158,419 research outputs found
Convergence to stable laws in the space
We study the convergence of centered and normalized sums of i.i.d. random
elements of the space of c{{\'a}}dl{{\'a}}g functions endowed
with Skorohod's topology, to stable distributions in . Our
results are based on the concept of regular variation on metric spaces and on
point process convergence. We provide some applications, in particular to the
empirical process of the renewal-reward process
Nonlinear Attitude Control of Spacecraft with a captured asteroid
One of the main control challenges of National Aeronautics and Space Administration’s
proposed Asteroid Redirect Mission (ARM) is to stabilize and control the attitude of the
spacecraft-asteroid combination in the presence of large uncertainty in the physical model of
a captured asteroid. We present a new robust nonlinear tracking control law that guarantees
global exponential convergence of the system’s attitude trajectory to the desired attitude trajectory.
In the presence of modeling errors and disturbances, this control law is finite-gain
L_p stable and input-to-state stable. We also present a few extensions of this control law,
such as exponential tracking control on SO(3) and integral control, and show its relation to
the well-known tracking control law for Euler-Lagrangian systems. We show that the resultant
disturbance torques for control laws that use feed-forward cancellation is comparable to
the maximum control torque of the conceptual ARM spacecraft and such control laws are
therefore not suitable. We then numerically compare the performance of multiple viable attitude
control laws, including the robust nonlinear tracking control law, nonlinear adaptive
control, and derivative plus proportional-derivative linear control. We conclude that under
very small modeling uncertainties, which can be achieved using online system identification,
the robust nonlinear tracking control law that guarantees globally exponential convergence
to the fuel-optimal reference trajectory is the best strategy as it consumes the least amount
of fuel. On the other hand, in the presence of large modeling uncertainties and actuator saturations,
a simple derivative plus proportional-derivative (D+PD) control law is effective, and
the performance can be further improved by using the proposed nonlinear tracking control
law that tracks a (D+PD)-control-based desired attitude trajectory. We conclude this paper
with specific design guidelines for the ARM spacecraft for efficiently stabilizing a tumbling
asteroid and spacecraft combination
On the convergence of a shock capturing discontinuous Galerkin method for nonlinear hyperbolic systems of conservation laws
In this paper, we present a shock capturing discontinuous Galerkin (SC-DG)
method for nonlinear systems of conservation laws in several space dimensions
and analyze its stability and convergence. The scheme is realized as a
space-time formulation in terms of entropy variables using an entropy stable
numerical flux. While being similar to the method proposed in [14], our
approach is new in that we do not use streamline diffusion (SD) stabilization.
It is proved that an artificial-viscosity-based nonlinear shock capturing
mechanism is sufficient to ensure both entropy stability and entropy
consistency, and consequently we establish convergence to an entropy
measure-valued (emv) solution. The result is valid for general systems and
arbitrary order discontinuous Galerkin method.Comment: Comments: Affiliations added Comments: Numerical results added,
shortened proo
Convergence to a L\'evy process in the Skorohod and topologies for nonuniformly hyperbolic systems, including billiards with cusps
We prove convergence to a Levy process for a class of dispersing billiards
with cusps. For such examples, convergence to a stable law was proved by Jung &
Zhang. For the corresponding functional limit law, convergence is not possible
in the usual Skorohod J_1 topology. Our main results yield elementary geometric
conditions for convergence (i) in M_1, (ii) in M_2 but not M_1.
In general, we show for a large class of nonuniformly hyperbolic systems how
to deduce functional limit laws once convergence to the corresponding stable
law is known.Comment: accepted versio
Central Limit Theorem with Exchangeable Summands and Mixtures of Stable Laws as Limits
The problem of convergence in law of normed sums of exchangeable random
variables is examined. First, the problem is studied w.r.t. arrays of
exchangeable random variables, and the special role played by mixtures of
products of stable laws - as limits in law of normed sums in different rows of
the array - is emphasized. Necessary and sufficient conditions for convergence
to a specific form in the above class of measures are then given. Moreover,
sufficient conditions for convergence of sums in a single row are proved.
Finally, a potentially useful variant of the formulation of the results just
summarized is briefly sketched, a more complete study of it being deferred to a
future work
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