219 research outputs found
Time-Optimal Trajectories of Generic Control-Affine Systems Have at Worst Iterated Fuller Singularities
We consider in this paper the regularity problem for time-optimal
trajectories of a single-input control-affine system on a n-dimensional
manifold. We prove that, under generic conditions on the drift and the
controlled vector field, any control u associated with an optimal trajectory is
smooth out of a countable set of times. More precisely, there exists an integer
K, only depending on the dimension n, such that the non-smoothness set of u is
made of isolated points, accumulations of isolated points, and so on up to K-th
order iterated accumulations
Converse Lyapunov Theorems for Switched Systems in Banach and Hilbert Spaces
We consider switched systems on Banach and Hilbert spaces governed by
strongly continuous one-parameter semigroups of linear evolution operators. We
provide necessary and sufficient conditions for their global exponential
stability, uniform with respect to the switching signal, in terms of the
existence of a Lyapunov function common to all modes
Pliability, or the whitney extension theorem for curves in carnot groups
The Whitney extension theorem is a classical result in analysis giving a
necessary and sufficient condition for a function defined on a closed set to be
extendable to the whole space with a given class of regularity. It has been
adapted to several settings, among which the one of Carnot groups. However, the
target space has generally been assumed to be equal to R^d for some d 1.
We focus here on the extendability problem for general ordered pairs
(G\_1,G\_2) (with G\_2 non-Abelian). We analyze in particular the case G\_1 = R
and characterize the groups G\_2 for which the Whitney extension property
holds, in terms of a newly introduced notion that we call pliability.
Pliability happens to be related to rigidity as defined by Bryant an Hsu. We
exploit this relation in order to provide examples of non-pliable Carnot
groups, that is, Carnot groups so that the Whitney extension property does not
hold. We use geometric control theory results on the accessibility of control
affine systems in order to test the pliability of a Carnot group. In
particular, we recover some recent results by Le Donne, Speight and Zimmermann
about Lusin approximation in Carnot groups of step 2 and Whitney extension in
Heisenberg groups. We extend such results to all pliable Carnot groups, and we
show that the latter may be of arbitrarily large step
On the stabilization of persistently excited linear systems
We consider control systems of the type , where
, is a controllable pair and is an unknown
time-varying signal with values in satisfying a persistent excitation
condition i.e., \int_t^{t+T}\al(s)ds\geq \mu for every , with
independent on . We prove that such a system is stabilizable
with a linear feedback depending only on the pair if the eigenvalues
of have non-positive real part. We also show that stabilizability does not
hold for arbitrary matrices . Moreover, the question of whether the system
can be stabilized or not with an arbitrarily large rate of convergence gives
rise to a bifurcation phenomenon in dependence of the parameter
On the controllability of quantum transport in an electronic nanostructure
We investigate the controllability of quantum electrons trapped in a
two-dimensional device, typically a MOS field-effect transistor. The problem is
modeled by the Schr\"odinger equation in a bounded domain coupled to the
Poisson equation for the electrical potential. The controller acts on the
system through the boundary condition on the potential, on a part of the
boundary modeling the gate. We prove that, generically with respect to the
shape of the domain and boundary conditions on the gate, the device is
controllable. We also consider control properties of a more realistic nonlinear
version of the device, taking into account the self-consistent electrostatic
Poisson potential
Growth rates for persistently excited linear systems
We consider a family of linear control systems where
belongs to a given class of persistently exciting signals. We seek
maximal -uniform stabilisation and destabilisation by means of linear
feedbacks . We extend previous results obtained for bidimensional
single-input linear control systems to the general case as follows: if the pair
verifies a certain Lie bracket generating condition, then the maximal
rate of convergence of is equal to the maximal rate of divergence of
. We also provide more precise results in the general single-input
case, where the above result is obtained under the sole assumption of
controllability of the pair
A characterization of switched linear control systems with finite L 2 -gain
Motivated by an open problem posed by J.P. Hespanha, we extend the notion of
Barabanov norm and extremal trajectory to classes of switching signals that are
not closed under concatenation. We use these tools to prove that the finiteness
of the L2-gain is equivalent, for a large set of switched linear control
systems, to the condition that the generalized spectral radius associated with
any minimal realization of the original switched system is smaller than one
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