Geodesic
First-order necessary conditions in the~problem of optimal control of a~differential inclusion with phase constraints
Sbornik. Mathematics, Tome 79 (1994) no. 1, pp. 117-139

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Nondegenerate first-order necessary conditions for optimality are obtained for the problem (1.1)–(1.4) under different assumptions about controllability at the endpoints. These necessary conditions are obtained in the Hamiltonian form of Clarke [1]. With the help of a smoothing technique [2] the perturbation method in [3] is used to carry the main results in [4] (there the case when the support function $H(x,t,\psi)=\sup_{y\in F(x,t)}\langle y,\psi\rangle$ depends smoothly on the variable $x$ is considered) over to the more natural class of problems with locally Lipschitz support function $H$. 
@article{SM_1994_79_1_a8,
     author = {A. V. Arutyunov and S. M. Aseev and V. I. Blagodatskikh},
     title = {First-order necessary conditions in the~problem of optimal control of a~differential inclusion with phase constraints},
     journal = {Sbornik. Mathematics},
     pages = {117--139},
     publisher = {mathdoc},
     volume = {79},
     number = {1},
     year = {1994},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1994_79_1_a8/}
}
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A. V. Arutyunov; S. M. Aseev; V. I. Blagodatskikh. First-order necessary conditions in the~problem of optimal control of a~differential inclusion with phase constraints. Sbornik. Mathematics, Tome 79 (1994) no. 1, pp. 117-139. http://geodesic.mathdoc.fr/item/SM_1994_79_1_a8/