AbstractThe value function (or the optimal result function) arising in optimal control problems with the Bolza pay-off functionals is studied as the unique minimax or viscosity solution of a corresponding boundary problem for the Hamilton-Jacobi-Bellman (the dynamic programming equiation) equation. It is obtained for one-dimensional state space, that the continuous value function is not differentiable on at the most countable set of the Rankine-Hugoniot lines. The structure of the set of singular points is important to constructions of optimal synthesis in optimal control problems.