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«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2025. Vol 54

On the Exact Form of V.A. Dykhta’s Feedback Minimum Principle in Nonlinear Control Problems

Author(s)

Nikolay I. Pogodaev1,  Olga N. Samsonyuk1,  Maksim V. Staritsyn1, 2

1 Matrosov Institute for System Dynamics and Control Theory SB RAS, Irkutsk, Russian Federation

2 National Research Irkutsk State Technical University, Irkutsk, Russian Federation

Abstract
This paper investigates a nonlinear optimal control problem for an ordinary differential equation (in the sense of Bochner) on a Banach space. The problem is posed in the class of conventional controls – measurable, essentially bounded functions of time – and takes the classical Mayer’s form with a free right endpoint of the trajectories. It is shown that the increment of the objective functional for such a problem, for any pair of admissible controls, can be represented exactly in terms of the cost function of the reference process – a solution to a linear transport equation. The restriction of this representation to the standard classes of needle-shaped and weak control perturbations plays the role of a functional variation of “infinite order”. A non-canonical necessary condition for optimality follows from the exact formula for the functional increment, which differs from both the Pontryagin principle and known higher-order conditions. This condition can be considered an exact nonlinear form of V.A. Dykhta’s feedback minimum principle.
About the Authors

Nikolay N. Pogodaev, Cand. Sci. (Phys.–Math.), Leading Researcher, Matrosov Institute for System Dynamics and Control Theory SB RAS, Irkutsk, 664033, Russian Federation, nickpogo@gmail.com

Olga N. Samsonyuk, Cand. Sci. (Phys.–Math.), Senior Researcher, Matrosov Institute for System Dynamics and Control Theory SB RAS, Irkutsk, 664033, Russian Federation, olga.samsonyuk@icc.ru

Maksim V. Staritsyn, Cand. Sci. (Phys.–Math.), Leading Researcher, Matrosov Institute for System Dynamics and Control Theory SB RAS, Irkutsk, 664033, Russian Federation, starmax@icc.ru

For citation
Pogodaev N. I., Samsonyuk O. N., Staritsyn M. V. On the Exact Form of V.A. Dykhta’s Feedback Minimum Principle in Nonlinear Control Problems. The Bulletin of Irkutsk State University. Series Mathematics, 2025, vol. 54, pp. 48–63. (in Russian) https://doi.org/10.26516/1997-7670.2025.54.48
Keywords
optimal control, necessary optimality conditions, feedback control, numerical algorithms
UDC
517.977
MSC
49J20
DOI
https://doi.org/10.26516/1997-7670.2025.54.48
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