# The Hamiltonian Inclusion for Nonconvex Velocity Sets

Abstract : Since Clarke's 1973 proof of the Hamiltonian inclusion for optimal control problems with convex velocity sets, there has been speculation (and, more recently, speculation relating to a stronger, partially convexified version of the Hamiltonian inclusion) as to whether these necessary conditions are valid in the absence of the convexity hypothesis. The issue was in part resolved by Clarke himself when, in 2005, he showed that $L^{\infty}$ local minimizers satisfy the Hamiltonian inclusion. In this paper it is shown, by counterexample, that the Hamiltonian inclusion (and so also the stronger partially convexified Hamiltonian inclusion) are not in general valid for nonconvex velocity sets when the local minimizer in question is merely a $W^{1,1}$ local minimizer, not an $L^{\infty}$ local minimizer. The counterexample demonstrates that the need to consider $L^{\infty}$ local minimizers, not $W^{1,1}$ local minimizers, in the proof of the Hamiltonian inclusion for nonconvex velocity sets is fundamental, not just a technical restriction imposed by currently available proof techniques. The paper also establishes the validity of the partially convexified Hamiltonian inclusion for $W^{1,1}$ local minimizers under a normality assumption, thereby correcting earlier assertions in the literature.
Type de document :
Article dans une revue
SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2014, 52 (2), pp.1237-1250. 〈10.1137/130917417〉

https://hal.inria.fr/hal-01067451
Contributeur : Estelle Bouzat <>
Soumis le : mardi 23 septembre 2014 - 14:27:51
Dernière modification le : lundi 21 mars 2016 - 17:51:27

### Citation

Richard Vinter. The Hamiltonian Inclusion for Nonconvex Velocity Sets. SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2014, 52 (2), pp.1237-1250. 〈10.1137/130917417〉. 〈hal-01067451〉

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