Abstract
For systems linear in control, we consider problems of recovering the dynamics and control from a posteriori statistics of trajectory sampling and known estimates for the sampling error. An optimal control problem of minimizing an integral regularized functional of dynamics and statistics residuals is introduced. Optimal synthesis is used to construct controls and trajectories that approximate a solution of the inverse problem. A numerical approximation method based on the method of characteristics for the Hamilton–Jacobi–Bellman equation and on the notion of minimax/viscosity solution is developed. Sufficient conditions are obtained under which the proposed approximations converge to a normal solution of the inverse problem under a matched vanishing of the approximation parameters (bounds for the sampling error, the regularizing parameter, the grid step in the state variable, and the integration step). Results of the numerical solution of identification and recovery problems for controls and trajectories are presented for a mechanical model of gravitation under given statistics of phase coordinate sampling.
Similar content being viewed by others
References
N. N. Krasovskii, Theory of Motion Control (Nauka, Moscow, 1968) [in Russian].
N. N. Krasovskii and A. I. Subbotin, Positional Differential Games (Nauka, Moscow, 1974) [in Russian].
N. N. Subbotina, E. A. Kolpakova, T. B. Tokmantsev, and L. G. Shagalova, The Method of Characteristics for Hamilton–Jacobi–Bellman Equations (RIO UrO RAN, Yekaterinburg, 2013) [in Russian].
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes (Fizmatgiz, Moscow, 1961; Wiley, New York, 1962).
R. Bellman, Dynamic Programming (Princeton Univ. Press, Princeton, 1957).
A. I. Subbotin, Generalized Solutions of First-Order Partial Differential Equations: The Dynamical Optimization Perspective (Birkhäuser, Boston, 1995; Regulyarn. Khaotich. Dinamika, Moscow–Izhevsk, 2003).
F. Clarke, Optimization and Nonsmooth Analysis (Wiley, New York, 1983; Nauka, Moscow, 1988).
N. N. Subbotina and T. B. Tokmantsev, “Classical characteristics of the Bellman equation in constructions of grid optimal synthesis,” Proc. Steklov Inst. Math. 271 (1), 246–264 (2010).
N. N. Subbotina and T. B. Tokmantsev, “On grid optimal feedbacks to control problems of prescribed duration on the plane,” in Advances in Dynamic Games (Birkhäuser, Boston, 2011), Ser. Annals of the International Society of Dynamic Games, Vol. 11, pp. 133–147.
Yu. S. Osipov, F. P. Vasil’ev, and M. M. Potapov, Foundations of the Dynamical Regularization Method (Izd. Mosk. Gos. Univ., Moscow, 1999) [in Russian].
E. A. Barbashin, Introduction to the Theory of Stability (Nauka, Moscow, 1967) [in Russian].
A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Nauka, Moscow, 1974; Dover, New York, 1999).
J. L. Davy, “Properties of the solution set of a generalized differential equation,” Bull. Austral. Math. Soc. 6 (3), 379–398 (1972).
N. N. Subbotina and T. B. Tokmantsev, “The method of characteristics in inverse problems of dynamics,” Univ. J. Control Automat. 1 (3),79–85 (2013).
E. B. Lee and L. Markus, Foundations of Optimal Control Theory (Wiley, New York, 1967; Nauka, Moscow, 1972).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © N.N. Subbotina, T.B.Tokmantsev, 2014, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2014, Vol. 20, No. 3.
Rights and permissions
About this article
Cite this article
Subbotina, N.N., Tokmantsev, T.B. A Study of the stability of solutions to inverse problems of dynamics of control systems under perturbations of initial data. Proc. Steklov Inst. Math. 291 (Suppl 1), 173–189 (2015). https://doi.org/10.1134/S0081543815090126
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0081543815090126