I.A. Chistyakov, P.A. Tochilin. Construction of discontinuous piecewise quadratic value functions in a target control problem ... P. 259-273

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


1.   Kurzhanski A.B. Comparison principle for equations of the Hamilton–Jacobi type in control theory. Proc. Steklov Inst. Math., 2006, vol. 253, no. 1, pp. S185–S195. doi: 10.1134/S0081543806050130 

2.   Kurzhanski A.B., Varaiya P. Dynamics and control of trajectory tubes. Cham: Birkhäuser, 2014, 445 p. (SCFA, vol. 85) doi: 10.1007/978-3-319-10277-1 

3.   Habets L.C.G.J.M., Collins P.J., van Schuppen J.H. Reachability and control synthesis for piecewise-affine hybrid systems on simplices. IEEE Trans. Autom. Control, 2006, vol. 51, no. 6, pp. 938–948. doi: 10.1109/TAC.2006.876952 

4.   Girard A., Martin S. Synthesis for constrained nonlinear systems using hybridization and robust controllers on simplices. IEEE Trans. Autom. Control, 2012, vol. 57, no. 4, pp. 1046–1051. doi: 10.1109/TAC.2011.2168874 

5.   Subbotin A.I. Generalized solutions of first order PDEs: The dynamical optimization perspective. Boston, MA: Birkhäuser, 1995, 312 p. doi: 10.1007/978-1-4612-0847-1 

6.   Fleming W.H., Soner H. Controlled Markov processes and viscosity solutions. New York, NY: Springer, 2006, 429 p. doi: 10.1007/0-387-31071-1 

7.   Chistyakov I.A., Tochilin P.A. Approximate solution of the target control problem with a nonlinearity depending on one state variable. Differential Equations, 2019, vol. 55, no. 11, pp. 1518–1530. doi: 10.1134/S0012266119110107 

8.   Chistyakov I.A., Tochilin P.A. Application of piecewise quadratic value functions to the approximate solution of a nonlinear target control problem. Differential Equations, 2020, vol. 56, no. 11, pp. 1513–1523. doi: 10.1134/S00122661200110129 

9.   Tochilin P. Piecewise affine feedback control for approximate solution of the target control problem. IFAC-PapersOnLine, 2020, vol. 53, no. 2, pp. 6127–6132. doi: 10.1016/j.ifacol.2020.12.1691 

10.   Tochilin P.A., Chistyakov I.A. On the construction of a discontinuous piecewise affine synthesis in a target control problem. Trudy Inst. Mat. i Mekh. UrO RAN, 2021, vol. 27, no. 3, pp. 194–210. doi: 10.21538/0134-4889-2021-27-3-194-210 (in Russian).

11.   Tarjan R. Depth-first search and linear graph algorithms. SIAM J. Comput., 1971, vol. 1, no. 2, pp. 146–160. doi: 10.1137/0201010 

12.   Sharir M. A strong-connectivity algorithm and its applications in data flow analysis. Comput. Math. with Appl., 1981, vol. 7, no. 1, pp. 67–72. doi: 10.1016/0898-1221(81)90008-0 

13.   Nesterov Yu., Nemirovskii A. Interior-point polynomial algorithms in convex programming.  Ser. SIAM studies in Appl. Math., vol. 13, Philadelphia: SIAM, 1994, 405 p.  doi: 10.1137/1.9781611970791\,.

14.   Vasil’ev F.P. Metody optimizatsii [Optimization methods]. Moscow: Factorial Press, 2002, 824 p. ISBN: 5-88688-056-9 .

15.   Reissig G. Computing abstractions of nonlinear systems. IEEE Trans. Autom. Control, 2011, vol. 56, no. 11, pp. 2583–2598. doi: 10.1109/TAC.2011.2118950 

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.