We consider a method for the approximate solution of solvability and control synthesis problems for a nonlinear autonomous system of ordinary differential equations on a fixed time interval. The proposed method is based on hybridization of equations and consideration of equivalent problems for a piecewise linear system. Next, the value function is constructed as an approximate solution of the Hamilton–Jacobi–Bellman equation, and the comparison principle is applied. The solution is chosen from the class of piecewise quadratic functions. To improve the accuracy of the method, the specified value function is assumed to have discontinuities on certain sets in the state space. We propose a numerical algorithm for feedback control calculation and obtain an a priori error estimate of reaching the target set for the original nonlinear system.
Keywords: nonlinear dynamics, control synthesis, dynamic programming, comparison principle, linearization, switched system, piecewise quadratic value function
Received May 19, 2022
Revised July 11, 2022
Accepted July 18, 2022
Funding Agency: The results of sections 1-3 and 6-9 were obtained by the first author with the financial support of the Ministry of Education and Science of the Russian Federation as part of the program of the Moscow Center for Fundamental and Applied Mathematics under the agreement № 075-15-2022-284. The results of sections 4-5 were obtained by the second author with the financial support of the Russian Science Foundation (project № 22-11-00042).
Ivan Aleksandrovich Chistyakov, doctoral student, Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics, Moscow, 119991 Russia, e-mail: chistyakov.ivan@yahoo.com
Pavel Aleksandrovich Tochilin, Cand. Sci. (Phys.-Math.), Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics, Moscow, 119991 Russia; V.A.Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, 117997 Russia,
e-mail: tochilin@cs.msu.ru
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Cite this article as: I.A. Chistyakov, P.A. Tochilin. Construction of discontinuous piecewise quadratic value functions in a target control problem. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 3, pp. 259–273; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2022, Vol. 319, Suppl. 1, pp. S98–S111.
