Geodesic
A Second-Order Maximum Principle in Optimal Control Under State Constraints
Serdica Mathematical Journal, Tome 39 (2013) no. 3-4, pp. 233-270.

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A second-order variational inclusion for control systems under state constraints is derived and applied to investigate necessary optimality conditions for the Mayer optimal control problem. A new pointwise condition verified by the adjoint state of the maximum principle is obtained as well as a second-order necessary optimality condition in the integral form. Finally, a new sufficient condition for normality of the maximum principle is proposed. Some extensions to the Mayer optimization problem involving a differential inclusion under state constraints are also provided. 2010 Mathematics Subject Classification: 49K15, 49K21, 34A60, 34K35.
Keywords: optimal control, second-order necessary optimality conditions, second-order tangents 
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Frankowska, Hélène; Hoehener, Daniel; Tonon, Daniela. A Second-Order Maximum Principle in Optimal Control Under State Constraints. Serdica Mathematical Journal, Tome 39 (2013) no. 3-4, pp. 233-270. http://geodesic.mathdoc.fr/item/SMJ2_2013_39_3-4_a4/