Unbounded viscosity solutions of hybrid control systems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 1, pp. 176-193.

We study a hybrid control system in which both discrete and continuous controls are involved. The discrete controls act on the system at a given set interface. The state of the system is changed discontinuously when the trajectory hits predefined sets, namely, an autonomous jump set A or a controlled jump set C where controller can choose to jump or not. At each jump, trajectory can move to a different euclidean space. We allow the cost functionals to be unbounded with certain growth and hence the corresponding value function can be unbounded. We characterize the value function as the unique viscosity solution of the associated quasivariational inequality in a suitable function class. We also consider the evolutionary, finite horizon hybrid control problem with similar model and prove that the value function is the unique viscosity solution in the continuous function class while allowing cost functionals as well as the dynamics to be unbounded.

DOI : 10.1051/cocv:2008076
Classification : 34H05, 34K35, 49L20, 49L25
Mots-clés : dynamic programming principle, viscosity solution, quasivariational inequality, hybrid control 
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Barles, Guy; Dharmatti, Sheetal; Ramaswamy, Mythily. Unbounded viscosity solutions of hybrid control systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 1, pp. 176-193. doi : 10.1051/cocv:2008076. http://archive.numdam.org/articles/10.1051/cocv:2008076/

[1] A. Back, J. Gukenheimer and M. Myers, A dynamical simulation facility for hybrid systems, in Workshop on Theory of Hybrid Systems, R.L. Grossman, A. Nerode, A.P. Rava and H. Rischel Eds., Lect. Notes Comput. Sci. 736, Springer, New York (1993) 255-267.

[2] G. Barles, Solutions de viscosité des équations de Hamilton Jacobi, Mathématiques et Applications 17. Springer, Paris (1994). | Zbl

[3] G. Barles, S. Biton and O. Ley, Uniqueness for Parabolic equations without growth condition and applications to the mean curvature flow in ℝ 2 . J. Differ. Equ. 187 (2003) 456-472. | Zbl

[4] M. Bardi and C. Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhauser, Boston (1997). | Zbl

[5] M.S. Branicky, Studies in hybrid systems: Modeling, analysis and control. Ph.D. Dissertation, Dept. Elec. Eng. Computer Sci., MIT Cambridge, USA (1995).

[6] M.S. Branicky, V. Borkar and S. Mitter, A unified framework for hybrid control problem. IEEE Trans. Automat. Contr. 43 (1998) 31-45. | Zbl

[7] M.G. Crandall, H. Ishii and P.L. Lions, User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Soc. 27 (1992) 1-67. | Zbl

[8] S. Dharmatti and M. Ramaswamy, Hybrid control system and viscosity solutions. SIAM J. Contr. Opt. 34 (2005) 1259-1288. | Zbl

[9] S. Dharmatti and M. Ramaswamy, Zero sum differential games involving hybrid controls. J. Optim. Theory Appl. 128 (2006) 75-102. | Zbl

[10] N.G. Galbraith and R.B. Vinter, Optimal control of hybrid systems with an infinite set of discrete states. J. Dyn. Contr. Syst. 9 (2003) 563-584. | Zbl

[11] O. Ley, Lower-bound gradient estimates for first-order Hamilton-Jacobi equations and applications to the regularity of propagating fronts. Adv. Differ. Equ. 6 (2001) 547-576. | Zbl

[12] P.P. Varaiya, Smart cars on smart roads: problems of control. IEEE Trans. Automat. Contr. 38 (1993) 195-207.

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