On the smoothness of an approximated optimization problem for a Goursat–Darboux system on a varied domain

We use an example of a controlled nonlinear Goursat–Darboux system to study a rather general approach to the approximate solution of optimal control problems associated with lumped and distributed parameter systems that are affine in control variables; the domain of independent variables can be fixed or varied. The main idea of this approach consists in the approximation of the original infinite-dimensional optimization problem by a smooth finite-dimensional mathematical programming problem of comparatively small dimension with the help of spline discontinuous interpolation of the desired control on a floating mesh. We establish the existence of partial derivatives for functions of the approximating problem and derive necessary formulas.