Unbounded viscosity solutions of hybrid control systems

Guy Barles; Sheetal Dharmatti; Mythily Ramaswamy

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 16, Issue: 1, page 176-193
  • ISSN: 1292-8119

Abstract

top
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.

How to cite

top

Barles, Guy, Dharmatti, Sheetal, and Ramaswamy, Mythily. "Unbounded viscosity solutions of hybrid control systems." ESAIM: Control, Optimisation and Calculus of Variations 16.1 (2010): 176-193. <http://eudml.org/doc/250742>.

@article{Barles2010,
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 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. },
author = {Barles, Guy, Dharmatti, Sheetal, Ramaswamy, Mythily},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Dynamic programming principle; viscosity solution; quasivariational inequality; hybrid control; dynamic programming principle},
language = {eng},
month = {1},
number = {1},
pages = {176-193},
publisher = {EDP Sciences},
title = {Unbounded viscosity solutions of hybrid control systems},
url = {http://eudml.org/doc/250742},
volume = {16},
year = {2010},
}

TY - JOUR
AU - Barles, Guy
AU - Dharmatti, Sheetal
AU - Ramaswamy, Mythily
TI - Unbounded viscosity solutions of hybrid control systems
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/1//
PB - EDP Sciences
VL - 16
IS - 1
SP - 176
EP - 193
AB - 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.
LA - eng
KW - Dynamic programming principle; viscosity solution; quasivariational inequality; hybrid control; dynamic programming principle
UR - http://eudml.org/doc/250742
ER -

References

top
  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 Applications17. Springer, Paris (1994).  
  3. G. Barles, S. Biton and O. Ley, Uniqueness for Parabolic equations without growth condition and applications to the mean curvature flow in 2 . J. Differ. Equ.187 (2003) 456–472.  Zbl1028.35067
  4. M. Bardi and C. Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhauser, Boston (1997).  Zbl0890.49011
  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.  Zbl0951.93002
  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.  
  8. S. Dharmatti and M. Ramaswamy, Hybrid control system and viscosity solutions. SIAM J. Contr. Opt.34 (2005) 1259–1288.  Zbl1108.49024
  9. S. Dharmatti and M. Ramaswamy, Zero sum differential games involving hybrid controls. J. Optim. Theory Appl.128 (2006) 75–102.  Zbl1099.91022
  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.  Zbl1026.49021
  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.  Zbl1015.35031
  12. P.P. Varaiya, Smart cars on smart roads: problems of control. IEEE Trans. Automat. Contr.38 (1993) 195–207.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.