Elliptic partial differential equations; Schwarz alternating method; Jacobi, Gauss-Seidel, SOR iterative methods

A method for finding the optimal control problems governed by linear parabolic equations using Legendre series is discussed. The optimal control of the one-dimensional diffusion equation is simplified into optimal control of linear system with a quadratic cost functional. The state variable, state rate and the control vector are expanded in the shifted Legendre series with unknown coefficients. The relation between the coefficients of the state rate and control vector with state variable is provided and the necessary condition of optimality is derived as a linear system of linear algebraic equations in terms of the unknown coefficient of the state vector. A numerical example is included to demonstrate the validity and the applicability of the technique.