# 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

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topBarles, 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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- 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
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