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| In optimization, second-order sufficient optimality conditions consist in the existence of a Lagrange multiplier and the definite positivity of a certain quadratic form on a cone, called critical cone. They ensure the local optimality of a feasible point.
Proving the local optimality in the case of an infinite number of constraints is difficult. In particular, for nonlinear optimal control problems with control contraints, the local optimality is difficult to obtain without a supplementary assumption, called strengthend Legendre-Clebsch assumption, implying that the Hessian of the Hamiltonian is definite positive for almost all time.
In this talk, I will prove the local optimality without the Legendre-Clebsch assumption, with a new technique combining a decomposition principle and a natural extension of the critical cone in the space of Young measures.
Reference : J.F. Bonnans, N.P. Osmolovskii, Second-order analysis of optimal control problems with control and final-state constraints, J. Convex Anal., 2010. |
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