Gianna Stefani

Strong Optimality for a Bang-Bang Trajectory

In this paper we give sufficient conditions for a bang-bang regular extremal to be a strong local optimum for a control problem in the Mayer form; strong means that we consider the C0 topology in the state space. The controls appear linearly and take values in a polyhedron, and the state space and the end point constraints are finite-dimensional smooth manifolds. In the case of bang-bang extremals, the kernel of the first variation of the problem is trivial, and hence the usual second variation, which is defined on the kernel of the first one, does not give any information. We consider the finite-dimensional subproblem generated by perturbing the switching times, and we prove that the sufficient second order optimality conditions for this finite-dimensional subproblem yield local strong optimality. We give an explicit algorithm to check the positivity of the second variation which is based on the properties of the Hamiltonian fields.

Controllability along a trajectory: a variational approach

This paper unifies and improves most sufficient conditions of local controllability both along a trajectory and at a point. This is accomplished by defining high-order variations that can be continuously summed. The peculiar property of these variations is that they may be generated by thin conditions as relations in the Lie algebra associated with a control system. This property leads to a variational interpretation of Sussmann-type sufficient conditions of local controllability (neutralization of obstructions), and it allows the use of different weights for neutralization.

Global stabilizability of homogeneous vector fields of odd degree

Abstract In this paper we deal with global stabilizability at the origin of a homogeneous vector field by means of homogeneous feedbacks of the same degree. We give a sufficient condition, which turns out to be also necessary in the linear case. For a particular class of systems, this condition is implied by a local controllability assumption.

Polynomial approximations to control systems and local controllability

This paper is concerned with studying some local properties of a control system via polynomial approximations induced by suitable filtrations of the Lie Algebra associated to the system. The approximations may be obtained by means of a finite algorithm.

Optimality Conditions for a Constrained Control Problem

This paper is devoted to the study of necessary or sufficient second-order conditions for a weak local minimum in an optimal control problem. The problem is stated in the Mayer form and includes equality constraints both on the endpoints and on the state-control trajectory. The second-order conditions are stated through an associated linear--quadratic problem.

State-local optimality of a bang–bang trajectory: a Hamiltonian approach

We give a sufficient condition for a bang–bang extremal to be a strong local optimizer for the minimum time problem with fixed endpoints. We underline that the conditions imply that the optimum is local with respect to the state and not necessarily to the final time. Moreover, it is given through a finite-dimensional minimization problem, hence is suited for numerical verification. A geometric interpretation through the projection of the Hamiltonian flow on the state space is also given.

An invariant second variation in optimal control

For an optimal control problem we define a second variation which is invariant under change of coordinates, and realize it as a linear-quadratic problem. When the strong Legendre condition is satisfied we give a complete Hamiltonian characterization of the index and of the nullity of the second variation.

On the relationship between global and local controllability

Given a pair(X0,D), whereX0 is a vector field andD is a family of vector fields on a manifoldM, we can define a system affine in the control, i.e., the new family of all the vector fields of the formX0 +uX, whereX ∈D andu is a real parameter. Such a system will be called globally controllable if each state is reachable from each other, whenever unbounded values foru are allowed. It is proved that a system affine in the control is globally controllable if and only if in any set of a suitable partition ofM there exist points locally controllable with bounded values foru. Further, it is proved that, under more restrictive assumptions, global controllability implies the existence of points locally controllable at a fixed time with bounded values foru. In the case of simply connected manifolds, a full equivalence among all the forms of controllability considered here is obtained.

Normal local controllability of order one

This paper gives a sufficient condition of normal local controllability at a point x0 for a family of analytical vector fields. The families considered are of the following

Sufficient conditions of local controllability

where Ω is a bounded subset of Rκ, The condition is of the first order, that is it depends only on the values of the fields and their first derivatives at x0. It is proved that this condition is the most general condition of this type. In fact, if this condition is not satisfied, there are families of vector fields which at the first order coincide with the given family, and which are not normally locally controllable at x0