Geodesic
Sensitivity relations for the Mayer problem with differential inclusions
Cannarsa, Piermarco1  ; Frankowska, Hélène2  ; Scarinci, Teresa1, 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

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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∈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 :
DOI : 10.1051/cocv/2014050
Classification : 34A60, 49J53
Keywords: Mayer problem, differential inclusions, optimality conditions, sensitivity relations

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 
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     title = {Sensitivity relations for the {Mayer} problem with differential inclusions},
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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

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