A Hamiltonian approach to strong minima in optimal control

Agrachev, Andrei; Stefani, Gianna; Zezza, P.

doi:10.1090/pspum/064/1654537

Cited by 22 publications

(29 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, where π : T * M → M is the standard projection (see [2] or [3, Ch.21]). Then γ ξ is strictly shorter than any connecting q 0 with q admissible path γ such that…”

Section: Proofsmentioning

confidence: 99%

Any sub-Riemannian metric has points of smoothness

Agrachev

2009

Dokl. Math.

View full text Add to dashboard Cite

We prove the result stated in the title; it is equivalent to the existence of a regular point of the sub-Riemannian exponential mapping. We also prove that the metric is analytic on an open everywhere dense subset in the case of a complete real-analytic sub-Riemannian manifold. PreliminariesLet M be a smooth (i. e. C ∞ ) Riemannian manifold and ∆ ⊂ T M a smooth vector distribution on M (a vector subbundle of T M). We denote by∆ the space of smooth sections of ∆ that is a subspace of the space VecM of smooth vector fields on M. The Lie bracket of vector fields X, Y is denoted by [X, Y ]. We assume that ∆ is bracket generating; in other words, ∀q ∈ M,Given q 0 , q 1 ∈ M, we define the space of starting from q 0 admissible paths:for almost all t} and the sub-Riemannian distance:where ℓ(γ) =

show abstract

“…, where π : T * M → M is the standard projection (see [2] or [3, Ch.21]). Then γ ξ is strictly shorter than any connecting q 0 with q admissible path γ such that…”

Section: Proofsmentioning

confidence: 99%

Any sub-Riemannian metric has points of smoothness

Agrachev

2009

Dokl. Math.

View full text Add to dashboard Cite

show abstract

“…For contact sub-Riemannian structures it is also known, although its proof exploits a little bit more sophisticated techniques than the proof of Theorem 1.1. There are various approaches and, in our opinion, a natural tool here is symplectic geometry (see [4] for an exposition and [6] for the detailed treatment). The following well-known fact indicates a point where the analogy with classical Riemannian geometry fails.…”

Section: Contact Sub-riemannian Geodesics Are Exactly Projections On mentioning

confidence: 99%

Exponential mappings for contact sub-Riemannian structures

Agrachev

1996

Journal of Dynamical and Control Systems

112

148

View full text Add to dashboard Cite

ABSTRACT. On sub-l:tiemannian manifolds any neighborhood of any point contains geodesics which are not length minimizers; the closures of the cut and the conjugate loci of any point q contain q. We study this phenomenon in the case of a contact distribution, essentially in the lowest possible dimension 3, where we extract differential invariants related to the singularities of the cut and the conjugate loci near q and give a generic classification of these singularities.

show abstract

“…In [ASZ98b] we stated sufficient conditions for strong local optimality for an optimal control problem in R n with unbounded controls, while in [ASZ98a] we gave an intrinsic expression of the accessory problem and studied the relations between the Hamiltonian flow and the index of the second variation. The geometric properties of the field of extremals necessary for proving sufficient conditions for strong optimality were studied in [ASZ99].…”

mentioning

confidence: 99%

“…Since the optimality can be lost only at these points, then the positivity of the second variation can be checked by an algorithm (see Lemma 2.8) which is based on the properties of the discrete flow of the bang-bang Jacobi system. For analogous conditions in the case of unbounded controls, see [ASZ98b].…”

mentioning

confidence: 99%

Strong Optimality for a Bang-Bang Trajectory

Agrachev¹,

Stefani²,

Zezza³

2002

SIAM J. Control Optim.

Self Cite

110

128

View full text Add to dashboard Cite

Abstract. 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 C 0 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.Key words. optimal control, bang-bang controls, sufficient optimality condition, strong local optima AMS subject classifications. Primary, 49K15; Secondary, 49K30, 58E25 PII. S036301290138866X1. Introduction. This paper is part of a general research program whose aim is to further extend the use of Hamiltonian methods in the study of optimal control problems. We believe that these methods can play a relevant role in control theory because they allow a general approach to sufficient conditions for strong local optimality, as we wish to show here.The Hamiltonian approach to strong optimality consists of constructing a field of state extremals covering a neighborhood of a given trajectory which has to be tested. This field of extremals is obtained by projecting on the state manifold M the flow H t of the maximized Hamiltonian emanating from the Lagrangian submanifold of the initial transversality conditions. If this projection admits a Lipschitz continuous local inverse, then we can estimate the variation of the cost function at a neighboring trajectory by a function ψ which depends only on the final point, and it is hence independent of the control differential equation; in this way we reduce the problem to a finite-dimensional one. The existence of a Lipschitz continuous local inverse is guaranteed by the surjectivity of the projection on M of the tangent map to the flow H t . This construction corresponds to the classical one of a nonselfintersecting family of state extremals. This is enough to obtain optimality if the final point is fixed since the submanifold of the final end points reduces to a singleton; otherwise we need some further optimality condition on the function ψ.We use the relations existing between a suitable second variation and the symplectic properties of the Hamiltonian flow to show that when this second variation is positive definite then the projection on the state manifold M of the tangent map to

show abstract

A Hamiltonian approach to strong minima in optimal control

Cited by 22 publications

References 0 publications

Any sub-Riemannian metric has points of smoothness

Any sub-Riemannian metric has points of smoothness

Exponential mappings for contact sub-Riemannian structures

Strong Optimality for a Bang-Bang Trajectory

Contact Info

Product

Resources

About