106 research outputs found
Necessary conditions involving Lie brackets for impulsive optimal control problems; the commutative case
In this article we study control problems with systems that are governed by
ordinary differential equations whose vector fields depend linearly in the time
derivatives of some components of the control. The remaining components are
considered as classical controls. This kind of system is called `impulsive
system'. We assume that the vector fields multiplying the derivatives of each
component of the control are commutative. We derive new necessary conditions in
terms of the adjoint state and the Lie brackets of the data functions
A note on systems with ordinary and impulsive controls
We investigate an everywhere defined notion of solution for control systems
whose dynamics depend nonlinearly on the control and state and are
affine in the time derivative For this reason, the input which
is allowed to be Lebesgue integrable, is called impulsive, while a second,
bounded measurable control is denominated ordinary. The proposed notion of
solution is derived from a topological (non-metric) characterization of a
former concept of solution which was given in the case when the drift is
-independent. Existence, uniqueness and representation of the solution are
studied, and a close analysis of effects of (possibly infinitely many)
discontinuities on a null set is performed as well.Comment: Article published in IMA J. Math. Control Infor
A Higher-order Maximum Principle for Impulsive Optimal Control Problems
We consider a nonlinear system, affine with respect to an unbounded control
which is allowed to range in a closed cone. To this system we associate a
Bolza type minimum problem, with a Lagrangian having sublinear growth with
respect to . This lack of coercivity gives the problem an {\it impulsive}
character, meaning that minimizing sequences of trajectories happen to converge
towards discontinuous paths. As is known, a distributional approach does not
make sense in such a nonlinear setting, where, instead, a suitable embedding in
the graph-space is needed.
We provide higher order necessary optimality conditions for properly defined
impulsive minima, in the form of equalities and inequalities involving iterated
Lie brackets of the dynamical vector fields. These conditions are derived under
very weak regularity assumptions and without any constant rank conditions
Necessary conditions involving Lie brackets for impulsive optimal control problems
We obtain higher order necessary conditions for a minimum of a Mayer optimal
control problem connected with a nonlinear, control-affine system, where the
controls range on an m-dimensional Euclidean space. Since the allowed
velocities are unbounded and the absence of coercivity assumptions makes big
speeds quite likely, minimizing sequences happen to converge toward
"impulsive", namely discontinuous, trajectories. As is known, a distributional
approach does not make sense in such a nonlinear setting, where instead a
suitable embedding in the graph space is needed. We will illustrate how the
chance of using impulse perturbations makes it possible to derive a Higher
Order Maximum Principle which includes both the usual needle variations (in
space-time) and conditions involving iterated Lie brackets. An example, where a
third order necessary condition rules out the optimality of a given extremal,
concludes the paper.Comment: Conference pape
limit solutions for control systems
For a control Cauchy problem on an interval , we
propose a notion of limit solution verifying the following properties: i)
is defined for (impulsive) inputs and for standard,
bounded measurable, controls ; ii) in the commutative case (i.e. when
for all ),
coincides with the solution one can obtain via the change of coordinates that
makes the simultaneously constant; iii) subsumes former concepts
of solution valid for the generic, noncommutative case.
In particular, when has bounded variation, we investigate the relation
between limit solutions and (single-valued) graph completion solutions.
Furthermore, we prove consistency with the classical Carath\'eodory solution
when and are absolutely continuous.
Even though some specific problems are better addressed by means of special
representations of the solutions, we believe that various theoretical issues
call for a unified notion of trajectory. For instance, this is the case of
optimal control problems, possibly with state and endpoint constraints, for
which no extra assumptions (like e.g. coercivity, bounded variation,
commutativity) are made in advance
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