On Consistent Regularities of Control and Value Functions

In this paper we deal with nonsmooth optimal control problems in the case when the control is allowed to be a discontinuous function. We analyse smoothness assumptions on an adjoint process in deterministic and stochastic cases. Possibilities of steep generalized space-gradients of the adjoint function imply the necessity of an approximation of the Hamiltonian. The key question of such an approximation is a relationship between the control and the value function. Under quite general assumptions it is proved that the performance measure for the original process is determined by the control function with possible discontinuities. 1 Introduction: mathematical models in topological spaces and ramiications from initial conditions given approximately .