Improved optimal conditions and iterative parameters for the optimal control problems with an integral constraint in square

A distributed optimal control problem is considered with an inequality constraint on the state variable. And the constraint reads that the integral of the state is not more than a given positive constant. An efficient approach is introduced to investigate optimality conditions of this problem. Based on the Uzawa algorithm, an efficient algorithm is designed and its convergence is discussed with details. Especially, the bounds of two iterative parameters are investigated. With the Legendre–Galerkin spectral method, numerical results show that the algorithm is highly feasible.