259 research outputs found
Asymptotic controllability and optimal control
We consider a control problem where the state must reach asymptotically a
target while paying an integral payoff with a non-negative Lagrangian. The
dynamics is just continuous, and no assumptions are made on the zero level set
of the Lagrangian. Through an inequality involving a positive number
and a Minimum Restraint Function --a special type of Control Lyapunov
Function-- we provide a condition implying that (i) the control system is
asymptotically controllable, and (ii) the value function is bounded above by
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
Minimum time problem with impulsive and ordinary controls
Given a nonlinear control system depending on two controls u and v, with dynamics affine in the (unbounded) derivative of u and a closed target set depending both on the state and on the control u, we study the minimum time problem with a bound on the total variation of u and u constrained in a closed, convex set U, possibly with empty interior. We revisit several concepts of generalized control and solution considered in the literature and show that they all lead to the same minimum time function T. Then we obtain sufficient conditions for the existence of an optimal generalized trajectory-control pair and study the possibility of Lavrentiev-type gap between the minimum time in the spaces of regular (that is, absolutely continuous) and generalized controls. Finally, under a convexity assumption on the dynamics, we characterize T as the unique lower semicontinuous solution of a regular HJ equation with degenerate state constraints
On L^1 limit solutions in impulsive control
We consider a nonlinear control system depending on two controls u and v, with dynamics affine in the (unbounded) derivative of u, and v appearing initially only in the drift term. Recently, motivated by applications to optimization problems lacking coercivity, Aronna and Rampazzo proposed a notion of generalized solution x for this system, called limit solution, associated to measurable u and v, and with u of possibly unbounded variation in [0, T ]. As shown by Aronna and Rampazzo, when u and x have bounded variation, such a solution (called in this case BV simple limit solution) coincides with the most used graph completion solution (see e.g. Rishel, Warga and Bressan and Rampazzo). In a previous paper we extended this correspondence to BVloc inputs u and trajectories (with bounded variation just on any [0,t] with t < T). Here, starting with an example of optimal control where the minimum does not exist in the class of limit solutions, we propose a notion of extended limit solution x, for which such a minimum exists. As a first result, we prove that extended BV (respectively, BVloc) simple limit solutions and BV (respectively, BVloc) simple limit solutions coincide. Then we consider dynamics where the ordinary control v also appears in the non-drift terms. For the associated system we prove that, in the BV case, extended limit solutions coincide with graph completion solution
Nondegenerate abnormality, controllability, and gap phenomena in optimal control with state constraints
In optimal control theory, infimum gap means that there is a gap between the
infimum values of a given minimum problem and an extended problem, obtained by
enlarging the set of original solutions and controls. The gap phenomenon is
somewhat "dual" to the problem of the controllability of the original control
system to an extended solution. In this paper we present sufficient conditions
for the absence of an infimum gap and for controllability for a wide class of
optimal control problems subject to endpoint and state constraints. These
conditions are based on a nondegenerate version of the nonsmooth constrained
maximum principle, expressed in terms of subdifferentials. In particular, under
some new constraint qualification conditions, we prove that: (i) if an extended
minimizer is a nondegenerate normal extremal, then no gap shows up; (ii) given
an extended solution verifying the constraints, either it is a nondegenerate
abnormal extremal, or the original system is controllable to it. An application
to the impulsive extension of a free end-time, non-convex optimization problem
with control-polynomial dynamics illustrates the results
Impulsive optimal control problems with time delays in the drift term
We introduce a notion of bounded variation solution for a new class of
nonlinear control systems with ordinary and impulsive controls, in which the
drift function depends not only on the state, but also on its past history,
through a finite number of time delays. After proving the well posedness of
such solutions and the continuity of the corresponding input output map with
respect to suitable topologies, we establish necessary optimality conditions
for an associated optimal control problem. The approach, which involves
approximating the problem by a non impulsive optimal control problem with time
delays and using Ekeland principle combined with a recent, nonsmooth version of
the Maximum Principle for conventional delayed systems, allows us to deal with
mild regularity assumptions and a general endpoint constraint
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