Geodesic
Extremal Problems for Differential Inclusions with State Constraints
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations. Certain mathematical problems of optimal control, Tome 233 (2001), pp. 5-70

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This paper is devoted to the study of optimal control problems for differential inclusions with state constraints. The main focus is placed on the derivation of the most complete first-order necessary optimality conditions that employ the specific features of both a differential constraint given by a differential inclusion and state constraints. For the problem considered, a generalization of the Pontryagin maximum principle is obtained that strengthens many known results in this field and contains an additional condition that the Hamiltonian (the maximum function) of the problem should be stationary. For the Lagrange multipliers entering the relations of the maximum principle, the properties primarily attributed to the state constraints are studied. In particular, the degeneracy of the necessary optimality conditions is analyzed and sufficient conditions are obtained for the regularity of the Lagrange multipliers. 
@article{TM_2001_233_a0,
     author = {S. M. Aseev},
     title = {Extremal {Problems} for {Differential} {Inclusions} with {State} {Constraints}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {5--70},
     publisher = {mathdoc},
     volume = {233},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2001_233_a0/}
}
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S. M. Aseev. Extremal Problems for Differential Inclusions with State Constraints. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations. Certain mathematical problems of optimal control, Tome 233 (2001), pp. 5-70. http://geodesic.mathdoc.fr/item/TM_2001_233_a0/