Stability of switched systems on the plane
We study the stability of the
origin of the dynamical system:
(1) \dot x(t)=u(t)Ax(t)+(1-u(t))Bx(t),
where A and B are two 2 x 2 real matrices with eigenvalues having
strictly negative real part, x \in R^2 and u(.) : [0,\infty[ \to [0,1] is
an arbitrary measurable control function. In particular we find a
necessary and sufficient condition on A and B under which the origin
of the system (1) is asymptotically stable for each function u(.).
The result is obtained studying the locus in
which the two vector fields Ax and Bx are collinear.
There are only
three relevant parameter: the first depends on the eigenvalues of A, the
second on the eigenvalues of B, the third contains the interrelation
between the two systems and it is the cross ratio of the four points in
the projective line CP^1 that correspond to the four eigenvectors
of A and B.
In the space of this parameters we study the shape and the convexity of
the stability
domain.
This bidimensional problem assumes particular
interest since systems of higher dimension can be reduced to our
situation.
Euler angles and time-optimal factorizations in SO(3)
We consider the following problem:
given two generators X, Y of the Lie algebra so(3) and a rotation
g \in SO(3) we are looking for a factorization
g=exp(t_1 A_1) ... exp(t_k A_k), with k \in N, t_i \in R, A_i \in {X,Y},
such that |t_1| + ... + |t_k| is minimal.
If X,Y are perpendicular (i.e., the corresponding axes are perpendicular),
then it is well known that every g \in SO(3) can be written as a product of
at most three factors, the parameters t_1, t_2, t_3 being nothing but
Euler angles.
This raises the following questions:
(1) Assume that X and Y are perpendicular and normalized.
Do Euler angles provide optimal factorizations?
(2) If not, which factorizations are optimal and how many factors
are needed?
(3) What is the situation like for SU(2)?
(4) What happens if one considers generators which are not
perpendicular to each other?
Considering the factorization problem as a control problem on a Lie group
we will obtain detailed answers to (1)--(3). In case of (4) we obtain
results suggesting a quite natural conjecture.
Quantized Systems
The aim of this talk is to introduce a new class of control
systems called quantized systems. These are precisely discrete systems
with a finite number of controls. Explicit application examples motivates
the interest in these systems. We will provide some basic definitions and
some results for the linear case and for some other particular cases.
Moreover, open research problems will be discussed.
Controllability of solvable Lie algebras
Let G be a real connected Lie group and L its Lie algebra (the set of
all right-invariant vector fields on G).
For right-invariant control systems on G
(1) dx/dt = A(x) + u B(x), x \in G, u \in R,
where A, B \in L, we are interested in the (global) cotrollability property:
when any pair of points in G can be connected by a trajectory of
system (1)?
This is equivalent to the following property: the subsemigroup of G
generated by the set
exp(A + R B) coincides with the Lie group G.
It turns out that for solvable simply connected Lie groups G the property of
controllability of systems (1) depends primarily not on vector fields A,
B, but on the Lie group G, i.e., on the Lie algebra L. We call a Lie
algebra L controllable if it contains elements A, B such that the
corresponding system (1) is controllable.
The talk will be devoted to the following questions:
Description of controllable solvable Lie algebras (a recent work of
D.Mittenhuber),
A complete list of low-dimensional controllable Lie algebras (up to dim 6).
Singular Lagrangian manifolds in Sub-Riemannian geometry
The construction of the ball of small radius in Sub-Riemannian geometry
shows the importance of Lagrangian stratifications when there exists
abnormal geodesics. We discuss this stratification using the example
of Martinet SR geometry. We show that such a stratification is also related
to the computations of limit cycles in 16-th Hilbert problem.
Car towing several trailers and Goursat distributions
In the first half of the 90's, through the work of Laumond,
Sordalen, Risler, Jean and others it had become clear that the kinematic
model of a car drawing a given number l of attached trailers represents
a rank--2 distribution D (on the configuration space
\Sigma = R^2 \times (S^1)^{l+1}) having the property that the Lie
square [D, D] of D is a rank--3 distribution, the Lie square of [D, D]
is a rank--4 distribution, and so on.
Distributions satisfying this condition are called Goursat.
Every Goursat distribution around a generic point behaves (locally)
in a unique way best visualised by chained model known in control
theory. For trailer systems this behaviour happens around positions where
NONE of the angles between: the car and 1st trailer, the 1st and 2nd
trailer, ... , the (l-2)th and (l-1)th, is right. All other positions
in \Sigma are singular. I will speak about:
1) a fundamental stratification of \Sigma by Jean '96 separating
different singular behaviours of the system and drawing attention to specific
values of the angles. This includes the computation of the growth vector at
a point, hence -- also the nonholonomy degree of any given position of the
system.
