35,450 research outputs found
Optimality conditions applied to free-time multi-burn optimal orbital transfers
While the Pontryagin Maximum Principle can be used to calculate candidate
extremals for optimal orbital transfer problems, these candidates cannot be
guaranteed to be at least locally optimal unless sufficient optimality
conditions are satisfied. In this paper, through constructing a parameterized
family of extremals around a reference extremal, some second-order necessary
and sufficient conditions for the strong-local optimality of the free-time
multi-burn fuel-optimal transfer are established under certain regularity
assumptions. Moreover, the numerical procedure for computing these optimality
conditions is presented. Finally, two medium-thrust fuel-optimal trajectories
with different number of burn arcs for a typical orbital transfer problem are
computed and the local optimality of the two computed trajectories are tested
thanks to the second-order optimality conditions established in this paper
-optimality conditions for circular restricted three-body problems
In this paper, the L1-minimization for the translational motion of a
spacecraft in a circular restricted three-body problem (CRTBP) is considered.
Necessary con- ditions are derived by using the Pontryagin Maximum Principle,
revealing the existence of bang-bang and singular controls. Singular extremals
are detailed, re- calling the existence of the Fuller phenomena according to
the theories developed by Marchal in Ref. [14] and Zelikin et al. in Refs. [12,
13]. The sufficient opti- mality conditions for the L1-minimization problem
with fixed endpoints have been solved in Ref. [22]. In this paper, through
constructing a parameterised family of extremals, some second-order sufficient
conditions are established not only for the case that the final point is fixed
but also for the case that the final point lies on a smooth submanifold. In
addition, the numerical implementation for the optimality conditions is
presented. Finally, approximating the Earth-Moon-Spacecraft system as a CRTBP,
an L1-minimization trajectory for the translational motion of a spacecraft is
computed by employing a combination of a shooting method with a continuation
method of Caillau et al. in Refs. [4, 5], and the local optimality of the
computed trajectory is tested thanks to the second-order optimality conditions
established in this paper
-Minimization for Mechanical Systems
Second order systems whose drift is defined by the gradient of a given
potential are considered, and minimization of the -norm of the control is
addressed. An analysis of the extremal flow emphasizes the role of singular
trajectories of order two [25,29]; the case of the two-body potential is
treated in detail. In -minimization, regular extremals are associated with
controls whose norm is bang-bang; in order to assess their optimality
properties, sufficient conditions are given for broken extremals and related to
the no-fold conditions of [20]. An example of numerical verification of these
conditions is proposed on a problem coming from space mechanics
Strong local optimality for generalized L1 optimal control problems
In this paper, we analyse control affine optimal control problems with a cost
functional involving the absolute value of the control. The Pontryagin
extremals associated with such systems are given by (possible) concatenations
of bang arcs with singular arcs and with inactivated arcs, that is, arcs where
the control is identically zero. Here we consider Pontryagin extremals given by
a bang-inactive-bang concatenation. We establish sufficient optimality
conditions for such extremals, in terms of some regularity conditions and of
the coercivity of a suitable finite-dimensional second variation.Comment: Journal of Optimization Theory and Applications, Springer Verlag, In
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Planar tilting maneuver of a spacecraft: singular arcs in the minimum time problem and chattering
In this paper, we study the minimum time planar tilting maneuver of a
spacecraft, from the theoretical as well as from the numerical point of view,
with a particular focus on the chattering phenomenon. We prove that there exist
optimal chattering arcs when a singular junction occurs. Our study is based on
the Pontryagin Maximum Principle and on results by M.I. Zelikin and V.F.
Borisov. We give sufficient conditions on the initial values under which the
optimal solutions do not contain any singular arc, and are bang-bang with a
finite number of switchings. Moreover, we implement sub-optimal strategies by
replacing the chattering control with a fixed number of piecewise constant
controls. Numerical simulations illustrate our results.Comment: 43 pages, 18 figure
Time Optimal Synthesis for Left--Invariant Control Systems on SO(3)
Consider the control system given by , where ,
and define two perpendicular left-invariant vector
fields normalized so that \|f\|=\cos(\al) and \|g\|=\sin(\al), \al\in
]0,\pi/4[. In this paper, we provide an upper bound and a lower bound for
, the maximum number of switchings for time-optimal trajectories.
More precisely, we show that N_S(\al)\leq N(\al)\leq N_S(\al)+4, where
N_S(\al) is a suitable integer function of \al which for \al\to 0 is of
order The result is obtained by studying the time optimal
synthesis of a projected control problem on , where the projection is
defined by an appropriate Hopf fibration. Finally, we study the projected
control problem on the unit sphere . It exhibits interesting features
which will be partly rigorously derived and partially described by numerical
simulations
Structural stability of bang-bang trajectories with a double switching time in the minimum time problem
In this paper we consider the problem of structural stability of strong local
optimisers for the minimum time problem in the case when the nominal problem
has a bang-bang strongly local optimal control which exhibits a double switch
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