Abstract:
We consider a dynamic control reconstruction problem for deterministic affine control systems. The reconstruction is performed in real time on the basis of known discrete inaccurate measurements of the observed trajectory of the system generated by an unknown measurable control with values in a given compact set. We formulate a well-posed reconstruction problem in the weak* sense and propose its solution obtained by the variational method developed by the authors. This approach uses auxiliary variational problems with a convex–concave Lagrangian regularized by Tikhonov's method. Then the solution of the reconstruction problem reduces to the integration of Hamiltonian systems of ordinary differential equations. We present matching conditions for the approximation parameters (accuracy parameters, the frequency of measurements of the trajectory, and an auxiliary regularizing parameter) and show that under these conditions the reconstructed controls are bounded and the trajectories of the dynamical system generated by these controls converge uniformly to the observed trajectory.
Sections 1–3 of the paper (development of the solution algorithm) are a part of research carried out at the Ural Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (contract no. 075-02-2021-1383). Sections 4–6 (proof of the convergence of the algorithm) are supported by the Russian Foundation for Basic Research (project no. 20-01-00362).
Citation:
N. N. Subbotina, E. A. Krupennikov, “Weak* Solution to a Dynamic Reconstruction Problem”, Optimal Control and Differential Games, Collected papers, Trudy Mat. Inst. Steklova, 315, Steklov Math. Inst., Moscow, 2021, 247–260; Proc. Steklov Inst. Math., 315 (2021), 233–246
\Bibitem{SubKru21}
\by N.~N.~Subbotina, E.~A.~Krupennikov
\paper Weak* Solution to a Dynamic Reconstruction Problem
\inbook Optimal Control and Differential Games
\bookinfo Collected papers
\serial Trudy Mat. Inst. Steklova
\yr 2021
\vol 315
\pages 247--260
\publ Steklov Math. Inst.
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm4220}
\crossref{https://doi.org/10.4213/tm4220}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2021
\vol 315
\pages 233--246
\crossref{https://doi.org/10.1134/S0081543821050187}
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Linking options:
https://www.mathnet.ru/eng/tm4220
https://doi.org/10.4213/tm4220
https://www.mathnet.ru/eng/tm/v315/p247
This publication is cited in the following 5 articles:
N. N. Subbotina, E. A. Krupennikov, “On a control reconstruction problem with nonconvex constraints”, Proc. Steklov Inst. Math. (Suppl.), 325, suppl. 1 (2024), S179–S193
P. G. Surkov, “Adaptivnyi algoritm ustoichivoi onlain identifikatsii pomekhi v sisteme drobnogo poryadka na beskonechnom vremennom gorizonte”, Tr. IMM UrO RAN, 29, no. 2, 2023, 172–188
N. Subbotina, E. Krupennikov, “Variational approach to construction of piecewise-constant approximations of the solution of dynamic reconstruction problem”, Differential Equations, Mathematical Modeling and Computational Algorithms, Springer Proceedings in Mathematics & Statistics, 423, 2023, 227–242
N. Subbotina, E. Krupennikov, N.S. Severina, M. Mikilyan, “To the dynamic reconstruction of sliding controls”, MATEC Web of Conferences, 362 (2022), 01030
N. N. Subbotina, E. A. Krupennikov, “On regularization of a variational approach to solving control reconstruction problems”, Lobachevskii J. Math., 43:6 (2022), 1428
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