Geodesic
Exact formulae for the increment of the objective functional and necessary optimality conditions, alternative to Pontryagin's maximum principle
Sbornik. Mathematics, Tome 215 (2024) no. 6, pp. 790-822

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The paper presents elements of the theory of local extremum in the problem of optimal control with free right end and, in general, uncertain initial position of trajectories on the basis of exact formulae for the increment (variations of infinite order) of the objective functional. Necessary conditions for optimality of ‘feedback’ type are obtained: their formulations involve auxiliary feedback controls which generate program descent controls (in the minimum problem). The conditions proposed in this work provide an alternative to the classical Pontryagin principle (and even improve it in some special cases) and open the way to constructing indirect methods for local search without procedures for adjustment of the parameters of ‘descent depth’. Bibliography: 26 titles.
Keywords: optimal control, exact formulae for the increment of the objective functional, feedback necessary conditions of optimality, Pontryagin's maximum principle, continuity equation. 
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N. I. Pogodaev; M. V. Staritsyn. Exact formulae for the increment of the objective functional and necessary optimality conditions, alternative to Pontryagin's maximum principle. Sbornik. Mathematics, Tome 215 (2024) no. 6, pp. 790-822. http://geodesic.mathdoc.fr/item/SM_2024_215_6_a4/