Geodesic
Quadratic conditions for a Pontryagin minimum in an optimal control problem linear in the control. II. Theorems on weakening equality constraints
Izvestiya. Mathematics, Tome 31 (1988) no. 1, pp. 121-141 Cet article a éte moissonné depuis la source Math-Net.Ru

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This is a continuation of Part I (Izv. Akad. Nauk SSSR Ser. Mat., 1986, V. 50, No 2, P. 284–312) and deals with the determination of a necessary condition for a Pontryagin minimum in the same optimal control problem (linear in the control and without constraints on the control) in the case when the Lyusternik condition does not hold for the extremal under investigation. This is accomplished by passing from the original problem to another problem in which some of the equality constraints are replaced by certain inequalities, while the rest satisfy the Lyusternik condition. The necessary condition in Part I is used for the new problem. In summary, it is shown that for the original problem the necessary condition in Part I is valid independently of whether the Lyusternik condition is satisfied. Bibliography: 5 titles. 
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A. V. Dmitruk. Quadratic conditions for a Pontryagin minimum in an optimal control problem linear in the control. II. Theorems on weakening equality constraints. Izvestiya. Mathematics, Tome 31 (1988) no. 1, pp. 121-141. http://geodesic.mathdoc.fr/item/IM2_1988_31_1_a4/

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