31 research outputs found
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
First and Second Order Optimality Conditions for the Control of Fokker-Planck Equations
In this article we study an optimal control problem subject to the
Fokker-Planck equation The control variable is time-dependent and
possibly multidimensional, and the function depends on the space variable
and the control. The cost functional is of tracking type and includes a
quadratic regularization term on the control. For this problem, we prove
existence of optimal controls and first order necessary conditions. Main
emphasis is placed on second order necessary and sufficient conditions
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State-constrained control-affine parabolic problems I: First and second order necessary optimality conditions
In this paper we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints. The state constraints are of integral type, the integral being with respect to the space variable. The control is multidimensional. The cost functional is of a tracking type and contains a linear term in the control variables. We derive second order necessary conditions relying on the concept of alternative costates and quasi-radial critical directions
State-constrained control-affine parabolic problems II: Second order sufficient optimality conditions
In this paper we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints. The state constraints are of integral type, the integral being with respect to the space variable. The control is multidimensional. The cost functional is of a tracking type and contains a linear term in the control variables. We derive second order sufficient conditions relying on the Goh transform