451 research outputs found
Quasi-optimal robust stabilization of control systems
In this paper, we investigate the problem of semi-global minimal time robust
stabilization of analytic control systems with controls entering linearly, by
means of a hybrid state feedback law. It is shown that, in the absence of
minimal time singular trajectories, the solutions of the closed-loop system
converge to the origin in quasi minimal time (for a given bound on the
controller) with a robustness property with respect to small measurement noise,
external disturbances and actuator noise
On the stabilization problem for nonholonomic distributions
Let be a smooth connected and complete manifold of dimension , and
be a smooth nonholonomic distribution of rank on . We
prove that, if there exists a smooth Riemannian metric on for which no
nontrivial singular path is minimizing, then there exists a smooth repulsive
stabilizing section of on . Moreover, in dimension three, the
assumption of the absence of singular minimizing horizontal paths can be
dropped in the Martinet case. The proofs are based on the study, using specific
results of nonsmooth analysis, of an optimal control problem of Bolza type, for
which we prove that the corresponding value function is semiconcave and is a
viscosity solution of a Hamilton-Jacobi equation, and establish fine properties
of optimal trajectories.Comment: accept\'e pour publication dans J. Eur. Math. Soc. (2007), \`a
para\^itre, 29 page
Convergence to consensus of the general finite-dimensional Cucker-Smale model with time-varying delays
We consider the celebrated Cucker-Smale model in finite dimension, modelling
interacting collective dynamics and their possible evolution to consensus. The
objective of this paper is to study the effect of time delays in the general
model. By a Lyapunov functional approach, we provide convergence results to
consensus for symmetric as well as nonsymmetric communication weights under
some structural conditions
The turnpike property in finite-dimensional nonlinear optimal control
Turnpike properties have been established long time ago in finite-dimensional
optimal control problems arising in econometry. They refer to the fact that,
under quite general assumptions, the optimal solutions of a given optimal
control problem settled in large time consist approximately of three pieces,
the first and the last of which being transient short-time arcs, and the middle
piece being a long-time arc staying exponentially close to the optimal
steady-state solution of an associated static optimal control problem. We
provide in this paper a general version of a turnpike theorem, valuable for
nonlinear dynamics without any specific assumption, and for very general
terminal conditions. Not only the optimal trajectory is shown to remain
exponentially close to a steady-state, but also the corresponding adjoint
vector of the Pontryagin maximum principle. The exponential closedness is
quantified with the use of appropriate normal forms of Riccati equations. We
show then how the property on the adjoint vector can be adequately used in
order to initialize successfully a numerical direct method, or a shooting
method. In particular, we provide an appropriate variant of the usual shooting
method in which we initialize the adjoint vector, not at the initial time, but
at the middle of the trajectory
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