23,179 research outputs found
High-Order Control Variations and Small-Time Local Controllability
The importance of “control variations” for obtaining local approximations
of the reachable set of nonlinear control systems is well known.
Heuristically, if one can construct control variations in all possible directions,
then the considered control system is small-time locally controllable
(STLC). Two concepts of control variations of higher order are introduced
for the case of smooth control systems. The relation between these variations
and the small-time local controllability is studied and a new sufficient
STLC condition is proved.* This work is partly supported by the Bulgarian Ministry of Science and Higher Education
– National Fund for Science Research under contract DO 02–359/2008
Configuration Controllability of Simple Mechanical Control Systems
In this paper we present a definition of 'configuration controllability' for mechanical systems whose Lagrangian is kinetic energy with respect to a Riemannian metric minus potential energy. A computable test for this new version of controllability is derived. This condition involves an object that we call the symmetric product. Of particular interest is a definition of 'equilibrium controllability' for which we are able to derive computable sufficient conditions. Examples illustrate the theory
Some results on second order controllability conditions
For a symmetric system, we want to study the problem of crossing an
hypersurface in the neighborhood of a given point, when we suppose that all of
the available vector fields are tangent to the hypersurface at the point.
Classically one requires transversality of at least one Lie bracket generated
by two available vector fields. However such condition does not take into
account neither the geometry of the hypersurface nor the practical fact that in
order to realize the direction of a Lie bracket one needs three switches among
the vector fields in a short time. We find a new sufficient condition that
requires a symmetric matrix to have a negative eigenvalue. This sufficient
condition, which contains either the case of a transversal Lie bracket and the
case of a favorable geometry of the hypersurface, is thus weaker than the
classical one and easy to check. Moreover it is constructive since it provides
the controls for the vector fields to be used and produces a trajectory with at
most one switch to reach the goal
Local controllability of 1D Schr\"odinger equations with bilinear control and minimal time
We consider a linear Schr\"odinger equation, on a bounded interval, with
bilinear control.
Beauchard and Laurent proved that, under an appropriate non degeneracy
assumption, this system is controllable, locally around the ground state, in
arbitrary time. Coron proved that a positive minimal time is required for this
controllability, on a particular degenerate example.
In this article, we propose a general context for the local controllability
to hold in large time, but not in small time. The existence of a positive
minimal time is closely related to the behaviour of the second order term, in
the power series expansion of the solution
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