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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. W. Demmel, Applied Numerical Linear Algebra (SIAM, Philadelphia, PA, 1997).
R. V. Gamkrelidze, Principles of Optimal Control Theory (Tbil. Univ., Tbilisi, 1977; Plenum Press, New York, 1978).
A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems (Nauka, Moscow, 1974; North-Holland, Amsterdam, 2009).
S. I. Kabanikhin and O. I. Krivorotko, “Identification of biological models described by systems of nonlinear differential equations,” J. Inverse Ill-Posed Probl. 23 (5), 519–527 (2015).
N. N. Krasovskii and A. I. Subbotin, Positional Differential Games (Nauka, Moscow, 1974); Engl. transl.: Game-Theoretical Control Problems (Springer, New York, 1988).
A. V. Kryazhimskii and Yu. S. Osipov, “Modelling of a control in a dynamic system,” Eng. Cybern. 21 (2), 38–47 (1984) [transl. from Izv. Akad. Nauk SSSR, Ser. Tekh. Kibern., No. 2, 51–60 (1983)].
Y.-C. Liu, Y.-W. Chen, Y.-T. Wang, and J.-R. Chang, “A high-order Lie groups scheme for solving the recovery of external force in nonlinear system,” Inverse Probl. Sci. Eng. 26 (12), 1749–1783 (2018).
J. R. Magnus and H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics (Wiley, Chichester, 1999).
Yu. S. Osipov and A. V. Kryazhimskii, Inverse Problems for Ordinary Differential Equations: Dynamical Solutions (Gordon and Breach, Amsterdam, 1995).
Yu. S. Osipov, A. V. Kryazhimskii, and V. I. Maksimov, “Some algorithms for the dynamic reconstruction of inputs,” Proc. Steklov Inst. Math. 275 (Suppl. 1), S86–S120 (2011) [transl. from Tr. Inst. Mat. Mekh. (Ekaterinburg) 17 (1), 129–161 (2011)].
Yu. S. Osipov, F. P. Vasil’ev, and M. M. Potapov, Foundations of the Dynamical Regularization Method (Mosk. Gos. Univ., Moscow, 1999) [in Russian].
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes (Fizmatgiz, Moscow, 1961; Pergamon, Oxford, 1964).
P. C. Sabatier, “Past and future of inverse problems,” J. Math. Phys. 41 (6), 4082–4124 (2000).
U. Schmitt, A. K. Louis, C. Wolters, and M. Vauhkonen, “Efficient algorithms for the regularization of dynamic inverse problems. II: Applications,” Inverse Probl. 18 (3), 659–676 (2002).
T. Schuster, B. Hahn, and M. Burger, “Dynamic inverse problems: Modelling—regularization—numerics,” Inverse Probl. 34 (4), 040301 (2018).
N. N. Subbotina, “Calculus of variations in solutions of dynamic reconstruction problems,” in Stability, Control and Differential Games: Proc. Int. Conf. SCDG2019, Yekaterinburg, 2019 (Springer, Cham, 2020), Lect. Notes Control Inf. Sci. – Proc., pp. 367–377.
N. N. Subbotina and E. A. Krupennikov, “The method of characteristics in an identification problem,” Proc. Steklov Inst. Math. 299 (Suppl. 1), S205–S216 (2017) [transl. from Tr. Inst. Mat. Mekh. (Ekaterinburg) 22 (2), 255–266 (2016)].
N. N. Subbotina and E. A. Krupennikov, “Hamiltonian systems for control reconstruction problems,” Minimax Theory Appl. 5 (2), 439–454 (2020).
N. N. Subbotina and E. A. Krupennikov, “Weak* approximations for the solution of a dynamic reconstruction problem,” Tr. Inst. Mat. Mekh. (Ekaterinburg) 27 (2), 208–220 (2021).
N. N. Subbotina, T. B. Tokmantsev, and E. A. Krupennikov, “Dynamic programming to reconstruction problems for a macroeconomic model,” in System Modeling and Optimization: Proc. Conf. CSMO 2015 (Cham, Springer, 2016), IFIP Adv. Inf. Commun. Technol. 494, pp. 472–481.
A. N. Tikhonov, “On the stability of inverse problems,” C. R. (Dokl.) Acad. Sci. URSS 39, 176–179 (1943) [transl. from Dokl. Akad. Nauk SSSR 39 (5), 195–198 (1943)].
V. V. Vasin, “Methods for solving nonlinear ill-posed problems based on the Tikhonov–Lavrentiev regularization and iterative approximation,” Eurasian J. Math. Comput. Appl. 4 (4), 60–73 (2016).
J. Warga, Optimal Control of Differential and Functional Equations (Academic, New York, 1972).
Funding
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).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2021, Vol. 315, pp. 247–260 https://doi.org/10.4213/tm4220.
Translated by I. Nikitin
Rights and permissions
About this article
Cite this article
Subbotina, N.N., Krupennikov, E.A. Weak* Solution to a Dynamic Reconstruction Problem. Proc. Steklov Inst. Math. 315, 233–246 (2021). https://doi.org/10.1134/S0081543821050187
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0081543821050187