2) how Jean's classification prompts a basic geometric classification
(if only rough; the local classification problem for Goursat objects is
a lot more hard) of singularities of Goursat objects.
3) whether the car with l trailers is a universal model for local
behaviours of Goursat distributions on manifolds of dimension l+3
(i.e., whether any Goursat germ is equivalent to the kinematic model
at certain point of \Sigma).
For l \le 5 the affirmative answer was known to me
in '97. Pasillas--Lepine informed me in '98 that a general answer
was also yes. In '99 this result was elegantly shown by Montgomery &
Zhitomirskii in their big work on Goursat. It is a byproduct of performing
-- over Goursat distributions -- so-called Cartan's prolongations as
described by Bryant & Hsu in their '93 paper. I will recall that
construction and explain the resulting universality of the trailer
systems.
4) [time permitting; briefly and only giving references] which
evolutions of the trailer systems are locally C^1--rigid.
Euler-Lagrange and Hamiltonian formalisms
in variational problems involving differential inclusions
Analysis of variational problems involving differential inclusions
dx/dt \in F(t, x)
is inevitably connected with nonsmoothness unless the set-valued mapping F
admits a smooth parametrization. Still it appears possible to develop a theory
of necessary conditions for such problems containing the corresponding
classical results and, moreover, extending the Hamiltonian formalism of
calculus of variations to such problems. In the talk we mainly discuss the
following questions:
(a) what is the analogue of the Euler-Lagrange equation for variational
problems with differential inclusions?
(b) what is the analogue of the Hamiltonian necessary condition for such
problems?
(c) how Euler-Lagrange and Hamiltonian conditions are connected?
We shall also discuss (if time permits) extensions for differential
inclusions involving partial differential operators and/or some problems
which remain open.
We are concerned with the problem of
constructing discontinuous state feedbacks that
asymptotically stabilize a nonlinear
control system.
We will first review the basic theory developed in the
past few years.
Next, we shall introduce a family of discontinuous,
piecewise smooth vector fields and
a class of ``patchy feedbacks''
which are obtained by patching together a locally finite
family of smooth controls.
Relying on the main properties of this class of feedback laws
we will show that,
if a system is asymptotically controllable at the origin, then
it can be stabilized by a piecewise constant
patchy feedback control.
Robustness properties of such feedback with respect to
external and internal perturbations will be also investigated.
Given a nonlinear system, we have already studied (previous lecture) the
observable cases where the state->output map is regular (they are of two
types, according to the fact that the number of observations is larger or
smaller than the number of controls). Now, we study the case where this
state->output map is degenerate.
As in the theory of differentiable functions, it looks that the right
notion of a degenerate mapping is related to the notion of a "finite
mapping".
In all situations (the degenerate and the non degenerate one), we can
construct nonlinear asymptotic observers, with prescribed arbitrary
exponential convergence, and we can use them to stabilise asymptotically
the system via dynamic output feedback.
The class of curves in the plane that L. Euler called elasticae in 1744 admit natural generalizations to arbitrary Riemannian manifolds M as the projections of solution curves of certain variational problems on the ortonormal frame bundle of M. For manifolds in which the frame bundle coincides with the isometry group, the elastic curves show remarkable connections with the movements of multi-dimensional mechanical tops. These, somewhat mysterious connections between tops, elastic curves and problems of optimal control constitute a major theme of the talk.
We consider two problems in planning and controlling a class of
vehicles along planar routes. The vehicle is supposed to move forward
only with a given velocity profile, and to have bounds on its turning
radius. This model, sometimes referred to as the ``Dubins' vehicle'',
is relevant to the kinematics of road vehicles as well as aircraft
cruising at constant altitude, or sea vessels.
We consider first the optimal control problem consisting in minimizing the
length travelled by the vehicle starting from a generic configuration
to connect to the specified route. A feedback law is proposed, such
that straight routes can be approached optimally, while system is
asymptotically stabilized. A generalization to curved paths is also
described.
In the second part, we will consider the problem of optimal
coordinated conflict management scheme for a simplified model of
traffic, where several Dubins' vehicles share their operational space
with possibly conflicting requirements on minimization of path length.
A technique to minimize total fuel consumption while guaranteeing
safety against collisions is described, in the hypothesis of full
cooperation among agents. A decentralized implementation of such a
scheme is then introduced, and its features as a hybrid control scheme
are described. Finally, we discuss the tradeoff between performance
and fault tolerance that goes with decentralization, and assess it by
extensive simulation trials. Links with work at U.C. Berkeley on
optimization under a non--cooperative assumption will be considered.
