Verification Theorems within the Framework of Viscosity Solutions

In optimal control theory, the verification technique plays an important role in testing for the optimality of a given control, and in constructing optimal feedback controls. However, the existing classical verification theorem is restrictive in that it requires the associated dynamic programming equation to have smooth solutions. In this paper, some verification theorems are presented within the framework of viscosity solutions under mild assumptions. These theorems are shown to have wider applicability than the classical verification theorem. As a relevant problem, some differences and relationships between the viscosity solution and Clarke′s generalized gradient are also discussed.