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

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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

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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

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