Unbounded viscosity solutions of hybrid control systems

¹ Laboratoire Mathématique et Physique Théorique, Fédération Denis Poisson, Université François Rabelais Tours, Parc de Grandmont, 37200, Tours, France. barles@lmpt.univ-tours.fr
² Laboratoire MIP, UMR CNRS 5640, Université Paul Sabatier, 31062 Toulouse Cedex 9, France. sheetal@mip.ups-tlse.fr
³ TIFR Centre for Applicable Mathematics, Sharada Nagar, Yelahanka New Town, Bangalore-560065, India. mythily@math.tifrbng.res.in

Received: 15 February 2008
Revised: 11 July 2008

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.