Sensitivity relations for the Mayer problem with differential inclusions

Cannarsa, Piermarco; Frankowska, Hélène; Scarinci, Teresa

doi:10.1051/cocv/2014050

Cannarsa, Piermarco ¹ ; Frankowska, Hélène ² ; Scarinci, Teresa ^1,²

¹ Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy
² CNRS, IMJ-PRG, UMR 7586, Sorbonne Universités, UPMC Univ Paris 06, Univ Paris Diderot, Sorbonne Paris Cité, Case 247, 4 Place Jussieu, 75252 Paris, France

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 3, pp. 789-814.

Résumé

In optimal control, sensitivity relations are usually understood as inclusions that identify the pair formed by the dual arc and the Hamiltonian as a suitable generalized gradient of the value function, evaluated along a given minimizing trajectory. In this paper, sensitivity relations are obtained for the Mayer problem associated with the differential inclusion $ẋ \in F (x)$ and applied to express optimality conditions. The first application of our results concerns the maximum principle and consists in showing that a dual arc can be constructed for every element of the superdifferential of the final cost as a solution of an adjoint system. The second and last application we discuss in this paper concerns optimal design. We show that one can associate a family of optimal trajectories, starting at some point $(t, x)$ , with every nonzero reachable gradient of the value function at $(t, x)$ , in such a way that families corresponding to distinct reachable gradients have empty intersection.

Reçu le : 2014-02-12

MR Zbl

DOI : 10.1051/cocv/2014050

Classification : 34A60, 49J53
Mots clés : Mayer problem, differential inclusions, optimality conditions, sensitivity relations

Affiliations des auteurs :

Cannarsa, Piermarco ¹ ; Frankowska, Hélène ² ; Scarinci, Teresa ^{1, 2}

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     title = {Sensitivity relations for the {Mayer} problem with differential inclusions},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {789--814},
     publisher = {EDP-Sciences},
     volume = {21},
     number = {3},
     year = {2015},
     doi = {10.1051/cocv/2014050},
     mrnumber = {3358630},
     zbl = {1319.49036},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2014050/}
}

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Cannarsa, Piermarco; Frankowska, Hélène; Scarinci, Teresa. Sensitivity relations for the Mayer problem with differential inclusions. ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 3, pp. 789-814. doi : 10.1051/cocv/2014050. http://archive.numdam.org/articles/10.1051/cocv/2014050/

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