We consider the problem of the dynamic reconstruction of an observed state trajectory $x^*(\cdot)$ of an affine deterministic dynamic system and the control that has generated this trajectory. The reconstruction is based on current information about inaccurate discrete measurements of $x^*(\cdot)$. A correct statement of the problem on the construction of approximations $u^l(\cdot)$ of the normal control $u^*(\cdot)$ generating $x^*(\cdot)$ is refined. The solution of this problem obtained using the variational approach proposed by the authors is discussed. Conditions on the input data and matching conditions for the approximation parameters (parameters of the accuracy and frequency of measurements of the trajectory and an auxiliary regularizing parameter) are given. Under these conditions, the reconstructed trajectories $x^l(\cdot)$ of the dynamical system converge uniformly to the observed trajectory $x^*(\cdot)$ in the space of continuous functions $C$ as $l\to\infty$. It is proved that the proposed controls $u^l(\cdot)$ converge weakly* to $u^*(\cdot)$ in the space of summable functions $L^1$.
Keywords: dynamic reconstruction problems, convex-concave discrepancy, problems of calculus of variations, Hamiltonian systems
Received February 26, 2021
Revised April 7, 2021
Accepted April 12, 2021
Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 20-01-00362).
Nina Nikolaevna Subbotina, Dr. Phys.-Math. Sci., RAS Corresponding Member, Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620002 Russia, e-mail: subb@uran.ru
Evgenii Aleksandrovich Krupennikov, Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620002 Russia, e-mail: krupennikov@imm.uran.ru
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Cite this article as: N.N. Subbotina, E.A. Krupennikov. Weak* approximations for the solution of a dynamic reconstruction problem, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 2, pp. 208–220; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2022, Vol. 317, Suppl. 1, pp. S71–S89.
