Geodesic
Bogolyubov's theorem under constraints generated by a~controlled second-order evolution system
Izvestiya. Mathematics , Tome 67 (2003) no. 5, pp. 1031-1060.

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We prove an analogue of Bogolyubov's theorem with constraints in the form of a controlled second-order evolution system. The main assertion of this theorem deals with relations between the values of an integral functional that is non-convex with respect to control on the solutions of a controlled system with non-convex constraints on the control and the values of the functional convexified with respect to control on the solutions of a controlled system with convexified constraints. This theorem also establishes relations between the solutions of non-convex and convexified controlled systems. We apply the theorem to the problem of minimizing a non-convex integral functional on the solutions of a non-convex controlled system. We consider in detail an example of a non-linear hyperbolic system. 
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A. A. Tolstonogov. Bogolyubov's theorem under constraints generated by a~controlled second-order evolution system. Izvestiya. Mathematics , Tome 67 (2003) no. 5, pp. 1031-1060. http://geodesic.mathdoc.fr/item/IM2_2003_67_5_a7/

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