We study the questions of stability and asymptotic stability for
time-varying systems described by ODE's with the right-hand sides of
the form $f(\omega t, x)$, where $f(t,x)$ is 1-periodic with respect
to $t$. The systems of this type draw much attention for a number of
reasons. Since the discovery of stabilizing effect of vibration in the
reverse pendulum example, there was much study regarding utilization
of time varying feedback laws for stabilization of the systems which
are not stabilizable by time-invariant feedback. Systems 'with a
deficit of control' or control systems for nonholonomic mechanical
objects (e.g. mobile robots) are natural examples of this kind.
Many questions related to the issue of stability are not yet answered
even for the linear time-varying systems. Two main approaches to the
stability issue - the one based on Lyapunov functions and another one
based on averaging - both encounter difficulties when dealing with the
problem of fast oscillations. The Lyapunov functions must be very
complicated in order to decrease along the trajectories generated by
fast-oscillating controls while the standard averaging procedures
result in systems with critical equilibria.
Our approach is kind of high-order averaging procedure which is based
and makes use of the tools of chronological calculus developed by
A.A.Agrachev and R.V.Gamkrelidze in 70's-80's. The method is
invariant - the high-order averagings are computed via Lie algebraic
brackets of vector fields (right-hand sides).
We apply the high-order averaging to study stability issues for both
linear and nonlinear systems. In particular we derive conditions of
stability for the second and third order linear differential equations
with periodic fast-oscillating coefficients, study output-feedback
stabilization of bilinear systems, consider averaging procedures for
nonlinear systems under homogeneity assumptions, study the problem of
stabilization of nonholonomic (control-linear) systems by means of
time-varying feedbacks.
A sub-Riemannian structure on a manifold $M$, dim$(M)=n$, is given by
a distribution $\Delta$ of $m$-planes, $m < n $ , together with a metric.
Admissible curves will be absolutely continuous curves having
a velocity almost everywhere tangent to the distribution.
The metric on $\Delta$ allows us to define length of admissible curves
and
the sub-Riemannian distance between two points is the infimum of
length of admissible curves joining those two points.
If the distribution and the metric are given by $m$ orthonormal vector
fields it turns out to be a problem of control optimal :
to minimize $\displaystyle \int_0^1 \sum_{i=1}^m u_i^2$ among all $u$
satisfying
$\gamma (0)=x_0$, $\gamma(1)=x_1$, where $\gamma$ is a solution of
$$\dot{\gamma}= \sum_{i=1}^m u_i X_i (\gamma).$$
We fix $x_0$ in $M$ and let $f$ be the function defined on $M$
by $x \mapsto d(x_0 ,x)$ where $d$ is the sub-Riemannian distance. At
the contrary of the Riemannian case, we know that $f$ fails to be
$C^1$ on any neighborhood of $x_0$.
We will give some results about the regularity of $f$, in particular
compare the singular support of $f$ with the cut locus and the set of
point reached from $x_0$ by abnormal minimizers.
For a generic single input planar control system
\dot x=F(x)+uG(x) (x belonging to R^2, F,G \in {\cal C}^3( R^2,R^2),
F(0)=0, |u| <= 1) we are concerning with the problem of
stabilizing the origin in minimum time.
We compute the extremal synthesis in the cotangent bundle
and we classifies the singularities of its projection on R^2.
Moreover we show that the extremal locus is a stratified set
in R^4 and we give a condition under which it is in fact a
piecewise--{\cal C}^1 manifold with boundary.
The most delicate trajectories are the Abnormal Extremals (i.e. the
trajectories such that the corresponding PMP--Hamiltonian vanishes). We
show that these trajectories can be classified through "words"
belonging to a language recognizable by an Automaton.
In this talk I want to report about the work which I do now with Prof
A. Agrachev.
Using Hamiltonian approach, one can reduce the problem
of finding feedback invariants of smooth control systems to the
problem of finding symplectic invariants of curve in Lagrangian
Grassmannian (the set of all Lagrangian subspaces of some symplectic
space).
For the problems with nonholonomic constrains, a special
class of curves in Lagrangian Grassmannian is obtained in this way. We
develope the differential geometry of these curves for $4$-dimensional
symplectic space ( the first nontrivial case): we introduce canonical
moving frame, find its structural equation, and obtain complete system
of invariants (so-called, metric and projective curvature).
One can apply the theory for developing differential geometry of some
concrete problems with nonholonomic constrains. As an example we take
contact structure of $3$- dimensional manifold provided with
sub-Riemannian metric and find connection and curvature tensor for
this problem. We are also able to write the equation for conjugate
points along the extremals of this problem and to obtain comparison
theorems.