Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-25T10:27:23.198Z Has data issue: false hasContentIssue false
  • English
  • Français

Unbounded viscosity solutions of hybrid control systems

Published online by Cambridge University Press:  19 December 2008

Guy Barles ,
Sheetal Dharmatti  and
Mythily Ramaswamy
Guy Barles
Affiliation:
Laboratoire Mathématique et Physique Théorique, Fédération Denis Poisson, Université François Rabelais Tours, Parc de Grandmont, 37200, Tours, France. [email protected]
Sheetal Dharmatti
Affiliation:
Laboratoire MIP, UMR CNRS 5640, Université Paul Sabatier, 31062 Toulouse Cedex 9, France. [email protected]
Mythily Ramaswamy
Affiliation:
TIFR Centre for Applicable Mathematics, Sharada Nagar, Yelahanka New Town, Bangalore-560065, India. [email protected]
Get access

Abstract

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 jumpset 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.

Keywords

Dynamic programming principleviscosity solutionquasivariational inequalityhybrid control
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 1 , January 2010 , pp. 176 - 193
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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.
G. Barles, Solutions de viscosité des équations de Hamilton Jacobi, Mathématiques et Applications 17. Springer, Paris (1994).
Barles, G., Biton, S. and Uniqueness, O. Ley for Parabolic equations without growth condition and applications to the mean curvature flow in $\mathbb{R}^2$ . J. Differ. Equ. 187 (2003) 456–472. CrossRef
M. Bardi and C. Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhauser, Boston (1997).
M.S. Branicky, Studies in hybrid systems: Modeling, analysis and control. Ph.D. Dissertation, Dept. Elec. Eng. Computer Sci., MIT Cambridge, USA (1995).
Branicky, M.S., Borkar, V. and Mitter, S., A unified framework for hybrid control problem. IEEE Trans. Automat. Contr. 43 (1998) 31–45. CrossRef
Crandall, M.G., Ishii, H. and Lions, P.L., User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Soc. 27 (1992) 1–67. CrossRef
Dharmatti, S. and Ramaswamy, M., Hybrid control system and viscosity solutions. SIAM J. Contr. Opt. 34 (2005) 1259–1288. CrossRef
Dharmatti, S. and Ramaswamy, M., Zero sum differential games involving hybrid controls. J. Optim. Theory Appl. 128 (2006) 75–102. CrossRef
Galbraith, N.G. and Vinter, R.B., Optimal control of hybrid systems with an infinite set of discrete states. J. Dyn. Contr. Syst. 9 (2003) 563–584. CrossRef
Lower-bound, O. Ley gradient estimates for first-order Hamilton-Jacobi equations and applications to the regularity of propagating fronts. Adv. Differ. Equ. 6 (2001) 547–576.
Varaiya, P.P., Smart cars on smart roads: problems of control. IEEE Trans. Automat. Contr. 38 (1993) 195–207. CrossRef