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

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In this paper, we give second-order sufficient conditions for strong local optimality of a bang-singular-bang extremal in the minimum time problem. The conditions are given in terms of some regularity assumptions on the extremal and of the coercivity of the extended second variation associated with the minimum time problem with fixed endpoints on the singular arc. The conditions are close to the necessary conditions in the usual sense; namely, we require strict inequalities where the necessary conditions have nonstrict inequalities.

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