Geodesic
Existence Theorem in the Optimal Control Problem on an Infinite Time Interval
Matematičeskie zametki, Tome 78 (2005) no. 4, pp. 503-518

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We consider the optimal control problem on an infinite time interval. The system is linear in the control, the functional is convex in the control, and the control set is convex and compact. We propose a new condition on the behavior of the functional at infinity, which is weaker than the previously known conditions, and prove the existence theorem for the solution under this condition. We consider several special cases and propose a general abstract scheme. 
@article{MZM_2005_78_4_a2,
     author = {A. V. Dmitruk and N. V. Kuz'kina},
     title = {Existence {Theorem} in the {Optimal} {Control} {Problem} on an {Infinite} {Time} {Interval}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {503--518},
     publisher = {mathdoc},
     volume = {78},
     number = {4},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2005_78_4_a2/}
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A. V. Dmitruk; N. V. Kuz'kina. Existence Theorem in the Optimal Control Problem on an Infinite Time Interval. Matematičeskie zametki, Tome 78 (2005) no. 4, pp. 503-518. http://geodesic.mathdoc.fr/item/MZM_2005_78_4_a2/