1,618 research outputs found
Lipschitzian Regularity of the Minimizing Trajectories for Nonlinear Optimal Control Problems
We consider the Lagrange problem of optimal control with unrestricted
controls and address the question: under what conditions we can assure optimal
controls are bounded? This question is related to the one of Lipschitzian
regularity of optimal trajectories, and the answer to it is crucial for closing
the gap between the conditions arising in the existence theory and necessary
optimality conditions. Rewriting the Lagrange problem in a parametric form, we
obtain a relation between the applicability conditions of the Pontryagin
maximum principle to the later problem and the Lipschitzian regularity
conditions for the original problem. Under the standard hypotheses of
coercivity of the existence theory, the conditions imply that the optimal
controls are essentially bounded, assuring the applicability of the classical
necessary optimality conditions like the Pontryagin maximum principle. The
result extends previous Lipschitzian regularity results to cover optimal
control problems with general nonlinear dynamics.Comment: This research was partially presented, as an oral communication, at
the international conference EQUADIFF 10, Prague, August 27-31, 2001.
Accepted for publication in the journal Mathematics of Control, Signals, and
Systems (MCSS). See http://www.mat.ua.pt/delfim for other work
Solving Optimal Control Problems for Delayed Control-Affine Systems with Quadratic Cost by Numerical Continuation
- In this paper we introduce a new method to solve fixed-delay optimal
control problems which exploits numerical homotopy procedures. It is known that
solving this kind of problems via indirect methods is complex and
computationally demanding because their implementation is faced with two
difficulties: the extremal equations are of mixed type, and besides, the
shooting method has to be carefully initialized. Here, starting from the
solution of the non-delayed version of the optimal control problem, the delay
is introduced by numerical homotopy methods. Convergence results, which ensure
the effectiveness of the whole procedure, are provided. The numerical
efficiency is illustrated on an example
Caratheodory-Equivalence, Noether Theorems, and Tonelli Full-Regularity in the Calculus of Variations and Optimal Control
We study, in a unified way, the following questions related to the properties
of Pontryagin extremals for optimal control problems with unrestricted
controls: i) How the transformations, which define the equivalence of two
problems, transform the extremals? ii) How to obtain quantities which are
conserved along any extremal? iii) How to assure that the set of extremals
include the minimizers predicted by the existence theory? These questions are
connected to: i) the Caratheodory method which establishes a correspondence
between the minimizing curves of equivalent problems; ii) the interplay between
the concept of invariance and the theory of optimality conditions in optimal
control, which are the concern of the theorems of Noether; iii) regularity
conditions for the minimizers and the work pioneered by Tonelli.Comment: 24 pages, Submitted for publication in a Special Issue of the J. of
Mathematical Science
Time Minimal Trajectories for a Spin 1/2 Particle in a Magnetic Field
In this paper we consider the minimum time population transfer problem for
the -component of the spin of a (spin 1/2) particle driven by a magnetic
field, controlled along the x axis, with bounded amplitude. On the Bloch sphere
(i.e. after a suitable Hopf projection), this problem can be attacked with
techniques of optimal syntheses on 2-D manifolds. Let be the two
energy levels, and the bound on the field amplitude. For
each couple of values and , we determine the time optimal synthesis
starting from the level and we provide the explicit expression of the time
optimal trajectories steering the state one to the state two, in terms of a
parameter that can be computed solving numerically a suitable equation. For
, every time optimal trajectory is bang-bang and in particular the
corresponding control is periodic with frequency of the order of the resonance
frequency . On the other side, for , the time optimal
trajectory steering the state one to the state two is bang-bang with exactly
one switching. Fixed we also prove that for the time needed to
reach the state two tends to zero. In the case there are time optimal
trajectories containing a singular arc. Finally we compare these results with
some known results of Khaneja, Brockett and Glaser and with those obtained by
controlling the magnetic field both on the and directions (or with one
external field, but in the rotating wave approximation). As byproduct we prove
that the qualitative shape of the time optimal synthesis presents different
patterns, that cyclically alternate as , giving a partial proof of a
conjecture formulated in a previous paper.Comment: 31 pages, 10 figures, typos correcte
Guidance, flight mechanics and trajectory optimization. Volume 4 - The calculus of variations and modern applications
Guidance, flight mechanics, and trajectory optimization - calculus of variations and modern application
Integral and measure-turnpike properties for infinite-dimensional optimal control systems
We first derive a general integral-turnpike property around a set for
infinite-dimensional non-autonomous optimal control problems with any possible
terminal state constraints, under some appropriate assumptions. Roughly
speaking, the integral-turnpike property means that the time average of the
distance from any optimal trajectory to the turnpike set con- verges to zero,
as the time horizon tends to infinity. Then, we establish the measure-turnpike
property for strictly dissipative optimal control systems, with state and
control constraints. The measure-turnpike property, which is slightly stronger
than the integral-turnpike property, means that any optimal (state and control)
solution remains essentially, along the time frame, close to an optimal
solution of an associated static optimal control problem, except along a subset
of times that is of small relative Lebesgue measure as the time horizon is
large. Next, we prove that strict strong duality, which is a classical notion
in optimization, implies strict dissipativity, and measure-turnpike. Finally,
we conclude the paper with several comments and open problems
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