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Article Dans Une Revue SIAM Journal on Control and Optimization Année : 2013

Pointwise second-order necessary optimality conditions for the Mayer problem with control constraints

Hélène Frankowska
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Daniela Tonon
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Résumé

This paper is devoted to second order necessary optimality conditions for the Mayer optimal control problem when the control set U is a closed subset of R^m. We show that, in the absence of endpoint constraints, if an optimal control is singular and integrable, then for almost every t such that the optimal control is in the interior of U, both the Goh and a generalized Legendre-Clebsch conditions hold true. Moreover, when the control set is a convex polytope, similar conditions are verified on the tangent subspace to U at the optimal control at time t for almost all t's such that the optimal control lies on the boundary of U. The same conditions are valid alsofor U having a smooth boundary at every t where the optimal control is singular, locally Lipschitz and on the boundary of U. In the presence of a smooth endpoint constraint, these second order necessary optimality conditions are satisfied whenever the Mayer problem is calm and the maximum principle is abnormal. If it is normal, then analogous results hold true on some smaller subspaces.
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Dates et versions

hal-00918491 , version 1 (16-12-2013)

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Hélène Frankowska, Daniela Tonon. Pointwise second-order necessary optimality conditions for the Mayer problem with control constraints. SIAM Journal on Control and Optimization, 2013, 51 (5), pp.3814-3843. ⟨10.1137/130906799⟩. ⟨hal-00918491⟩
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