Viscosity Solutions of Hybrid Game Problems with Unbounded Cost Functionals

This paper analyzes zero sum game involving hybrid controls using viscosity solution theory where both players use discrete as well as continuous controls. We study two problems, one in finite horizon and other in infinite horizon. In both cases, we allow the cost functionals to be unbounded with certain growth, hence the corresponding lower and upper value functions defined in Elliot–Kalton sense can be unbounded. We characterize the value functions as the unique viscosity solution of the associated lower and upper quasi variational inequalities in a suitable function class. Further we find a condition under which the game has a value for both games. The major difficulties arise due to unboundedness of value function. In infinite horizon case we prove uniqueness of viscosity solution by converting the unbounded value function into bounded ones by suitable transformation. In finite horizon case an argument is based on comparison with a supersolution.