Application of fuzzy systems on the numerical solution of the elliptic PDE-constrained optimal control problems

‎This paper presents a numerical fuzzy indirect method based on the fuzzy basis functions technique to solve an optimal control problem governed by Poisson's differential equation‎. The considered problem may or may not be accompanied by a control box constraint‎. ‎The first-order necessary optimality conditions have been derived, which may contain a variational inequality in function space‎. ‎In the presented method‎, ‎the obtained optimality conditions have been discretized using fuzzy basis functions and a system of equations introduced as the discretized optimality conditions‎. ‎The derived system mostly contains some nonsmooth equations and conventional system solvers fail to solve it‎. A fuzzy-system-based semi-smooth Newton method has also been introduced‎ ‎to deal with the obtained system‎. ‎Solving optimality systems by the presented method gets us unknown fuzzy quantities on the state and control fuzzy expansions‎. ‎Finally‎, ‎some test problems‎ ‎have been studied to demonstrate the efficiency and accuracy of the presented fuzzy numerical technique‎.