Search results for "conjugate points"

Second order optimality conditions with applications

2007

International audience; The aim of this article is to present the algorithm to compute the first conjugate point along a smooth extremal curve. Under generic assump- tions, the tra jectory ceases to be optimal at such a point. An implementation of this algorithm, called cotcot, is available online and based on recent devel- opments in geometric optimal control. It is applied to analyze the averaged optimal transfer of a satellite between elliptic orbits.

Multiplicity results for fourth order two-point boundary value problems with asymmetric nonlinearities

1998

Second order optimality conditions in the smooth case and applications in optimal control

2007

International audience; The aim of this article is to present algorithms to compute the first conjugate time along a smooth extremal curve, where the trajectory ceases to be optimal. It is based on recent theoretical developments of geometric optimal control, and the article contains a review of second order optimality conditions. The computations are related to a test of positivity of the intrinsic second order derivative or a test of singularity of the extremal flow. We derive an algorithm called COTCOT (Conditions of Order Two and COnjugate times), available on the web, and apply it to the minimal time problem of orbit transfer, and to the attitude control problem of a rigid spacecraft. …

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…

Geodesic ray transform with matrix weights for piecewise constant functions

2019

We show injectivity of the geodesic X-ray transform on piecewise constant functions when the transform is weighted by a continuous matrix weight. The manifold is assumed to be compact and nontrapping of any dimension, and in dimension three and higher we assume a foliation condition. We make no assumption regarding conjugate points or differentiability of the weight. This extends recent results for unweighted transforms.

Jacobi Fields, Conjugate Points

2001

Let us go back to the action principle as realized by Jacobi, i.e., time is eliminated, so we are dealing with the space trajectory of a particle. In particular, we want to investigate the conditions under which a path is a minimum of the action and those under which it is merely an extremum. For illustrative purposes we consider a particle in two-dimensional real space.

The energy minimization problem for two-level dissipative quantum systems

2010

In this article, we study the energy minimization problem of dissipative two-level quantum systems whose dynamics is governed by the Kossakowski–Lindblad equations. In the first part, we classify the extremal curve solutions of the Pontryagin maximum principle. The optimality properties are analyzed using the concept of conjugate points and the Hamilton–Jacobi–Bellman equation. This analysis completed by numerical simulations based on adapted algorithms allows a computation of the optimal control law whose robustness with respect to the initial conditions and dissipative parameters is also detailed. In the final section, an application in nuclear magnetic resonance is presented.

Second order optimality conditions in optimal control with applications

2006

The aim of this article is to present the algorithm to compute the first conjugate point along a smooth extremal curve. Under generic assumptions, the trajectory ceases to be optimal at such a point. An implementation of this algorithm, called \texttt{cotcot}, is available online and based on recent developments in geometric optimal control. It is applied to analyze the averaged optimal transfer of a satellite between elliptic orbits.