Application of Finite Volume and Finite Element methods to distributed optimal control of semi-linear elliptic equations

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In this paper we are concerned with distributed optimal control problems governed by a second-order linear and semi-linear elliptic partial differential equations (PDE), where the control is distributed on the domain ?. The semi-linear elliptic boundary value problem is analysed by proving the existence of a unique solution and optimal control as well as deriving necessary optimality conditions. The problems are discretized using Finite Volume and Finite Element methods. However, the PDE being semi-linear, causes an additional issue, therefore the Newton’s method is introduced to linearize the equation. The aim of this paper is to present and apply different optimization methods and discretization techniques to find state and control which minimize the corresponding cost functional for linear and semi-linear PDEs. In the first optimization method the linear optimal control problem is transformed into reduced quadratic optimization problem and solved using ”quadprog” algorithm in MATLAB. In the second method both problems are optimized using ”fmincon” algorithm in two separate ways: with and without gradient of the reduced cost functional. After comparing the results, the first way was proved to be much faster as a lot of time is saved having the gradient calculated and supplied prior optimization. Finally, the projected gradient method is introduced. Overall, this last method was proved to be the most efficient.