The optimal control of a stochastic system with both complete and partial observations is considered. In the completely observable case, because the cost function is, in the terminology of Meyer, a “semimartingale speciale,” a dynamic programming condition for the optimal control is obtained in terms of a certain Hamiltonian. The partially observable case is then discussed from first principles, and it is shown that, almost surely, the optimum control should minimize the conditional expectation of a certain Hamiltonian, with respect to an optimum measure and the observed $\sigma $-field.