Differentiation of operators and optimality conditions in category interpretation

The general extremum theory essentially uses properties of operator derivatives. As an example we consider a system described by a nonlinear elliptic equation. In this system with large values of the nonlinearity parameter and large dimension of the domain the control-state mapping is not Gateaux differentiable. For this reason one cannot immediately differentiate the optimality criterion and establish the necessary optimality conditions by classical methods. However the mentioned mapping is extendedly differentiable. This allows one to obtain optimality conditions without constraints on parameters of the system. Concluding the paper, we interpret the optimality conditions with classical and extended derivatives within the theory of categories.