Search results for "PROB"
showing 10 items of 8859 documents
Asymptotics of accessibility sets along an abnormal trajectory
2001
We describe precisely, under generic conditions, the contact of the accessibility set at time $T$ with an abnormal direction, first for a single-input affine control system with constraint on the control, and then as an application for a sub-Riemannian system of rank 2. As a consequence we obtain in sub-Riemannian geometry a new splitting-up of the sphere near an abnormal minimizer $\gamma$ into two sectors, bordered by the first Pontryagin's cone along $\gamma$, called the $\xLinfty$-sector and the $\xLtwo$-sector. Moreover we find again necessary and sufficient conditions of optimality of an abnormal trajectory for such systems, for any optimization problem.
On local optima in minimum time control of the restricted three-body problem
2016
International audience; The structure of local minima for time minimization in the controlled three-body problem is studied. Several homotopies are systematically used to unfold the structure of these local minimizers, and the resulting singularity of the path associated with the value function is analyzed numerically.
Optimal control of an ensemble of Bloch equations with applications in MRI
2016
International audience; The optimal control of an ensemble of Bloch equations describing the evolution of an ensemble of spins is the mathematical model used in Nuclear Resonance Imaging and the associated costs lead to consider Mayer optimal control problems. The Maximum Principle allows to parameterize the optimal control and the dynamics is analyzed in the framework of geometric optimal control. This lead to numerical implementations or suboptimal controls using averaging principle.
Regularization of chattering phenomena via bounded variation controls
2018
In control theory, the term chattering is used to refer to strong oscillations of controls, such as an infinite number of switchings over a compact interval of times. In this paper we focus on three typical occurences of chattering: the Fuller phenomenon, referring to situations where an optimal control switches an infinite number of times over a compact set; the Robbins phenomenon, concerning optimal control problems with state constraints, meaning that the optimal trajectory touches the boundary of the constraint set an infinite number of times over a compact time interval; the Zeno phenomenon, referring as well to an infinite number of switchings over a compact set, for hybrid optimal co…
On the optimal control of the circular restricted three body problem
2011
The context of this work is space mechanics. More precisely, we aim at computing low thrust transfers in the Earth-Moon system modeled by the circular restricted three-body problem. The goal is to calculate the optimal steering of the spacecraft engine with respect to two optimization criteria: Final time and fuel consumption. The contributions of this thesis are of two kinds. Geometric, first, as we study the controllability of the system together with the geometry of the transfers (structure of the command) by means of geometric control tools. Numerical, then, different homotopic methods being developed. A two-three body continuation is used to compute minimum time trajectories, and then …
Minimum fuel control of the planar circular restricted three-body problem
2012
The circular restricted three-body problem is considered to model the dynamics of an artificial body submitted to the attraction of two planets. Minimization of the fuel consumption of the spacecraft during the transfer, e.g. from the Earth to the Moon, is considered. In the light of the controllability results of Caillau and Daoud (SIAM J Control Optim, 2012), existence for this optimal control problem is discussed under simplifying assumptions. Thanks to Pontryagin maximum principle, the properties of fuel minimizing controls is detailed, revealing a bang-bang structure which is typical of L1-minimization problems. Because of the resulting non-smoothness of the Hamiltonian two-point bound…
Constrained differential inclusions with nonlocal initial conditions
2017
International audience; We show existence for the perturbed sweeping process with nonlocal initial conditions under very general hypotheses. Periodic, anti-periodic, mean value and multipoints conditions are included in this study. We give abstract results for differential inclusions with nonlocal initial conditions through bounding functions and tangential conditions. Some applications to differential complementarity systems and to vector hysteresis are given.
Solving chance constrained optimal control problems in aerospace via Kernel Density Estimation
2017
International audience; The goal of this paper is to show how non-parametric statistics can be used to solve some chance constrained optimization and optimal control problems. We use the Kernel Density Estimation method to approximate the probability density function of a random variable with unknown distribution , from a relatively small sample. We then show how this technique can be applied and implemented for a class of problems including the God-dard problem and the trajectory optimization of an Ariane 5-like launcher.
Minimum Time Control of the Restricted Three-Body Problem
2012
The minimum time control of the circular restricted three-body problem is considered. Controllability is proved on an adequate submanifold. Singularities of the extremal flow are studied by means of a stratification of the switching surface. Properties of homotopy maps in optimal control are framed in a simple case. The analysis is used to perform continuations on the two parameters of the problem: The ratio of the masses, and the magnitude of the control.
On some Riemannian aspects of two and three-body controlled problems
2009
The flow of the Kepler problem (motion of two mutually attracting bodies) is known to be geodesic after the work of Moser [20], extended by Belbruno and Osipov [2, 21]: Trajectories are reparameterizations of minimum length curves for some Riemannian metric. This is not true anymore in the case of the three-body problem, and there are topological obstructions as observed by McCord et al. [19]. The controlled formulations of these two problems are considered so as to model the motion of a spacecraft within the influence of one or two planets. The averaged flow of the (energy minimum) controlled Kepler problem with two controls is shown to remain geodesic. The same holds true in the case of o…