2000 character limit reached
On constructive extractability of measurable selectors of set-valued maps (2403.05858v1)
Published 9 Mar 2024 in eess.SY, cs.SY, and math.DS
Abstract: This paper investigates the possibility of constructive extraction of measurable selector from set-valued maps which may commonly arise in viability theory, optimal control, discontinuous systems etc. For instance, existence of solutions to certain differential inclusions, often requires iterative extraction of measurable selectors. Next, optimal controls are in general non-unique which naturally leads to an optimal set-valued function. Finally, a viable control law can be seen, in general, as a selector. It is known that selector theorems are non-constructive and so selectors cannot always be extracted. In this work, we analyze under which particular conditions selectors are constructively extractable. An algorithm is derived from the theorem and applied in a computational study with a three-wheel robot model.
- P. Kachroo, “Existence of solutions to a class of nonlinear convergent chattering-free sliding mode control systems,” IEEE transactions on automatic control, vol. 44, no. 8, pp. 1620–1624, 1999.
- V. Azhmyakov, “Optimal control of sliding mode processes: A general approach,” in 2010 11th International Workshop on Variable Structure Systems (VSS). IEEE, 2010, pp. 504–509.
- A. Levant, M. Livne, and D. Lunz, “On discretization of high-order sliding modes,” Recent trends in sliding mode control, pp. 177–202, 2016.
- H. Frankowska, “The maximum principle for a differential inclusion problem,” in Analysis and Optimization of Systems. Springer, 1984, pp. 517–531.
- Y. S. Ledyaev and E. D. Sontag, “A lyapunov characterization of robust stabilization,” Nonlinear Analysis-Series A Theory and Methods and Series B Real World Applications, vol. 37, no. 7, pp. 813–840, 1999.
- B. Barmish, “Measurable selection theorems and their application to problems of guaranteed performance,” IEEE Transactions on Automatic Control, vol. 23, no. 4, pp. 685–687, 1978.
- O. Hernández-Lerma and J. Lasserre, “Error bounds for rolling horizon policies in discrete-time markov control processes,” IEEE Transactions on Automatic Control, vol. 35, no. 10, pp. 1118–1124, 1990.
- E. Ryan, “A nonlinear universal servomechanism,” IEEE Transactions on Automatic Control, vol. 39, no. 4, pp. 753–761, 1994.
- W. Wu, A. Arapostathis, and S. Shakkottai, “Optimal power allocation for a time-varying wireless channel under heavy-traffic approximation,” IEEE Transactions on Automatic Control, vol. 51, no. 4, pp. 580–594, 2006.
- N. Papageorgiou, V. Rădulescu, and D. Repovš, “Sensitivity analysis for optimal control problems governed by nonlinear evolution inclusions,” Advances in Nonlinear Analysis, vol. 6, no. 2, pp. 199–235, 2017.
- D. Possamaï, X. Tan, and C. Zhou, “Stochastic control for a class of nonlinear kernels and applications,” The Annals of Probability, vol. 46, no. 1, pp. 551–603, 2018.
- A. V. Proskurnikov and M. Mazo, “Lyapunov event-triggered stabilization with a known convergence rate,” IEEE Transactions on Automatic Control, 2019.
- P. Tsiotras and M. Mesbahi, “Toward an algorithmic control theory,” Journal of Guidance, Control, and Dynamics, vol. 40, no. 2, pp. 194–196, 2017.
- E. V. Denardo, “Contraction mappings in the theory underlying dynamic programming,” Siam Review, vol. 9, no. 2, pp. 165–177, 1967.
- P. Osinenko, L. Beckenbach, and S. Streif, “Practical sample-and-hold stabilization of nonlinear systems under approximate optimizers,” IEEE Control Systems Letters, vol. 2, no. 4, pp. 569–574, 2018.
- D. Bridges, F. Richman, and W. Yuchuan, “Sets, complements and boundaries,” Indagationes Mathematicae, vol. 7, no. 4, pp. 425–445, 1996.
- P. R. Wolenski, “The exponential formula for the reachable set of a lipschitz differential inclusion,” SIAM Journal on Control and Optimization, vol. 28, no. 5, pp. 1148–1161, 1990.
- D. Angeli, B. Ingalls, E. Sontag, and Y. Wang, “Uniform global asymptotic stability of differential inclusions,” Journal of Dynamical and Control Systems, vol. 10, no. 3, pp. 391–412, 2004.
- Y. Tanaka, “On the maximum theorem: a constructive analysis,” Int. J. Comp. and Math. Sciences, vol. 6, no. 2, pp. 173–175, 2012.
- P. Osinenko and S. Streif, “Analysis of extremum value theorems for function spaces in optimal control under numerical uncertainty,” IMA Journal of Mathematical Control and Information, vol. 2, no. 4, pp. 569–574, 2018.
- ——, “A constructive version of the extremum value theorem for spaces of vector-valued functions,” Journal of Logic and Analysis, 2018, 06 June 2018.
- A. Filippov, “On certain questions in the theory of optimal control,” Journal of the Society for Industrial and Applied Mathematics, Series A: Control, vol. 1, no. 1, pp. 76–84, 1962.
- R. Brockett, “Asymptotic stability and feedback stabilization,” Differential geometric control theory, vol. 27, no. 1, pp. 181–191, 1983.
- P. Braun, L. Grüne, and C. Kellett, “Feedback design using nonsmooth control lyapunov functions: A numerical case study for the nonholonomic integrator,” in Proceedings of the 56th IEEE Conference on Decision and Control, 2017.
- S. Kimura, H. Nakamura, and Y. Yamashita, “Control of two-wheeled mobile robot via homogeneous semiconcave control lyapunov function,” IFAC Proceedings Volumes, vol. 46, no. 23, pp. 92–97, 2013.
- ——, “Asymptotic stabilization of two-wheeled mobile robot via locally semiconcave generalized homogeneous control lyapunov function,” SICE Journal of Control, Measurement, and System Integration, vol. 8, no. 2, pp. 122–130, 2015.
- F. Clarke, “Lyapunov functions and discontinuous stabilizing feedback,” Annual Reviews in Control, vol. 35, no. 1, pp. 13–33, 2011.
Sponsor
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.