Abstract: Complex systems are not only affected by many elements but also constrained by some other conditions. In order to find out the optimal performance index of the system state equation constrained by other conditions under fixed boundary conditions， this paper presented a method of constructing auxiliary functions. Firstly，the performance indexes of the system with and without plastic constraints were transformed into the expression of the performance indexes without conditions. Then，the necessary conditions for the system to reach the optimum and their proof were obtained by using the Lagrange equation theorem in the classical variational method.

Key words: optimal control, functional variation, functional extremism, constraint condition, fixed boundary