  We present and analyze a new method for solving optimal control problems for
Volterra integral equations, based on approximating the controlled Volterra
integral equations by a sequence of systems of controlled ordinary differential
equations. The resulting approximating problems can then be solved by dynamic
programming methods for ODE controlled systems. Other, straightforward versions
of dynamic programming, are not applicable to Volterra integral equations. We
also derive the connection between our version of dynamic programming and the
Hamiltonian equations for Volterra controlled systems.
