The optimal periodic control problem for a system described by first order partial differential equations is approximated by a sequence of discretized optimization problems. Trigonometric polynomials in two variables are used in the latter problems to approximate the state trajectory, the control and functions appearing in differential equations and in the criterion of the basic problem. The state equations and the instantaneous constraints on the state and the control are taken into account by the mixed exterior-interior penalty function. Sufficient conditions are given for the convergence of solutions of discretized problems to the optimal solution of the basic problem. The possibility of applying the method to a class of optimal periodic control problems in chemical engineering is emphasized.