Abstract
Hybrid systems are a general framework which can model a large class of control systems arising whenever a set of continuous and discrete dynamics are mixed in a single system. In this paper, we study the convergence of monotone numerical approximations of value functions associated to control problems governed by hybrid systems. We discuss also the feedback reconstruction and derive a convergence result for the approximate feedback control law. Some numerical examples are given to show the robustness of the monotone approximation schemes.
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References
Goebel, R., Sanfelice, R.G., Teel, A.R.: Hybrid dynamical systems. IEEE Control Systems 29, 28–93 (2009)
Liberzon, D.: Switching in Systems and Control. Springer, New York (2003)
Sussmann, H.: A maximum principle for hybrid optimal control problems. In: Proceedings of the 38th IEEE Conference on Decision and Control 1, 425–430 (1999)
Dharmatti, S., Ramaswamy, M.: Hybrid control system and viscosity solutions. SIAM J. Control Optim. 34, 1259–1288 (2005)
Zhang, H., James, R.: Optimal control of hybrid systems and a system of quasi-variational inequalities. SIAM J. Control Optim. 45, 722–761 (2006)
Barles, G., Dharmatti, S.: Unbounded viscosity solutions of hybrid control systems. ESAIM: COCV 16, 176–193 (2010)
Ferretti, R., Zidani, H.: Monotone numerical schemes and feedback construction for hybrid control systems. HAL preprint. http://hal.archives-ouvertes.fr/hal-00989492/ (2014). Accessed 4 Aug 2014
Branicky, M.S., Borkar, V., Mitter, S.: A unified framework for hybrid control problem. IEEE Trans. Autom. Control 43, 31–45 (1998)
Bensoussan, A., Menaldi, J.L.: Hybrid control and dynamic programming. Dyn. Contin. Discret. Impulse Syst. 3, 395–442 (1997)
Granato, G., Zidani, H.: Level-set approach for reachability analysis of hybrid systems under lag constraints. SIAM J. Control Optim. 52, 606–628 (2014)
Ishii, K.: Viscosity solutions of nonlinear second order elliptic PDEs associated with impulse control problems II. Funkcialaj Ekvacioj 38, 297–328 (1995)
Barles, G., Souganidis, P.E.: Convergence of approximation schemes for fully nonlinear second order equations. Asymptot. Anal. 4, 271–283 (1991)
Crandall, M.G., Lions, P.-L.: Two approximations of solutions of Hamilton–Jacobi equations. Math. Comp. 43, 1–19 (1984)
Kushner, H.J., Dupuis, P.G.: Numerical Methods for Stochastic Control Problems in Continuous Time. Springer, New York (2001)
Camilli, F., Falcone, M.: An approximation scheme for the optimal control of diffusion processes. RAIRO Modélisation Mathématique et Analyse Numérique 29, 97–122 (1995)
Falcone, M., Ferretti, R.: Semi-Lagrangian schemes for Hamilton–Jacobi equations, discrete representation formulae and Godunov methods. J. Comput. Phys. 175, 559–575 (2002)
Falcone, M., Ferretti, R.: Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations. SIAM, Philadelphia (2013)
Crandall, M.G., Tartar, L.: Some relations between nonexpansive and order preserving mappings. Proc. Am. Math. Soc. 78, 385–390 (1980)
Acknowledgments
This work has been partially supported by the EU under the 7th Framework Program Marie Curie Initial Training Network “FP7-PEOPLE-2010-ITN”, GA number 264735-SADCO.
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Ferretti, R., Zidani, H. Monotone Numerical Schemes and Feedback Construction for Hybrid Control Systems. J Optim Theory Appl 165, 507–531 (2015). https://doi.org/10.1007/s10957-014-0637-0
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DOI: https://doi.org/10.1007/s10957-014-0637-0