basis of the Riemannian metric.
Our aim is first to generalize the Gaussian curvature to 2 - dimensional optimal contro
l problems
and see that its ``control" analogue reflects similar properties; second to apply the o
btained results to Zermelo's navigation problem.
-
October 3, 2001 at 14:30:
M. Zelikin, Moscow State University
The Structure of Fibre Bundles in Optimal Chattering Syntheses
The term chattering control means the optimal control with countable
number of switches on a finite time interval. The theory of chattering
control is gathering force as a separate branch of the geometrical
optimal control theory. Up to now the main attention was paid to
control systems with single input. The multiinput control problems
with chattering arcs are as yet little understood. It was found an
explicite analitic expression for some optimal trajectories of
linear-quadratic problems having control in the unit disk. The part of
switching accumulation points play here points of discontinuity of the
second kind of the control function.
-
October 3, 2001 at 16:30:
A.S. Demidov, Moscow State University
The Inverse Problem for the Grad-Shafranov Equation with an
Affine Right-hand Side
Consider a possibility of an identification real parametres $a$ and $b$ in the
equation
(1) \Delta u=au+b\ge 0 in \Omega \subset \R^2
for a function
$u\in C^2(\overline\Omega)$ which satisfies the boundary conditions:
(2.a) u|_{\partial\Omega}=0
{2.b) \frac{\partial u}{\partial \nu} |_{\partial\Omega}=\Phi
with some fixed function $\Phi\in C^1(\partial\Omega)$.
It is obvious that $a\ge -\lambda_1$, where $\lambda_1=\lambda_1(\Omega)$
is first eigenvalue of Dirichlet problem for Laplace operator.
Moreover, $b=b(a,\Phi)\ge 0$, and also $b=0\iff a=-\lambda_1$.
It's clear that $\Phi =$ constant if the domain $\Omega$ is a disk $D$.
So (in the case: $\Omega = D$) the problem (1)-(2.a,b) has a solution
(expressed by Bessel functions) for any
$a\ge-\lambda_1$ and corresponding $b=b(a,\Phi)$.
Theorem.
Assume that $\Omega\ne D$, $\partial\Omega\in C^{3,\lambda}$.
Then there exist at most finitely many different pairs $(a,b)$
satisfying the conditions (1)-(2.a,b).