On differentiation of functional in problem on parametric coefficient optimization in semilinear global electric circuit equation

For the problem on parametric optimization with respect to an integral criterion of the coefficient and the right-hand side of the semilinear global electric circuit equation, we obtain formulae for the first partial derivatives of the cost functional with respect to control parameters. The problem on reconstructing unknown parameters of the equation by the observed data of local sensors can be represented in such form. In the paper we generalize a similar result obtained earlier by the author for the case of linear global electric circuit equation. But it is commonly believed by experts that the right hand side treated as the volumetric density of external currents of the equation depends on the gradient, with respect to the collection of space variables, of the unknown electric potential function. Because of this, it is necessary to study the case of a semilinear equation. We use the conditions of preserving global solvability of the semilinear global electric circuit equation and the estimates for the increment of the solutions, which we have obtained formerly. The mathematical novelty of presented research is due to the fact that, unlike the earlier studied linear case, now the right hand side depends nonlinearly on the state, which, in its turn, depends on the controlled parameters. Such more complicated nonlinear dependence of the state on the control parameters requirs, in particular, the development of a special technique to estimate the additional terms arising in the increment of solutions of the controlled equation.