Geodesic
A~method of smooth approximation in the theory of necessary optimality conditions for differential inclusions
Izvestiya. Mathematics , Tome 61 (1997) no. 2, pp. 235-258

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In this paper we develop a constructive method of approximation of a differential inclusion by a sequence of smooth control systems. Combining this with other methods of approximation [7], [17], we reduce the optimal control problem for a differential inclusion with state constraints to the classical optimal control problem without constraints on state or endpoints. New necessary optimality conditions for differential inclusions with state constraints are developed. These conditions involve both the refined Euler–Lagrange inclusion [8] and the stationarity condition for the Hamiltonian [15], [16]. 
@article{IM2_1997_61_2_a0,
     author = {S. M. Aseev},
     title = {A~method of smooth approximation in the theory of necessary optimality conditions for differential inclusions},
     journal = {Izvestiya. Mathematics },
     pages = {235--258},
     publisher = {mathdoc},
     volume = {61},
     number = {2},
     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1997_61_2_a0/}
}
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S. M. Aseev. A~method of smooth approximation in the theory of necessary optimality conditions for differential inclusions. Izvestiya. Mathematics , Tome 61 (1997) no. 2, pp. 235-258. http://geodesic.mathdoc.fr/item/IM2_1997_61_2_a0/