Unbounded Viscosity Solutions of Hybrid Control Systems

Guy Barles (1) , Sheetal Dharmatti (2) , Mythily Ramaswamy (3)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 MIP - Mathématiques pour l'Industrie et la Physique
3 TIFR-CAM - Center for Applicable Mathematics [Bangalore]

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.

Mots clés: Dynamic programming Principle; viscosity solution; quasivariational inequality; hybrid control

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Guy Barles, Sheetal Dharmatti, Mythily Ramaswamy. Unbounded Viscosity Solutions of Hybrid Control Systems. ESAIM: Control, Optimisation and Calculus of Variations, 2010, 16 (01), pp.176--193. ⟨10.1051/cocv:2008076⟩. ⟨hal-00255763⟩