The authors present a mathematical foundation for the algorithmic solution of free- and fixed-time optimal control problems with evolution equation dynamics, finite-dimensional controls, and constraints on the controls and end points. In particular, (i) expressions for the derivatives of the solutions of the evolution equations are developed with respect to controls in L(2)(m)[0,1] and to the final time; (ii) the solutions of the relaxed evolution equations are shown to have a certain kind of directional derivative; (iii) algorithmic optimality conditions are developed with respect to both ordinary and relaxed controls and the final time; and (iv) an approximation theory is presented that shows that finite-dimensional minimax, and methods of centers-type algorithms can be used to obtain arbitrarily good approximations to stationary controls for optimal control problems with evolution equation dynamics and various constraints.