The classical calculus of variations approach to the design of finite-dimensional optimal control systems is extended to a general class of fully non-linear distributed-parameter systems with distributed and/or boundary control inputs. As in lumped theory the result is a distributed-parameter two-time-point boundary-value problem in the canonical form of Hamilton, which in the linear case reduces to a partial differential equation of the Riccati type. A computational approximation technique for integrating the canonical equations is given which is based on the Riccati equation and does not require any hill-climbing procedure. The results are finally extended to an important class of mixed distributed-parameter and lumped-parameter systems.